Boundary actions of lattices and $C^0$ local semi-rigidity

TL;DR

This paper studies the stability of boundary actions of lattices in semisimple Lie groups, showing that small perturbations are semi-conjugate to the original action, but also constructing non semi-conjugate perturbations.

Contribution

It extends semi-rigidity results to boundary actions of lattices, demonstrating continuous semi-conjugacy under perturbations and providing counterexamples for higher regularity.

Findings

Perturbations of standard boundary actions are semi-conjugate to the original.

The semi-conjugacy is close to the identity in the homeomorphism group.

Counterexamples show some perturbations are not semi-conjugate at the $C^0$ level.

Abstract

We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate to the original action.

Equations

$$x\mapsto {\mathcal{B}}_{\mu }(x)=\frac{1}{2}{\int }_{F}d(x,y{)}^{2}d\mu (y).$$

$$\mathrm{bar}(\mu )\in B_{2ϵ}(x).$$

$$2ϵ\le d(y,x)\le d(y,z)+d(z,x)<d(y,z)+ϵ.$$

$$G(x_0,\mu )={\nabla }_{x=x_0}{B}_{\mu }(x)={\int }_{F}d(x_0,y){\nabla }_{x=x_0}d(x,y)d\mu (y)=0,$$

$$\mathrm{bar}(f_{\ast }\mu )=f(\mathrm{bar}(\mu )).$$

$$0=G(\mathrm{bar}(\mu ))={\int }_M{v}_{\mathrm{bar}(\mu ),y}d\mu (y).$$

$${\int }_N{v}_{f(\mathrm{bar}(\mu )),y}d(f_{\ast }\mu )(y)={\int }_M{v}_{f(\mathrm{bar}(\mu )),f(y)}d\mu (y)={\int }_{M}d{f}_{\mathrm{bar}(\mu )}(v_{\mathrm{bar}(\mu ),y})d\mu (y),$$

$${\int }_N{v}_{f(\mathrm{bar}(\mu )),y}d(f_{\ast }\mu )(y)=df\left({\int }_M{v}_{\mathrm{bar}(\mu ),y}d\mu (y)\right)=df(0)=0.$$

$$f(\mathrm{bar}(\mu ))=\mathrm{bar}(f_{\ast }\mu ).$$

$$D_{x}G(x,w,z)=\sum _{i=1}^n{w}_i\left(D_{x}d(x,z_i)\otimes {\nabla }_{x}d(x,z_i)+d(x,z_i)D_x{\nabla }_{x}d(x,z_i)\right).$$

$$H={\left\{(s,{\phi }_{\xi }(s)):s\in B(0,{ϵ}_0)\right\}}_{\xi \in F},$$

$$b(s):={\mathrm{bar}}_s(\mu (s)))$$

$$d_{(s_0,b(s_0))}^{\mathrm{Gr}}\left(T_{(s_0,b(s_0))}G,\mathcal{D}\left(s_0,b(s_0)\right)\right)<ϵ.$$

$$\Phi (z):=(p(z),{\pi }_{p(z)}(z))$$

$$({\mathrm{pr}}_F\circ \Phi {)}^{-1}(\xi )={\Phi }^{-1}(X\times \xi )$$

$${\gamma }_x:={\pi }_{\gamma x}^{-1}\circ {\rho }_0(\gamma )\circ {\pi }_x$$

$$({\mathrm{pr}}_F\circ \Phi {)}^{-1}(\xi )={\Phi }^{-1}(X\times \{\xi \})=\left\{v\in T^{1}X:{\pi }_{p(v)}(v)=\xi \right\}$$

$$X_F:=\left\{v\in T^{1}X:{\gamma }_v(\infty )\in G\cdot \mathfrak{X}\right\}.$$

$$\Phi :={\Phi }_0{|}_{X_F}.$$

$$\underset{\rho \in \text{leavevmode}U}{\sup }\text{leavevmode}\underset{\xi \in F}{\sup }\text{leavevmode}{d}_F({\pi }_x(\tilde{f}(x,\xi )),\xi )<ϵ,$$

$$\underset{y\in B_R(x)}{\sup }{d}_{F_y}\left(\tilde{f}(y,\xi ),({\mathrm{pr}}_F\circ \Phi {)}^{-1}(\eta )\cap F_y\right)<ϵ,$$

$$\underset{\rho \in \text{leavevmode}U}{\sup }\left(\underset{(x,\xi )\in X\times F}{\sup }{d}_{\tilde{f}(x,\xi )}^{\mathrm{Gr}}\left(T_{\tilde{f}(x,\xi )}\tilde{f}(X\times \xi ),T_{\tilde{f}(x,\xi )}({\mathrm{pr}}_F\circ \Phi {)}^{-1}({\eta }_x)\right)\right)<ϵ,$$

$${\psi }_{{\mathcal{U}}_i}(x,\xi )={\pi }_x^{-1}(\xi ).$$

$${\Psi }_{{\mathcal{W}}_j}(\gamma x,\rho (\gamma )\xi )={\hat{\rho }}_0(\gamma ){\psi }_{{\mathcal{U}}_i}(x,\xi ),$$

$$W_j={\gamma }_j{U}_i$$

$${\xi }_j:={\rho }_0({\gamma }_j)\rho ({\gamma }_j^{-1})(\xi ).$$

$${\Psi }_{{\mathcal{W}}_j}(x,\xi )={\hat{\rho }}_0({\gamma }_j){\Psi }_{{\mathcal{U}}_i}({\gamma }_j^{-1}x,\rho ({\gamma }_j{)}^{-1}\xi )={\pi }_x^{-1}({\rho }_0({\gamma }_j)\rho ({\gamma }_j{)}^{-1}(\xi ))={\pi }_x^{-1}({\xi }_j).$$

$$\displaystyle\widetilde{f}(x,\xi):=\operatorname{bar}_{x}\left(\sum_{j=1}^{m}\widetilde{\sigma_{j}}(x)\delta_{\pi_{x}^{-1}(\xi_{j})}\right)=\operatorname{bar}_{x}\bigg{(}\sum_{j=1}^{m}\widetilde{\sigma}_{j}(x)\delta_{\Psi_{\mathcal{W}_{j}}(x,\xi)}\bigg{)},$$

$$\tilde{f}(\gamma x,\rho (\gamma )\xi )$$

$$\displaystyle=\operatorname{bar}_{\gamma x}\left(\sum_{i=1}^{m}\widetilde{\sigma}_{i}(x)\delta_{\widehat{\rho}_{0}(\gamma)\Psi_{\mathcal{W}_{i}}(x,\xi)}\right)=\widehat{\rho}_{0}(\gamma)\left(\operatorname{bar}_{x}\bigg{(}\sum_{i=1}^{m}\widetilde{\sigma}_{i}(x)\delta_{\Psi_{\mathcal{W}_{i}}(x,\xi)}\bigg{)}\right)$$

$$={\hat{\rho }}_0(\gamma )\tilde{f}(x,\xi ),$$

Full text

† Supported in part by Simons Foundation grant #965245

‡ Supported in part by Emmy Noether Project 427903332 (funded by the DFG)

‡† Supported in part by NSF grant DMS 2003712.

1. 1 Introduction
2. 2 Preliminaries I: Barycenters
3. 3 Standard Actions and the Generalization of a Lemma of Bowden-Mann
4. 4 Preliminaries II: General Results about Topological Actions
5. 5 Preliminaries III: Symmetric Spaces and Furstenberg Boundaries
6. 6 Coarse Geometry of Symmetric Spaces
7. 7 $C^{0}$ Local Semi-rigidity of Boundary Actions of Lattices
8. 8 Applications of the Main Theorem: Differentiable and Lipschitz Local Rigidity
9. 9 Actions on the Geodesic Boundary
10. A Barycenters and Estimates
11. B Some Lemmas about Foliations

1. Introduction

Local rigidity of actions has played an important role in dynamics over the last few decades. This was most explored for smooth actions on manifolds with perturbations that are close in the $C^{1}$ or possibly $C^{k}$ topology to the original action.

Classically one investigated structural stable systems. These are actions such that a sufficiently small perturbation in the $C^{1}$-topology is $C^{0}$-conjugate or at least $C^{0}$-orbit-equivalent to the unperturbed action. There are many such systems in dynamics, in particular diffeomorphisms and flows. Indeed, much attention was paid to characterizing the latter in the 1970s and 1980s. The simplest structurally stable diffeomorphisms and flows come from uniformly hyperbolic dynamics, in particular Anosov systems, as was proved by Anosov in the 1960s. This class includes geodesic flows of closed manifolds $M$ of negative sectional curvature. There one also has a natural action of the fundamental $\pi_{1}M$ on the geodesic compactification $\partial\widetilde{M}$. For constant curvature $-1$ manifolds $M$, Sullivan proved local rigidity of these boundary actions using structural stability of the geodesic flows by a duality argument [Sul85]. This was followed by work of Ghys, Kanai, Katok-Spatzier, and recently by Kapovich, Kim and Lee [Ghy93, Kan96, KS97, KKL22]. The latter three in particular consider projective actions of uniform lattices in a higher rank semisimple Lie group, items of great classical interest.

These results were an exploit of structural stability of hyperbolic actions and required $C^{1}$ dynamics for the perturbation. Recently, Bowden and Mann investigated $C^{0}$ perturbations of boundary actions of cocompact groups of isometries of a simply connected Riemannian manifold $X$ of negative curvature [BM22]. In particular, this includes the action of a uniform lattice in a rank one semisimple Lie group on its Furstenberg boundary.

Remarkably, Bowden and Mann proved a $C^{0}$ local rigidity result for these actions, i.e. for perturbations in the homeomorphism group of the manifold close in the compact open topology. Note that in this generality, while the boundary $\partial_{\infty}X$ of $X$ is a compact manifold, in fact topologically a sphere, it does not always carry a differentiable structure for which the boundary action is $C^{1}$. Thus $C^{1}$-local rigidity of such actions does not even make sense. Later, Mann and Manning generalized their work to Gromov hyperbolic groups acting on their boundaries assuming the latter are homeomorphic to a sphere [MM21]. Later yet, Mann, Manning and Weisman dealt with the general case of Gromov hyperbolic groups [MMW22]. In a work in progress, they are generalizing these results further to relatively hyperbolic groups for the action on the Bowditch boundary.

The main goal of this paper is to establish similar results for actions of uniform lattices in higher rank semisimple Lie groups on their Furstenberg boundaries (Theorem 1.1). These results fit in with the general program of studying actions of higher rank semisimple Lie groups $G$ and their lattices $\Gamma$ on manifolds. This quest was initiated by R. Zimmer in the 1980s and is now called the Zimmer program [Zim83, Zim84, Zim87]. It followed his remarkable generalization of Margulis’ superrigidity theorem (for homomorphisms from lattices into Lie groups) to cocycles over $G$ or $\Gamma$ actions over finite measure preserving actions, e.g. volume preserving actions on compact manifolds. Much has been achieved since then; see the surveys by Fisher [Fis20, Fis07]. In particular, Brown, Fisher and Hurtado made major progress towards Zimmer’s conjecture that lattices $\Gamma$ cannot act on “low-dimensional” manifolds [BFH22, BFH20, BFH21]. This follows earlier special results by Farb-Shalen, Polterovich and Franks-Handel [FS99, Pol02, FH03].

At this point we have many examples of actions of such groups which are not of algebraic nature per se, even though they are constructed from algebraic actions via a blow-up construction [KL96, Ben00, BF05, Fis07]. The classification problem however persists, in a softer form, asked by Zimmer, namely that the actions are homogeneous on a dense open set. Labourie in [Lab98] and later Margulis in his list of problems for the new century [Mar00] included a precise formulation .

Establishing the desired classification under natural dynamical or geometric hypotheses has also been pursued with vigor. For example, we now have good understanding of “hyperbolic” actions of $G$. For instance, Brown, Rodriguez Hertz and Wang classified actions of higher rank lattices on tori and nilmanifolds [BRHW17]. More recently, Damjanovic, Spatzier, Vinhage and Xu [DSVX22] proved algebraicity of volume preserving actions of $G$ with many Anosov elements on any compact manifold.

Zimmer’s program mostly discussed volume preserving actions. Actions on boundaries of $G$ or $\Gamma$ action on $G/Q$, $Q$ a parabolic, however cannot preserve any probability measure. Still they are important building blocks in understanding general actions of these groups on compact manifolds. Hence we are keen on understanding their rigidity properties, both local and global ones. For $C^{k}$ ($k\geq 1$) or Lipschitz actions of uniform lattices, local rigidity was resolved in [Kan96, KS97, KKL22], as discussed above. Brown, Rodriguez Hertz and Wang even get global rigidity results for certain groups and parabolics [BRHW].

However, $C^{0}$-local rigidity for the standard $\Gamma$ actions on boundaries has not been studied in the higher rank case. This is precisely what we achieve in Theorem 1.1 - our main result in this paper - for uniform lattices. This opens up a new direction in the Zimmer program, namely studying continuous actions of $G$ and $\Gamma$. So far the only known and fabulous results for higher rank lattices are for actions on the circle or the real line due to Morris [Wit94] for $SL(n,\mathbb{Z})$ and other lattices of full $\mathbb{Q}$-rank. This was proven in full for all higher rank lattices by Deroin and Hurtado [DH20] recently. These works followed earlier results for $C^{1}$-actions by Ghys, Burger-Monod and Navas [Ghy99, BM99, Nav02].

Naturally, one can now ask questions about $C^{0}$ local semi-rigidity for other algebraic actions, e.g. ones preserving an affine structure. For these, local rigidity properties are known under higher regularity hypotheses. Indeed, Fisher and Margulis prove $C^{k}$ local rigidity for a suitable $k$ depending on the type of action and the dimension of the manifold [FM03, FM05, FM09]. These follow earlier work by Margulis and Qian [MQ01]. Are these $C^{0}$-locally semi-rigid? Is there a general $C^{0}$ Zimmer program? We refer to Weinberger’s survey for further constructions, results, questions and conjectures [Wei11, FS00].

We note that such actions are usually not $C^{0}$ locally rigid. We actually find a general construction of $C^{0}$ perturbations for any $C^{1}$ action of a countable group on a compact manifold with a dense orbit; see 4.11. Even better, in our Theorem 1.4, we construct new $C^{\infty}$ actions of $G$ and $\Gamma$ on spheres, deforming the boundary action of $G$ on the sphere at infinity of its locally symmetric space. These are not $C^{0}$ semi-conjugate to the unperturbed action. Our constructions generalize further, cf. Remark 9.1. Let us also note that Uchida and Kuroki had constructed other $C^{0}$ actions of $SL(3,\mathbb{R})$ on $S^{4}$ [Uch85, Kur05]. These are neither deformations of nor semi-conjugate to our actions; indeed they only have finitely many orbits. Finally, we mention the recent work by Fisher and Melnick [FM22] who construct many new examples of real analytic actions of $\mathrm{SL}(n,\mathbb{R})$ on manifolds of dimension $n$ which form a discrete set of $C^{0}$ conjugacy classes. Their real analytic conjugacy classes however range over a continuum.

We now informally discuss the main ingredients of our work. Hyperbolicity will again play a fundamental role, just as in the local rigidity results alluded to at the beginning of the paper. While we do not have simple north-south dynamics anymore, individual hyperbolic elements act by Morse-Smale diffeomorphisms, a more complicated expression of hyperbolicity. The magic glue however is provided by the geodesic boundary of the symmetric space that all the Furstenberg boundaries naturally sit in. This is where the structure of the flats at infinity comes in. In the end, we will exploit to a large extent that quasiflats in higher rank symmetric spaces are close to a finite union of flats (Lemma 6.1). This approach mirrors the work of Bowden and Mann which centrally uses quasi-geodesics and Morse lemma techniques.

To our knowledge, this is the first instance of the use of quasi-isometric rigidity in higher rank dynamics, a connection long sought after. In fact, it appears that this is the first application of these techniques outside of coarse geometry.

To start, let us first formulate a precise definition of local semi-rigidity. Given a topological space $F$, endow its group of homeomorphisms $\operatorname{Homeo}(F)$ with the compact open topology. Suppose $\Gamma$ is a finitely generated group with finite generating set $\Gamma_{0}$. Suppose $\rho_{0}$ is an action of $\Gamma$ on a topological space $F$ by homeomorphisms.

We call $\rho_{0}$ * $C^{0}$-locally semi-rigid* if there is a neighborhood $N$ of the $\operatorname{id}\in\operatorname{Homeo}(F)$ such that: if $\rho:\Gamma\to\operatorname{Homeo}(F)$ is an action of $\Gamma$ on $F$ such that for all $\gamma\in\Gamma_{0}$, $\rho(\gamma)\cdot\rho_{0}(\gamma)^{-1}\in N$, then there exists a continuous surjective map $f:F\to F$ which intertwines $\rho$ and $\rho_{0}$ (i.e. $f\circ\rho(\gamma)=\rho_{0}(\gamma)\circ f$ for all $\gamma\in\Gamma$). In other words, $\rho_{0}$ is a continuous factor of $\rho$. One also calls such a map $f$ a semi-conjugacy, or more precisely, a $(\rho,\rho_{0})$-semi-conjugacy.

Obtaining semi-conjugacies is often the optimal local rigidity result one can obtain for $C^{0}$-perturbations, as we will see in 4.11. There we adapt the well-known Denjoy construction to construct $C^{0}$-perturbations of any $C^{1}$-action $\rho_{0}$ of a countable group on a compact manifold with a dense orbit. These perturbations are not $C^{0}$ conjugate to $\rho_{0}$. Hence in the context of lattice actions on boundaries, the next result, our main theorem, is the best possible.

Theorem 1.1** (Theorem 7.1).**

Suppose $\Gamma$ is a uniform lattice in a connected linear semisimple Lie group $G$ without compact factors. Let $\rho_{0}$ be the action of $\leavevmode\nobreak\ \Gamma$ on $G/Q$ by left multiplication. Then any $C^{0}$-action $\leavevmode\nobreak\ \rho$ of $\Gamma$ that is sufficiently close to $\rho_{0}$ in the $C^{0}$ topology is semi-conjugate to $\rho_{0}$. Moreover, the semi-conjugacy is close to $\operatorname{id}$ in $\operatorname{Homeo}(F)$.

We note that in the rank one case, our result is a special case of Bowden and Mann’s work [BM22]. Moreover, our proof essentially simplifies to their argument. Furthermore, it follows from Thereom 7.1 that pre-images of points under the semi-conjugacy have small diameter, showing the Denjoy construction is, in essence, the only obstruction to conjugacy. We remark that our proof does not cover the case of non-uniform lattices. We naturally conjecture that the statements still hold true.

From our Main Theorem, in Section 8, we recover prior local rigidity results that yield conjugacies, under additional assumptions on the regularity of the perturbations. Kapovich, Kim and Lee [KKL22] prove this for Lipschitz perturbations of the standard action, using a structural stability approach à la Sullivan.

Corollary 1.2**.**

[KKL22, Theorem 1.2]** Let $\Gamma<G$ be a uniform lattice in a connected linear semisimple Lie group $G$ of higher rank and without compact factors. Let $Q$ be a parabolic subgroup of $G$, and let $\rho_{0}$ be the action of $\Gamma$ on $G/Q$ by left translation. Then there is a neighborhood $U$ of $\rho_{0}$ in Lipschitz topology such that every $\rho\in U$ is $C^{0}$-conjugate to $\rho_{0}$.

For smooth $C^{1}$ close perturbations, Kanai under more restrictive assumptions and Katok-Spatzier in general obtained the following, with very different proofs:

Corollary 1.3**.**

[Kan96, KS97]** Let $\Gamma<G$ be a uniform lattice in a connected linear semisimple Lie group $G$ of higher rank and without compact factors. Let $Q$ be a parabolic subgroup of $G$, and let $\rho_{0}$ be the action of $\Gamma$ on $G/Q$ by left translation. Then there is a neighborhood $U$ of $\rho_{0}$ of smooth actions that are close to $\rho_{0}$ in $C^{1}$-topology such that every $\rho\in U$ is $C^{\infty}$-conjugate to $\rho_{0}$.

In up-and-coming work [BRHW], Brown, Rodriguez Hertz and Wang obtain a $C^{k}$-version of Corollary 1.3. Remarkably, their work even applies to non-uniform lattices. For minimal Furstenberg boundaries of split simple groups, they even prove a global rigidity result [BRHW, BRHW22].

Naturally one asks how rigid the actions of lattices $\Gamma\subset G$ are on other types of boundaries. The most natural is the geodesic boundary. In the higher rank case, it turns out that we can always deform nontrivially and these deformations can be made as smooth as we wish. For $k\geq 0$, we will say that $\rho_{0}$ is not locally semi-rigid among $C^{k}$-actions if any sufficiently small neighborhood of $\rho_{0}$ in the space of $C^{k}$ actions contains an action that is not $C^{0}$-semi-conjugate to $\rho_{0}$.

Theorem 1.4**.**

*(see Section 9)

Let $\Gamma$ be a lattice for a higher rank symmetric space of noncompact type $X$, not necessarily uniform. Then the standard action of $\Gamma$ on $\partial_{\infty}X$ is not locally semi-rigid among $C^{k}$ actions for any $k\in[0,\infty]$.*

We note that, in a similar vein, Fisher and Melnick have constructed new actions of $G$ on circle bundles over projective spaces [FM22]. They even find real analytic actions. We however do not know if the perturbations in Theorem 1.4 can be made real analytic.

One wonders if analogous results hold more generally in non-positive curvature. More precisely, consider a compact non-positively curved manifold $M$ without local Euclidean de Rham factor together with the action of its fundamental group $\pi_{1}(M)$ on the geodesic boundary $\partial_{\infty}{\widetilde{M}}$ of the universal cover $\widetilde{M}$. Is this action $C^{0}$ locally semi-rigid? We suspect that local semi-rigidity holds for manifolds with isolated flats while we believe that local semi-rigidity fails for graph manifolds. It seems rather difficult to formulate a precise condition when local semi-rigidity holds.

Outline of the arguments:

We first give an outline for the proof of Theorem 1.1. Consider the standard action of a uniform lattice $\Gamma$ on $G/Q$ for a parabolic subgroup $Q$. We employ the suspension construction to get a Weyl chamber bundle $\Gamma\backslash G/M_{Q}$ over the locally symmetric space associated to $\Gamma$. The Weyl chamber flow gives rise to center stable and center unstable manifolds. Each of them, on the Weyl chamber bundle over the universal cover, is isometric to the symmetric space associated to $G$. Given a perturbed action on $G/Q$, we construct a new foliation on the Weyl chamber bundle that is a perturbation of the center stable foliation. In the smooth setting, this construction can be done by pulling back appropriate foliations by a diffeomorphism between suspensions. This diffeomorphism is constructed by using the Implicit Function Theorem to solve certain coboundary equations, see for example [CNS22, Appendix]. A key requirement in our argument here is that the perturbed center stable foliation is continuous but with $C^{1}$ leaves. The argument in [CNS22] in the aforementioned smooth setting does not guarantee this. Our approach in this paper is to construct each leaf of the foliation on the Weyl chamber bundle over the universal cover by developing from a local leaf. A local leaf is constructed by taking a barycenter of finitely many center stable leaves with weights varying smoothly.

The next main step is establishing a weak form of structural stability. The intersection of each center unstable leaf with a perturbed center stable leaf is a smooth manifold and it has the same dimension as a center leaf for the Weyl chamber flow. Locally, this intersection is a biLipschitz image of a center leaf. We can improve it to a global biLipschitz image of a center leaf. Then we use quasi-isometric rigidity to prove that the intersection is shadowing an actual center leaf for the Weyl chamber flow. This is done first by showing that a $L$-biLipschitz flat (i.e. the image of a flat under an $L$-biLipschitz map) is uniformly close to a single flat when $L$ is sufficiently close to $1$. The global shadowing conclusion follows from how different flats union to a symmetric totally geodesic subspace that is isometric to a center leaf.

After establishing a weak form of structural stability on the large scale, we can start constructing a semiconjugacy. For this, we use the correspondence - via shadowing - between intersections of center unstable leaves with perturbed center stable leaves and center stable leaves respectively. The proof that the semi-conjugacy is well-defined and continuous is similar to the proof that a quasi-isometric embedding extends to a continuous map on the boundaries.

We finish this section with an outline for the proof of Theorem 1.4. First note that the standard action $\rho_{0}$ of $G$ on the geodesic boundary of the associated symmetric space preserves the Tits metric. In other words, Weyl chambers map to Weyl chambers isometrically under $G$. The perturbed action $\rho$ that we construct will still map Weyl chambers to Weyl chambers, exactly as $\rho_{0}$ does. However, we will move in the Weyl chamber directions, moving either towards the center of the chamber or the Weyl chamber faces.

Acknowledgements

We thank Katie Mann, David Fisher, Zhiren Wang and Aaron Brown for helpful discussions and conversations. CC and TN thank University of Michigan, CC thanks Heidelberg University, CC and MI thank Karlsruhe Institute of Technology, and CC and TN thank Max Planck Institute for Mathematics for hosting them at various periods during the course of this project.

2. Preliminaries I: Barycenters

2.1. Definition of barycenter

Let $F$ be a $C^{k}$ Riemannian manifold for some $k\geq 2$. We say a set $K\subset F$ is geodesically convex or totally convex if for every pair of points $x,y\in K$ there is a unique minimizing geodesic segment of $F$ between $x$ and $y$ and this segment is contained in $K$. As is well known, normal neighborhoods in Riemannian manifolds are geodesically convex [dC92].

Suppose $\mu$ is a probability measure with $\operatorname{supp}(\mu)\subset K$ where $K$ a geodesically convex subset of $F$. Consider the function

[TABLE]

The barycenter of $\mu$ in $K$, denoted by $\operatorname{bar}(\mu)$ in $K$, is the unique point in $K$ where the minimum of the function occurs. Note that the existence of a minimum is guaranteed by the compactness of $K$ while the uniqueness depends on strict convexity of $\operatorname{\mathcal{B}}_{\mu}$ (and hence depends on $d$ and $\operatorname{supp}(\mu)$).

Remark 2.1**.**

Suppose $\mu$ is a probability measure such that $\operatorname{supp}(\mu)\subset B_{\epsilon}(x)$ where $B_{\epsilon}(x)$ is a metric ball of radius $\epsilon$ (in the metric $d$) which is geodesically convex. Assume that $\operatorname{bar}(\mu)$ exists. Then,

[TABLE]

Indeed, for any point $y\not\in B_{2\epsilon}(x)$ and any $z\in B_{\epsilon}(x)$, we have by the triangle inequality

[TABLE]

So $d(y,z)\geq\epsilon>d(x,z)$ and thus $\mathcal{B}_{\mu}(x)<\mathcal{B}_{\mu}(y)$. Thus $y$ cannot be the barycenter of $\mu$.

We now introduce the following condition which is sufficient for the barycenter map to be well-defined.

*Condition $(\star)$: * We will say that a measure $\mu$ on $F$ satisfies Condition $(\star)$ if

• •

$\operatorname{supp}(\mu)$ is contained in a compact geodesically convex subset $K_{\mu}\subset K$ of $F$,

• •

$\operatorname{\mathcal{B}}_{\mu}(x)$ has one critical point in $K_{\mu}$, i.e. there exists $x_{0}\in K_{\mu}$ such that

[TABLE]

• •

$D_{x}G(x,\mu)$, the Hessian of $\operatorname{\mathcal{B}}_{\mu}(x)$, exists and is positive definite on $K_{\mu}$ (cf. Remark 2.2).

Note that if $\mu$ satisfies Condition $(\star)$, then $\operatorname{bar}(\mu)$ in $K_{\mu}$ is well-defined. Moreover, measures satisfying this condition are well-behaved under affine diffeomorphisms.

Remark 2.2**.**

Since the Riemannian metric on $F$ is $C^{2}$, then $y\mapsto d(x_{0},y)^{2}$ is $C^{2}$ on metric balls around $x_{0}$ of sufficiently small radius (see Lemma A.7). Thus the Hessian of $\operatorname{\mathcal{B}}_{\mu}(x)$ exists on a sufficiently small neighborhood of $x$. Hence the last condition in Condition $(\star)$ is satisfied by $\mu$ provided $\operatorname{diam}(K_{\mu})$ is sufficiently small.

Lemma 2.3**.**

Suppose $M,N$ are $C^{2}$-Riemannian manifolds. Let $f:M\to N$ be an affine diffeomorphism with respect to the respective Riemannian connections. If $\mu$ and $f_{*}\mu$ are measures on $M$ and $N$ satisfying Condition $(\star)$ with $\operatorname{diam}(K_{\mu})$ and $\operatorname{diam}K_{f_{*}\mu}$ sufficiently small, then

[TABLE]

Proof.

By Remark 2.2, $\operatorname{\mathcal{B}}_{\mu}$ and $\operatorname{\mathcal{B}}_{f_{*}\mu}$ are $C^{2}$ functions on $K_{\mu}$ and $K_{f_{*}\mu}$. Note that it suffices to prove that if $G(x,\mu)=0$, then $G(f(x),f_{*}\mu)=0$. Indeed, since $\mu$ (resp. $f_{*}\mu$) satisfies Condition $(\star)$, it has a unique point of minimum for the function $x\mapsto\operatorname{\mathcal{B}}_{\mu}(x)$ (resp. $x\mapsto\operatorname{\mathcal{B}}_{f_{*}\mu}(x)$) in $K_{\mu}$ (resp. $K_{f_{*}\mu}$). We only need to show that $f$ maps one critical point to another.

In order to verify this, note that $d(z,y)\nabla_{z}d(z,y)=v_{z,y}$ where $v_{z,y}$ is the tangent at $z\in M$ to the geodesic $\sigma$ with $\sigma(0)=z$ and $\sigma(1)=y$. Then $\operatorname{bar}(\mu)$ is the unique point such that

[TABLE]

Then

[TABLE]

where the second equality uses the fact that $f$ is affine (and maps geodesic segments to geodesics segments) and the first equality is the change of variables formula. Then

[TABLE]

Thus $G(f(\operatorname{bar}(\mu)),f_{*}\mu)=0$. By uniqueness of the critical point of $G$ in $K_{f_{*}(\mu)}$ (see Condition $(\star)$),

[TABLE]

∎

Finally let us explain the following special case of barycenter maps that we will be interested in. Let $\mu=\mu_{z,w}$ be a finite weighted sum of Dirac measures at points $z=\left\{z_{1},\dots,z_{n}\right\}\subset F$ with weights $w=\left\{w_{1},\dots,w_{n}\right\}\in\mathbb{R}^{n}$. Then writing $G(x,w,z):=G(x,\mu_{z,w})=\sum_{i=1}^{n}w_{i}d(x,z_{i})\nabla_{x}d(x,z_{i})$ we have the implicit equation $G(\operatorname{bar}(\mu_{z,w}),z,w)=0$. Denoting the covariant derivative at $x=\operatorname{bar}(\mu_{z,w})$ by $D_{x}$, we obtain the Hessian of $\mathcal{B}_{\mu_{z,w}}$,

[TABLE]

In Appendix A, we will show that if $\operatorname{diam}_{F}\left\{z_{1},\dots,z_{n}\right\}$ is sufficiently small, then $\operatorname{bar}(\mu_{z,w})$ is well-defined and is well-behaved under affine diffeomorphisms.

Remark 2.4**.**

1. (1)

If $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}$ has sufficiently small diameter, then $\mu_{z,w}$ satisfies Condition $(\star)$ and $\operatorname{bar}(\mu_{z,w})$ is well-defined. 2. (2)

Suppose $F^{\prime}$ is another $C^{3}$ Riemannian manifold and $f:F\to F^{\prime}$ is an affine diffeomorophism. If $\operatorname{diam}\left\{f(z_{1}),\dots,f(z_{n})\right\}$ is also sufficiently small, then $\operatorname{bar}(f_{*}\mu_{z,w})=\operatorname{bar}(\mu_{z,w})$.

*The smallness of the diameter that we require in the above remark is quite explicit and depends only on the Riemannian metrics on $F$ and $F^{\prime}$. See Lemma A.3 and Corollary A.4 for details.

Part (1) of the remark follows from Lemma A.3, because this lemma shows that $D_{x}G(x,w,z)$ is positive definite in a neighborhood of $x=\operatorname{bar}(\mu_{z,w})$. In order to prove part (2) of the remark, we first apply part (1) to $f_{*}\mu_{z,w}$ and then apply Lemma 2.3.*

2.2. $C^{1,2}$ regularity

Next, we will discuss the regularity of the barycenter maps when the metric varies. For this, the appropriate regularity of the family of metrics will be $C^{1,2}$ that we now define. But we first introduce the notion of $C^{\ell,2}$ foliations and functions.

Definition 2.5**.**

Suppose $N$ is a smooth manifold and $\operatorname{\mathcal{F}}$ is a continuous foliation of $N$ with smooth leaves. Let $\ell\geq 1$.

1. (1)

The foliation $\operatorname{\mathcal{F}}$ is $C^{\ell,2}$ if $\operatorname{\mathcal{F}}$ is a $C^{\ell}$ foliation and the leaf-wise 2-jets of the leaf inclusions vary transversally in a $C^{\ell}$ way. In particular, $p:N\to L$ is a $C^{\ell,2}$-fiber bundle provided $p:N\to L$ is a $C^{0}$ fiber bundle and the foliation $\operatorname{\mathcal{F}}=\left\{p^{-1}(x):x\in L\right\}$ of $N$ by fibers is a $C^{\ell,2}$ foliation. 2. (2)

Suppose $\operatorname{\mathcal{F}}$ is a $C^{\ell,2}$-foliation and $f:N\to\operatorname{\mathbb{R}}$ is a continuous function. Then $f$ is a $C^{\ell,2}$-function provided the leaf-wise derivatives up to order $2$ exist and vary transversally in a $C^{\ell}$ way (i.e. the tangential derivatives up to order 2 are transversally differentiable up to order $\ell$).

In particular, let us explain what we will mean by a $C^{1,2}$ family of metrics on a manifold $F$. Let $B(0,\epsilon_{0})\subset\operatorname{\mathbb{R}}^{m}$ and let $\left\{g_{s}:s\in B(0,\epsilon_{0})\right\}$ be a family of $C^{2}$ Riemannian metrics on $F$. Consider the product foliation $\operatorname{\mathcal{F}}=\left\{F\times\left\{s\right\}:s\in B(0,\epsilon_{0})\right\}$ of $F\times B(0,\epsilon_{0})$ that is $C^{\ell,2}$ for any $\ell\geq 1$.

Definition 2.6**.**

We will say that the family $\left\{g_{s}:s\in B(0,\epsilon_{0})\right\}$ is a $C^{1,2}$ family of metrics if the function $F\times B(0,\epsilon_{0})\ni(x,s)\mapsto g_{s}(X_{x},X_{x})$ is a $C^{1,2}$ function for any smooth vector field $X$ on $F$.

A consequence of having a $C^{1,2}$ family of metrics is the following regularity of distance functions. See the moreover part of Lemma A.7 for a proof.

Lemma 2.7** (Lemma A.7).**

Suppose $\left\{g_{s}:s\in B(0,\epsilon_{0})\right\}$ is a $C^{1,2}$ family of metrics on a compact manifold $F$ such that the sectional curvature of each $g_{s}\in[-a^{2},b^{2}]$ where $a,b>0$ are two fixed constants. Then, for all $z$ and $x$ with $d_{s}(x,z)$ sufficiently small, $d_{s}(x,z)^{2}$ is $C^{2}$ in $x$ and $z$, $C^{1}$ in $s$ and $\nabla_{x}^{s}d_{s}(x,z)^{2}$ is $C^{1}$ in $s$.

2.3. Regularity of the barycenter maps when the metric varies

In Appendix A, we will prove several results on the regularity of the barycenter map as the Riemannian metric on $F$ as well as the measures on $F$ vary. For the convenience of the reader, we now quote some of the key results from Appendix A that we will require later in the paper.

Fix constants $\epsilon_{0}>0$, $a,b\geq 0$, $m,n\in\operatorname{\mathbb{N}}$ and $k\geq 3$. Let $F$ be a smooth manifold and $B(0,\epsilon_{0})\subset\operatorname{\mathbb{R}}^{m}$. Suppose that

1. (1)

$B(0,\epsilon_{0})\ni s\mapsto g_{s}$ is a $C^{1,2}$ family of Riemannian metrics on $F$ with sectional curvatures $\kappa_{s}\in[-a^{2},b^{2}]$, 2. (2)

there are $C^{1}$ functions $w_{i}:B(0,\epsilon_{0})\to\operatorname{\mathbb{R}}$ and $z_{i}:B(0,\epsilon_{0})\to F$ for $1\leq i\leq n$ such that $\mu(s):=\sum_{1}^{n}w_{i}(s)\delta_{z_{i}(s)}$ are probability measures on $F$.

Let $r(s)=\operatorname{diam}\left\{z_{1}(s),\dots,z_{n}(s)\right\}$.

Proposition 2.8**.**

Suppose the setting is as above. Then there exists a constant $R>0$ (that depends only on $a$, $b$, and injectivity) such that whenever $r(s_{0})<R$, $\operatorname{bar}_{s_{0}}(\mu(s_{0}))$ exists and the map $s\mapsto\operatorname{bar}_{s}(\mu(s))$ is $C^{1}$.

Proof.

This is Proposition A.10. ∎

The next proposition gives a finer estimate on the derivative of the barycenter map as the metrics $g_{s}$ and the measures $\mu(s)$ vary. In this next result, we will use $d^{\operatorname{Gr}}_{(s,x)}$ to denote the Grassmannian distance on $\operatorname{Gr}(T_{(s,x)}(B(0,\epsilon_{0})\times F))$ induced by the Riemannian metric $ds^{2}+d_{g_{s}}^{2}$ on $B(0,\epsilon_{0})\times F$.

Proposition 2.9**.**

Suppose the setting is as above. Suppose that there is a continuous foliation $\operatorname{\mathcal{H}}$ on $B(0,\epsilon_{0})\times F$ such that:

1. (a)

$\operatorname{\mathcal{H}}$* has $C^{1}$ leaves which transversely vary continuously in the leafwise $C^{1}$-topology,* 2. (b)

for each $\xi\in F$, there is a $C^{1}$ function $\varphi_{\xi}:B(0,\epsilon_{0})\to F$ satisfying $\varphi_{\xi}(0)=\xi$ and the foliation $\operatorname{\mathcal{H}}$ is given by

[TABLE] 3. (c)

the tangent distribution of $\operatorname{\mathcal{H}}$ is denoted by $\operatorname{\mathcal{D}}$.

Fix $\xi_{1},\dots,\xi_{n}\in F$. Consider $z_{i}(s):=\varphi_{\xi_{i}}(s)$ and the measures $\mu(s):=\sum_{i=1}^{n}w_{i}(s)\delta_{z_{i}(s)}$. Suppose $\operatorname{b}:B(0,\epsilon_{0})\to F$ is the the barycenter map

[TABLE]

and let $G=\left\{(s,\operatorname{b}(s)):s\in B(0,\epsilon_{0})\right\}$ be the graph of $\operatorname{b}$. Then:

1. (1)

$D_{s=s_{0}}\operatorname{b}(s)$* and $T_{(s_{0},\operatorname{b}(s_{0}))}G$ vary continuously in $s_{0},\xi_{1},\dots,\xi_{n}$,* 2. (2)

for every $\epsilon>0$ and $c\in[0,1)$, there exists $\delta>0$ with the following property: if $s_{0}\in B(0,c\epsilon_{0})$ and $r(s_{0})<\delta$, then

[TABLE]

Proof.

This is Corollary A.13. ∎

3. Standard Actions and the Generalization of a Lemma of Bowden-Mann

Our primary goal in this section is to formulate and prove a generalization of a lemma of Bowden and Mann [BM22]. We will obtain this generalization for a class of actions that we will call standard actions.

3.1. Standard Actions

Suppose $M$ is a compact non-positively curved Riemannian manifold. Then $\pi_{1}(M)$ acts on the geodesic boundary of the universal cover $\widetilde{M}$ of $M$, which we denote by $\partial_{\infty}\widetilde{M}$. This boundary action is a primary example of a standard action. Note that $\pi_{1}(M)$ also acts on $T^{1}\widetilde{M}$ by isometries of the Sasaki metric. The relationship between these two aforementioned actions is the motivation behind the definition of a standard action (Definition 3.1). This definition, which may appear strange at first sight, is an abstraction of what happens in this case; see Lemma 3.5 for details.

We will now informally explain this relationship. Let $\widehat{\rho_{0}}$ be the action of $\pi_{1}(M)$ on $\widetilde{M}\times\partial_{\infty}\widetilde{M}$, where $\pi_{1}(M)$ acts on $\widetilde{M}$ by deck transformations and on $\partial_{\infty}\widetilde{M}$ by the boundary action. Consider the map $\Phi:T^{1}\widetilde{M}\to\widetilde{M}\times\partial_{\infty}\widetilde{M}$ defined by the foot-point projection to $\widetilde{M}$ and the radial projection to infinity in $\partial_{\infty}\widetilde{M}$. The map $\Phi$ maps each fiber in $T^{1}\widetilde{M}$ homeomorphically to $\partial_{\infty}\widetilde{M}$. The most crucial point is that $\Phi$ is a homeomorphism that intertwines the two actions. The map $\Phi$ also has some regularity: $\Phi^{-1}(\widetilde{M}\times\left\{\xi\right\})$ is a $C^{1}$-submanifold of $T^{1}\widetilde{M}$ for any $\xi\in\partial_{\infty}\widetilde{M}$. In fact, these are the center-unstable manifolds for the geodesic flow in $T^{1}\widetilde{M}$. In Definition 3.1, the intertwining of the product action (on $\widetilde{M}\times\partial_{\infty}\widetilde{M}$) with an isometric action (on $T^{1}\widetilde{M}$) is the content of part (3) while regularity of the map $\Phi$ is the content of part (2). Note that generally $\Phi^{-1}(\widetilde{M}\times\left\{\xi\right\})$ is only continuous in $\xi$; the definition makes no further assumptions about this ‘transversal’ regularity (see Remark 3.3).

We will now precisely define standard actions. Recall the definition of $C^{\ell,2}$ fiber bundles from Section 2.2. We will denote by $\operatorname{pr}_{X_{i}}$ the natural $i$-th coordinate projection map $X_{1}\times X_{2}\to X_{i}$ for $i=1,2$.

Definition 3.1**.**

Suppose $F$ and $X$ are differentiable (hence smooth) manifolds, $\Gamma\leq\mathrm{Isom}(X)$ is a discrete subgroup and $\ell\geq 1$. We will say that the action $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ is a standard action provided there exists a $C^{\ell,2}$-fiber bundle $p:X_{F}\to X$ with fiber $F$ such that:

1. (1)

there exists a family of homeomorphisms $\{\pi_{x}:F_{x}:=p^{-1}(x)\to F\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in X\}$, 2. (2)

the map $\Phi:X_{F}\to X\times F$ defined by

[TABLE]

is a homeomorphism and

[TABLE]

is a $C^{\ell}$-submanifold of $X_{F}$ for every $\xi\in F$, 3. (3)

there exists a family of $C^{\ell,2}$-Riemannian metrics $\left\{g_{x}:x\in X\right\}$ on the fibers $F_{x}$ in $X_{F}$ such that the map $\gamma_{x}:(F_{x},g_{x})\to(F_{\gamma x},g_{\gamma x})$ defined by

[TABLE]

is affine (with respect to the respective Riemannian connections) for any $\gamma\in\Gamma$ and $x\in X$.

If $X_{F},p,\Phi,\{\pi_{x}\},\{g_{x}\},\ell$ are as above, we will call $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ a standard action with associated data $(X_{F},p,\Phi,\{\pi_{x}\},\{g_{x}\},\ell)$.

Remark 3.2**.**

Note that the metric $\left\{g_{x}\right\}$ is determined by the maps $\left\{\pi_{x}\right\}$ in the following sense. Suppose $\left\{h_{x}\right\}$ is another family of metrics such that $\gamma_{x}^{\prime}:=\pi_{\gamma x}^{-1}\circ\rho_{0}(\gamma)\pi_{x}$ is also affine. Then $\operatorname{id}:=(\gamma^{\prime}_{x})^{-1}\circ\gamma_{x}:(F_{x},g_{x})\to(F_{x},h_{x})$ is an affine map.

In many cases, e.g. $F=S^{n}$, this implies that $g_{x}$ and $h_{x}$ differ only by a scaling factor. But this is not always the case. For instance, if $A\in SL(2,\operatorname{\mathbb{Z}})$ and $g$ is the Euclidean metric on a torus $\mathbb{T}^{2}$, then $\operatorname{id}:(\mathbb{T}^{2},g)\to(\mathbb{T}^{2},A^{*}g)$ is an affine map but the metrics aren’t multiples of each other.

Remark 3.3**.**

Note that there is no assumption on the regularity of the sub-manifolds $(\operatorname{pr}_{F}\circ\Phi)^{-1}(\xi)$ when $\xi$ varies.

We will discuss examples in the next subsection. First, however, let us introduce the notion of induced action: given a standard action $\rho_{0}$, we can induce an action $\widehat{\rho}_{0}$ on the fiber bundle $X_{F}$.

Definition 3.4**.**

Suppose $X,F$ are as above and $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ is a standard action with associated data $(X_{F},p,\Phi,\{\pi_{x}\},\{g_{x}\},\ell)$. Then the induced action on $X_{F}$, denoted by $\widehat{\rho}_{0}$, is the pullback by $\Phi:X_{F}\to X\times F$ of the action of $\Gamma$ on $X\times F$ where the action on $X$ is by isometries and the action on $F$ is by $\rho_{0}$ (i.e. $\gamma\in\Gamma$ acts by $\gamma\cdot(x,\xi)=(\gamma\cdot x,\rho_{0}(\gamma)\cdot\xi)$).

In this language, we say that $\rho_{0}$ is a standard action provided the action $\widehat{\rho_{0}}$ preserves the fibers and the fiberwise action is affine with respect to the family of metrics $\{g_{x}:x\in X\}$.

3.2. Examples of Standard Action

In this section, we discuss several examples of standard actions.

3.2.1. Geodesic boundary actions:

Our first example of a standard action will be on a sphere and fundamental to our proof of Theorem 1.1, the geodesic boundary of a simply connected non-positively curved Riemannian manifold.

Lemma 3.5**.**

Let $X$ be a simply connected complete smooth Riemannian manifold of non-positive sectional curvature. Suppose $\Gamma\leq\mathrm{Isom}(X)$ be any subgroup. Let $\rho_{0}$ be the standard action of $\Gamma$ on the geodesic boundary $\partial_{\infty}X$ of $X$, which is homeomorphic to ${\mathbb{S}}^{\dim(X)-1}$. Then $\rho_{0}$ is a standard action.

Proof.

Let $X_{F}=T^{1}X$ and $F=\partial_{\infty}X$. We will show that the definition of the standard action is satisfied for $\ell=1$. The standard foot-point projection $p:T^{1}X\to X$ gives $T^{1}X$ the structure of a smooth fiber bundle. Thus the fiber bundle is $C^{\ell,2}$ (in fact $C^{\ell,\infty}$) for any $\ell\geq 1$. Consider the family of homeomorphisms $\pi_{x}:T_{x}^{1}X\to\partial_{\infty}X$ given by the radial projections. Then $\Phi:T^{1}X\to X\times\partial_{\infty}X$ defined by $\Phi(v)=(p(v),\pi_{p(v)}(v))$ is clearly a homeomorphism. For any $\xi\in\partial_{\infty}X$,

[TABLE]

is a center-stable manifold of the geodesic flow on $T^{1}X$. We claim that $({\rm pr}_{F}\circ\Phi)^{-1}(\xi)$ is a $C^{1}$-submanifold of $X_{F}$ for any $\xi\in F$. Indeed, each center-stable manifold is given by evaluating the gradient of the Busemann function at all points of $X$. By [HIH77, Proposition 3.1], the gradient of the Busemann function is $C^{1}$ when $X$ is as in this Lemma. Thus each center-stable manifold is $C^{1}$.

Let $h$ be the given Riemannian metric on $X$. For each $x\in X$, let $g_{x}$ be the Riemannian metric on $T_{x}^{1}X$ induced by $h$. The fiber $(T_{x}^{1}X,g_{x})$ is isometric to the standard unit sphere. In fact, this metric is the restriction of the Sasaki metric on $T^{1}X$ on the fiber $T_{x}^{1}X$. Since the Sasaki metric is smooth whenever $h$ is smooth, $x\mapsto g_{x}$ varies smoothly (and hence $C^{\ell,\infty}$ for any $\ell\geq 1$). If $\gamma\in\mathrm{Isom}(X)$, then $\gamma$ acts by an isometry of the Sasaki metric. Hence $\gamma_{x}$ is an isometry. ∎

Note that $({\rm pr}_{F}\circ\Phi)^{-1}(\xi)$ only varies continuously in $\xi$ in general. But as remarked above, this lack of transversal regularity of the weak-stable manifolds $({\rm pr}_{F}\circ\Phi)^{-1}(\xi)$ is not a problem since the definition does not require it.

Remark 3.6**.**

Suppose $X$ and $\Gamma$ is as above. If $({\rm pr}_{F}\circ\Phi)^{-1}(\xi)$ is $C^{\infty}$ for each $\xi\in F$, then the above proof shows that $\rho_{0}$ is in fact a standard action with $\ell=\infty.$ In particular we mention three such cases of interest:

1. (1)

when $X/\Gamma$ is a compact manifold with negative sectional curvature. 2. (2)

when $X$ is a Hilbert geometry with strictly convex boundary, a similar construction works as long as we allow the fibers to have non-Riemannian metrics. 3. (3)

when $X$ is a Riemannian symmetric space.

The latter will be of particular interest to us later in the paper and we discuss it below.

3.2.2. Boundary actions on $G/Q$:

We will now consider the case of a lattice in a semisimple Lie group acting on the $G/Q$ boundaries, which are subsets of the geodesic boundary of the corresponding Riemannian symmetric space. For notation used in this section, we refer the reader to Section 5.

Lemma 3.7**.**

Suppose $G$ is a connected semisimple Lie group with finite center and without compact factors, $Q<G$ is a parabolic subgroup and $\Gamma<G$ is a discrete subgroup. Then the action of $\Gamma$ by left multiplication on $F=G/Q$ is a standard action.

Proof.

Let $K$ be a maximal compact subgroup of $G$. Then $X=G/K$, equipped with a left $G$-invariant Riemannian metric, is a Riemannian symmetric space with $\mathrm{Isom}(X)^{0}=G$. Here $F=G/Q$ and $X_{F}$ will be a space diffeomorphic to $G/M_{Q}$. In Section 5, we will see that $G/M_{Q}$ is the Weyl chamber $Q$-face bundle.

We will use the notation: for any $v\in T^{1}X$, let $\gamma_{v}$ be the geodesic in $X$ determined by $v$. Consider the continuous trivialization of the unit tangent bundle $\Phi_{0}:T^{1}X\to X\times S^{\dim(X)-1}$ defined by $\Phi_{0}(v)=(\pi(v),\gamma_{v}(\infty))$. Recall that the $\Gamma$ action on $\partial_{\infty}X$ is a standard action.

Fix $\mathfrak{X}\in\partial_{\infty}X$ be such that $Q=G_{\mathfrak{X}}$, so that $G/Q$ is diffeomorphic to $G\cdot\mathfrak{X}\subset\partial_{\infty}X$ via the orbit map $gQ\mapsto g\cdot\mathfrak{X}$. In particular, $G\cdot\mathfrak{X}$ is a smooth sub-manifold of $\partial X$ as $G$ acts smoothly on $\partial_{\infty}X$ ([Bre72, Theorem 1.2 Pg 302]).

Let

[TABLE]

Define $\Phi:X_{F}\to X\times F$ by restriction of $\Phi_{0}$, i.e

[TABLE]

Then $\Phi$ is a continuous trivialization of $X_{F}$ as a fiber bundle over $X$ with fibers $F=G/Q$.

We claim that $X_{F}$ is a smooth submanifold of $T^{1}X$ that is diffeomorphic to $G/M_{Q}$. To prove the claim, fix $v_{0}\in T^{1}_{eK}X$ such that $\gamma_{v_{0}}(\infty)=\mathfrak{X}$. Consider the orbit map $G\to X_{F}\subset T^{1}X$ defined by $g\mapsto g\cdot v_{0}$. Note that the stabilizer of $v_{0}$ is $K\cap Q$, i.e. $M_{Q}$. Further, $G$ acts transitively on $X_{F}$. Indeed, if $v\in X_{F}$, then there exists $g\in G$ such that $gv\in T_{eK}X$ and $\gamma_{v}(\infty)=g\cdot\gamma_{v_{0}}(\infty)$ (since $G$ acts as transitively on $X$ and and $\gamma_{v}(\infty)\in G\cdot\mathfrak{X}$). Thus $X_{F}$ is diffeomorphic to $G/M_{Q}$ via the above map. Since $G$ action on $T^{1}X$ is smooth and $X_{F}$ is the $G$-orbit of $v_{0}$, $X_{F}$ is an immersed submanifold of $T^{1}X$ (again by [Bre72, ch.VI, Theorem 1.2]). But since the $G$ action on $T^{1}X$ is properly discontinuous, this implies that $X_{F}$ is locally closed and hence, an embedded submanifold of $T^{1}X$.

We now finish the proof that the action on $G/Q$ is a standard action, by restricting all the data from $T^{1}X$ and $\partial_{\infty}X$ to the smooth subbundle $X_{F}\subset T^{1}X$ and $F=G\cdot\mathfrak{X}\subset\partial_{\infty}X$ respectively. Note that with the above claim, it becomes evident that $X_{F}$ has the structure of a smooth fiber bundle over $X=G/K$ with fibers homeomorphic to $G\cdot\mathfrak{X}\cong G/Q$. Let $p:T^{1}X\to X$ and $\{\pi_{x}:x\in X\}$ be the maps as in Lemma 3.5 and define $p^{\prime}:=p|_{X_{F}}$ and $\pi^{\prime}_{x}:=\pi_{x}|_{X_{F}}$ for all $x\in X$. The $G$-invariant Riemannian metric on $T^{1}X$ induces a family of metrics ${g^{\prime}_{x}:x\in X}$ on the fibers of $X_{F}$. Then the $\Gamma$ action on $G/Q$ is a standard action with the data $(X_{F},p^{\prime},\Phi,\left\{\pi^{\prime}_{x}\right\},\left\{g^{\prime}_{x}\right\},\ell)$ for any $\ell\geq 1$. ∎

Remark 3.8**.**

As seen in the proof above, $X_{F}$ is diffeomorphic with $G/M_{Q}$. Thus by pre-composing with this diffeomorphism, one can also show that $\Gamma$ has a standard action on $G/M_{Q}$ with appropriate associated data. In fact, this is the associated data that we will use in Proposition 7.3. Also see Section 5.6 for further discussion about this action.

3.2.3. Factor actions:

This is a degenerate example. Let $X_{F}=X\times F$ where $F$ is any compact Riemannian manifold and $\Gamma$ acts by the identity on $X$ and by an isometric action $\rho_{0}$ on the $F$-factor. More generally, we can take $\rho_{0}$ to be an affine action on F with respect to the Levi-Civita connection on $F$. Note that $X$ can be any smooth manifold here; in particular, a point.

3.2.4. Pull-back actions:

Restriction of standard actions to a closed invariant submanifold $F^{\prime}\subset F$ is another natural way of getting new examples. This is essentially a generalization of Lemma 3.7 to more general invariant submanifolds. However, for this restriction to be a standard action, one has to carefully ensure that the corresponding submanifolds in $X_{F}$ are $C^{\ell}$ for an appropriate $\ell$.

3.3. Generalized Bowden-Mann Lemma

In this section we generalize [BM22, Lemma 3.1] of Bowden and Mann (later adapted by Mann and Manning [MM21] for Gromov hyperbolic groups). We formulate our generalization in the language of standard actions. Recall the definition of standard action $\rho_{0}$ (Definition 3.1) and the induced action $\widehat{\rho_{0}}$ (Definition 3.4).

Proposition 3.9**.**

(cf. [BM22, Lemma 3.1]) Suppose $X$ is a Riemannian manifold, $\Gamma<\mathrm{Isom}(X)$ is a discrete subgroup acting freely such that $\Gamma\backslash X$ is compact, and $F$ is a smooth manifold. Suppose $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ is a standard action with associated data $(X_{F},p,\Phi,\{\pi_{x}\},\{g_{x}\},\ell)$ where $\ell\geq 1$. For any $\rho:\Gamma\to\operatorname{Homeo}(F)$ that is sufficiently $C^{0}$-close to $\rho_{0}$, there is a continuous map $\widetilde{f}:X\times F\to X_{F}$ such that:

1. (1)

$p(\widetilde{f}(x,\xi))=x$* and $\widetilde{f}(\gamma\cdot x,\rho(\gamma)\xi)=\widehat{\rho}_{0}(\gamma)\widetilde{f}(x,\xi)$,* 2. (2)

the map $\widetilde{f}(\cdot,\xi):X\to\widetilde{f}(X\times\{\xi\})$ is a $C^{1}$ diffeomorphism for every $\xi\in F$. Moreover, for any $(x_{0},\xi_{0})\in X\times F$, $(\rho,y,\xi)\mapsto D_{y}\widetilde{f}(y,\xi)$ is continuous in all three variables at $(\rho_{0},x_{0},\xi_{0})$, 3. (3)

for every $\epsilon>0$ and for every $x\in X$, there exists a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(x,\epsilon)$ of $\rho_{0}$ such that

[TABLE] 4. (4)

suppose that for any compact set $\operatorname{\mathcal{K}}\subset X$, there exist constants $c,\alpha$ depending only on $\operatorname{diam}(\operatorname{\mathcal{K}})$ such that for all $x,z\in\operatorname{\mathcal{K}}$, $\pi_{z}^{-1}\circ\pi_{x}:F_{x}\to F_{z}$ is Hölder with Hölder constants $c$ and $\alpha$. Then for any $\epsilon,R>0$ and $(x,\xi)\in X\times F$, there is a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon,R,x,\xi)$ of $\rho_{0}$ such that: if $\rho\in\operatorname{\mathcal{U}}$ and $\eta:=\pi_{x}(\widetilde{f}(x,\xi))\in F$, then

[TABLE]

where $d_{F_{y}}$ is the distance on the fiber $F_{y}$ (induced by the Riemannian metric $g_{y}$), 5. (5)

Suppose $h$ is a Riemannian metric on $X_{F}$ such that $\widehat{\rho_{0}}(\Gamma)<\mathrm{Isom}(X_{F},h)$ and $h|_{F_{x}}=g_{x}$ for all $x\in X$. For any $\epsilon>0$, there exists a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon)$ of $\rho_{0}$ such that

[TABLE]

where $\eta_{x}:=\pi_{x}(\widetilde{f}(x,\xi))$.

Remark 3.10**.**

Part (4) of this proposition also holds under the following alternate assumption. Suppose $X_{F}$ carries a Riemannian metric $h^{\prime}$ such that $h^{\prime}|_{F_{x}}=g_{x}$ and each fiber $(F_{x},g_{x})$ is a totally geodesic submanifold of $(X_{F},h^{\prime})$ (or even more generally, there exists $0<L<1$ such that $d_{X_{F}}(u,v)\geq Ld_{F_{x}}(u,v)$ for any $u,v,\in F_{x}$).

Note that the two assumptions do not imply each other. For instance, consider the case where $X=\widetilde{M}$ is a simply connected Riemannian manifold and $X_{F}=T^{1}\widetilde{M}$. If $X$ has a negatively curved metric, then the H’́older assumption (in part (4)) as well the totally geodesic assumption (mentioned in the above paragraph) is satisfied. But if $X$ is non-positively curved then the alternate assumption stated in this remark is satisfied while the Hölder assumption is not.

Proof.

Without loss of generality, we can assume that $\ell=1$. Since $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ is a standard action with associated data $(X_{F},p,\Phi,\{\pi_{x}\},\{g_{x}\},1)$, we have:

1. (a)

a $C^{1,2}$ fiber bundle $p:X_{F}\to X$ with fiber $F$, 2. (b)

a family of homeomorphisms $\{\pi_{x}:x\in X\}$ where $\pi_{x}:F_{x}\to F$, 3. (c)

a homeomorphism $\Phi:X_{F}\to X\times F$ given by $\Phi(z)=(p(z),\pi_{p(z)}(z))$, 4. (d)

$(\operatorname{pr}_{F}\circ\Phi)^{-1}(\xi)$ is a $C^{1}$-submanifold of $X_{F}$ for every $\xi\in F$, 5. (e)

a $C^{1,2}$-smoothly varying family of Riemannian metrics $\left\{g_{x}:x\in X\right\}$ on the fibers $F_{x}$ such that $\gamma_{x}:=\pi_{\gamma x}^{-1}\circ\rho_{0}(\gamma)\circ\pi_{x}$ is affine for any $\gamma\in\Gamma$ and $x\in X$.

Moreover $\widehat{\rho}_{0}$ denotes the action on $X_{F}$ induced by $\rho_{0}$.

**Construction of $\widetilde{f}$: ** Let $D\subset X$ be a pre-compact connected fundamental domain for the action of $\Gamma$ on $X$. Let $p_{\Gamma}:X\to\Gamma\backslash X$ be the covering map and let $\left\{\mathcal{O}_{i}\right\}$ be a finite cover of $\Gamma\backslash X$ by connected open sets. Since $\Gamma$ acts freely and properly discontinuously, we may assume that there exists an open cover $\left\{\widetilde{\mathcal{O}}_{j}\right\}$ of $X$ such that: for each $j$, there is an $i$ for which the restriction of the covering map $p_{\Gamma}|_{\widetilde{\operatorname{\mathcal{O}}_{j}}}:\widetilde{\mathcal{O}}_{j}\to\Gamma\backslash X$ is an isometry onto $\operatorname{\mathcal{O}}_{i}$. Indeed, we can obtain such a cover by shrinking the diameter of $\operatorname{\mathcal{O}}_{i}$ sufficiently and taking their pre-images under the covering map.

Let $\operatorname{\mathcal{W}}=\{\operatorname{\mathcal{W}}_{j}:1\leq j\leq m^{\prime}\}$ be the finite sub-family of $\left\{\widetilde{\mathcal{O}}_{i}\right\}$, consisting of open sets intersecting $\overline{D}$ non-trivially. Let $\left\{\sigma_{i},\mathcal{O}_{i}\right\}$ be a partition of unity on $\Gamma\backslash X$ subordinate to the open cover $\left\{\mathcal{O}_{i}\right\}$ and and let $\widetilde{\sigma_{i}}:X\to\mathbb{R}$ be the lift of $\sigma_{i}$.

The $\Gamma$-action on $\left\{\widetilde{\mathcal{O}}_{i}\right\}$ introduces an equivalence relation $\sim_{\Gamma}$ on $\mathcal{W}$, namely $\operatorname{\mathcal{W}}_{i}\sim_{\Gamma}\operatorname{\mathcal{W}}_{j}$ if and only if $\operatorname{\mathcal{W}}_{i}=\gamma\operatorname{\mathcal{W}}_{j}$ for some $\gamma\in\Gamma$. Now we choose one representative from each equivalence class in $\operatorname{\mathcal{W}}$ and label them as $\operatorname{\mathcal{U}}_{1},\dots,\operatorname{\mathcal{U}}_{m^{\prime\prime}}$. Note that we specially mark these $\operatorname{\mathcal{U}}_{i}$’s because we will define our map on these $\operatorname{\mathcal{U}}_{i}$’s and extend it equivariantly.

First, for each of the open sets $\operatorname{\mathcal{U}}_{1},\dots,\operatorname{\mathcal{U}}_{m^{\prime\prime}}$, we define a map $\psi_{\operatorname{\mathcal{U}}_{i}}:\operatorname{\mathcal{U}}_{i}\times F\to X_{F}$ by

[TABLE]

That is, for each $\xi\in F$, the map $\psi_{\mathcal{U}_{i}}(\cdot,\xi)$ is a local section $\mathcal{U}_{i}\to X_{F}$.

As $\mathcal{W}$ is a finite set, there is a finite subset $S_{\operatorname{\mathcal{W}}}\subset\Gamma$ such that: if $1\leq j\leq m^{\prime}$, then $\mathcal{W}_{j}=\gamma\mathcal{U}_{i}$ for some $\gamma\in S_{\operatorname{\mathcal{W}}}$ and $1\leq i\leq m^{\prime\prime}$.

Then for $1\leq j\leq m^{\prime}$, if $\mathcal{W}_{j}=\gamma\mathcal{U}_{i}$ we define $\Psi_{{\operatorname{\mathcal{W}}}_{j}}:{\mathcal{W}}_{j}\times F\to X_{F}$ to be

[TABLE]

where $(x,\xi)\in\mathcal{U}_{i}\times F$. We note that if $\mathcal{W}_{i}=\gamma^{\prime}\mathcal{W}_{j}$ then we have $\Psi_{\mathcal{W}_{i}}(\gamma^{\prime}x,\rho(\gamma^{\prime})\xi)=\widehat{\rho}_{0}(\gamma^{\prime})\Psi_{\mathcal{W}_{j}}(x,\xi)$ for any $x\in\mathcal{W}_{j}$ and any $\xi\in F$.

We consider the neighborhood $\Omega$ of $\overline{D}$ defined by: $x\in\Omega$ provided $\{\widetilde{\mathcal{O}_{i}}:x\in\widetilde{\mathcal{O}_{i}}\}\subset\operatorname{\mathcal{W}}$, i.e. all the lifts $\widetilde{\mathcal{O}}_{i}$ of the open cover $\{\mathcal{O}_{i}\}$ containing $x$ belong to $\operatorname{\mathcal{W}}$. Fix any $x\in\Omega$. We will now define the map $\widetilde{f}(x,\cdot)$ (see (3.4)). Suppose that, possibly after re-indexing the $\mathcal{W}_{i}$’s, $\mathcal{W}_{1},\dots,\mathcal{W}_{m}$ are all the $\mathcal{W}_{i}$’s that contain $x$. By the remark above, each $\operatorname{\mathcal{W}}_{j}$ corresponds to $\gamma_{j}\in\operatorname{\mathcal{S}}_{\operatorname{\mathcal{W}}}$ such that

[TABLE]

for some $\operatorname{\mathcal{U}}_{i}$. For any $\xi\in F$, set

[TABLE]

Then,

[TABLE]

For every $\xi\in F$ define,

[TABLE]

where $\delta_{z}$ is the Dirac measure at $z$, and $\operatorname{bar}_{x}$ is the barycenter operation on the fiber $F_{x}$. Since $x$ belongs to a compact set, it follows that $\pi_{x}^{-1}(\xi_{j})$ is close to $\pi_{x}^{-1}(\xi)$ when $\rho$ is sufficiently close to $\rho_{0}$, and hence the barycenter is well-defined.

In order to extend to a $(\widehat{\rho},\widehat{\rho}_{0})$-equivariant map $\widetilde{f}:X\times F\to X_{F}$, we need to check that the map $\widetilde{f}$ defined on $\Omega\times F$ is equivariant. That is, if both $x$ and $\gamma x$ are in $\Omega$ for some $\gamma\in\Gamma$ then $\widetilde{f}(\gamma x,\rho(\gamma)\xi)=\widehat{\rho}_{0}(\gamma)\widetilde{f}(x,\xi)$, for every $\xi\in F$. Indeed, if $\mathcal{W}_{1},\dots,\mathcal{W}_{m}$ are the open sets containing $x$ then $\gamma\mathcal{W}_{1},\dots,\gamma\mathcal{W}_{m}$ are the open sets containing $\gamma x$. By construction,

[TABLE]

where the second to last equality follows from the fact that the fiber-wise barycenter construction is $\widehat{\rho}_{0}$-equivariant as $\rho_{0}$ is a standard action (cf. Remark 2.4). Therefore the map $\widetilde{f}$ is equivariant on $\Omega\times F$. Since $\Omega$ is an open neighborhood of $\overline{D}$, $\widetilde{f}$ extends to a map $\widetilde{f}:X\times F\to X_{F}$ that is $(\widehat{\rho},\widehat{\rho}_{0})$-equivariant. This finishes the construction of $\widetilde{f}$. Now we proceed to prove all parts of the proposition.

Proof of Part (1): This follows from the construction of $\widetilde{f}.$

Proof of Part (2): Fix any $\xi\in F$. Note that by definition, $p\circ\widetilde{f}=\operatorname{id}$ and hence $\widetilde{f}|_{X\times\{\xi\}}$ is injective. Then for proving (2), it suffices to show that $\widetilde{f}$ is $C^{1}$ along the horizontal leaf $X\times\{\xi\}$. Indeed, as $\widetilde{f}$ covers the identity, this implies that $\widetilde{f}|_{X\times\{\xi\}}$ is a diffeomorphism onto its image $\widetilde{f}(X\times\{\xi\})$. Because $\widetilde{f}$ is equivariant, it is enough to verify the regularity of $\widetilde{f}(\cdot,\xi)$ on the neighborhood $\Omega$ of $\overline{D}$ for every $\xi\in F$.

As $\widehat{\rho_{0}}(\Gamma)\backslash X_{F}$ is a $C^{1,2}$-fiber bundle, we can choose an atlas of $C^{1,2}$-smooth local trivializations. Refining the open cover $\{\mathcal{O}_{i}\}$ if necessary, we can assume that the local trivializations of $\widehat{\rho_{0}}(\Gamma)\backslash X_{F}$ are of the form $\{\mathcal{O}_{i}\times F\}$. Let $\{\mathcal{W}_{i}\times F\}$ be the finite subset of the corresponding local trivializations, as defined before in Eq. 3.2. For $\xi^{\prime}\in F$, consider the map $\operatorname{\mathcal{W}}_{i}\ni y\mapsto\phi_{\xi^{\prime}}(y):=\pi_{y}^{-1}(\xi^{\prime})$. By definition of standard action, each $\phi_{\xi^{\prime}}$ is a $C^{1}$ map.

Each local trivialization has a continuous foliation $\operatorname{\mathcal{H}}$ whose leaves are graphs of $\phi_{\xi^{\prime}}$ for every $\xi^{\prime}\in F$, i.e. $\left\{(y,\pi_{y}^{-1}(\xi^{\prime})):y\in\operatorname{\mathcal{W}}_{i}\right\}$ and $\xi^{\prime}\in F$. The weights $\widetilde{\sigma}_{i}(y)$ also vary $C^{1}$ in $y$. On each local trivialization $\mathcal{W}_{i}\times F$, by pulling back the metrics on $F_{y}$ to $F$, we obtain a $C^{1,2}$-family of metrics $\left\{g_{y}:y\in\operatorname{\mathcal{W}}_{i}\right\}$ on $F$. For $1\leq i\leq q$, let $\xi_{i}:=\rho_{0}(\gamma_{i})\rho(\gamma_{i})^{-1}\xi$, as in Eq. 3.3. Then Eq. 3.4 implies that on each local trivialization $\operatorname{\mathcal{W}}_{i}\times F$,

[TABLE]

where $\operatorname{bar}_{y}$ is the fiber-wise barycenter with respect to the metric $g_{y}$.

We now claim that $\widetilde{f}(X\times\xi)$ is a $C^{1}$ submanifold of $X_{F}$. The proof of this claim is an application of Proposition 2.8. Each $\mathcal{W}_{i}$, via a coordinate chart, can be thought as an open ball in some Euclidean space. In this coordinate chart, $\widetilde{f}(y,\xi)$ is then a fiber-wise barycenter map on $F_{y}$ with respect to the metrics $g_{y}$. Then Proposition 2.8 implies that for a fixed $\xi$, $\widetilde{f}(y,\xi)$ varies in a $C^{1}$ way with respect to the base point $y\in\mathcal{W}_{i}$. The same holds after pushing back to $X_{F}$ under the trivialization. Now observe that for any $y\in X$, there exists $\gamma\in\Gamma$ and $y_{0}\in\operatorname{\mathcal{W}}_{i}$ such that $\widehat{\rho_{0}}(\gamma)\widetilde{f}(y_{0},\xi)=\widetilde{f}(y,\xi)$. Then the claim again follows because $\widehat{\rho_{0}}(\gamma)$ acts by smooth diffeomorphisms.

We now prove the second claim: if $(x,\xi)\in X\times F$, then there exists a neighborhood of $(\rho_{0},x,\xi)$ where the map $(\rho,y,\xi^{\prime})\mapsto D_{y}\widetilde{f}(y,\xi^{\prime})$ is continuous in all variables. The proof is an application of Proposition 2.9. Since $\widetilde{f}$ is $\widehat{\rho_{0}}(\Gamma)$ equivariant, it suffices to prove it on a local trivialization chart $\left\{\operatorname{\mathcal{W}}_{i}\times F\right\}$. By Proposition 2.9, $D_{y}\widetilde{f}(y,\xi)$ varies continuously with respect to $\xi_{i}$. Thus $D_{y}\widetilde{f}(y,\xi)$ varies continuously as $\rho\to\rho_{0}$ or as $\xi$ varies continuously. Furthermore $D_{y}\widetilde{f}(y,\xi)$ is continuous in $y$ by part (2). Thus the moreover part holds on $\left\{\operatorname{\mathcal{W}}_{i}\times F\right\}$.

Proof of Part (3): By part (2), $(\rho,x,\xi)\mapsto\widetilde{f}(x,\xi)$ is continuous in all three variables. As $\pi_{x}$ is also continuous, the conclusion follows.

Proof of Part (4): Since $\widetilde{f}$ is $(\widehat{\rho},\widehat{\rho_{0}})$-equivariant, note that it suffices to prove the claim for $x\in\overline{D}$. Fix $\epsilon,R>0$. Let $x\in\overline{D}$ and $\xi\in F$.

Let $\epsilon^{\prime}>0$. We first claim that there exists $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon^{\prime},R,x,\xi)$ such that

[TABLE]

provided $\epsilon^{\prime}$ is sufficiently small. We now prove this claim. From the proof of part (3), we know that $(\rho,y,\xi)\mapsto\widetilde{f}(y,\xi)$ is a continuous map. Since the metrics $\left\{g_{y}\right\}$ are a $C^{1,2}$ family of metrics on the fibers $F_{y}$, $d_{F_{y}}(u,v)$ varies continuously in $y$ provided $d_{F_{y}}(u,v)$ is sufficiently small; see Lemma 2.7. Then $(\rho,y,\xi)\mapsto\widetilde{d}_{F_{y}}(\widetilde{f}(y,\xi),\pi_{y}^{-1}(\xi))$ is a continuous function in all variables near $(\rho_{0},x_{0},\xi_{0})$ provided $d_{F_{x_{0}}}(\widetilde{f}(x_{0},\xi),\pi_{x_{0}}^{-1}(\xi_{0}))$ is sufficiently small.

Suppose $\epsilon^{\prime}>0$ is sufficiently small. Let $y\in B_{R}(x)$. Then part (3) and the discussion in the previous paragraph imply that there exists a neighborhood $U_{y}\subset X$ of $y$ and a neighborhood $\operatorname{\mathcal{U}}^{\prime}=\operatorname{\mathcal{U}}^{\prime}(\epsilon^{\prime},y,\xi)$ of $\rho_{0}$ such that:

[TABLE]

Now since $B_{R}(x)$ is pre-compact, we can find $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon^{\prime},R,x,\xi)$ such that Eq. 3.6 is satisfied and this finishes the proof of the claim.

Let $\eta:=\pi_{x}(\widetilde{f}(x,\xi))$. Then by Eq. 3.6, $d_{F_{x}}(\pi_{x}^{-1}(\xi),\pi_{x}^{-1}(\eta))<\epsilon^{\prime}$. Moreover for any $y\in B_{R}(x)$, $(\operatorname{pr}_{F}\circ\Phi)^{-1}(\eta)\cap F_{y}=\pi_{y}^{-1}(\eta)$. We now claim that after refining $\operatorname{\mathcal{U}}$,

[TABLE]

To prove this claim, let $\left\{s_{1},\dots,s_{d}\right\}$ be the fixed finite generating set of $\Gamma$ corresponding to the compact fundamental domain $\overline{D}$. For any $y\in B_{R}(x)$, there exists $\gamma\in\Gamma$ such that $y\in\gamma\overline{D}$. Let $K^{\prime}=K^{\prime}(y)$ be the word length of $\gamma$ with respect to this generating set, i.e. $\gamma=s_{i_{1}}\cdots s_{i_{K^{\prime}}}$. Note that $K_{0}:=\max_{y\in B_{R}(x)}K^{\prime}(y)$ exists and depends only on $R$ and the choice of generating set (i.e. choice of $\overline{D}$).

By the assumption in part (4), there exist $c>1$ and $0<\alpha\leq 1$ such that for any $x,z\in\gamma\cdot\overline{D}$ and $u,v\in F_{x}$, the map $\pi_{z}^{-1}\circ\pi_{x}:F_{x}\to F_{z}$ satisfies

[TABLE]

Note that by assumption, $c,\alpha$ depend only on $\operatorname{diam}(\overline{D})$ (and hence are independent on which $\Gamma$-translate of $\overline{D}$ the points $x$ and $z$ lie in). Then

[TABLE]

Recall that $K_{0}=\max_{y\in B_{R}(x)}K^{\prime}(y)$ and $K_{0}$ is independent of $c,\alpha$ and $\epsilon^{\prime}$. For fixed constants $c,\alpha$ and $K_{0}$, the function $M(\epsilon^{\prime}):=\left(\max_{1\leq K^{\prime}\leq K_{0}}c^{K^{\prime}}(\epsilon^{\prime})^{\alpha^{K^{\prime}}}\right)$ is continuous in $\epsilon^{\prime}$ and it goes to [math] when $\epsilon^{\prime}\to 0$. So we can choose $\epsilon^{\prime}=:\epsilon_{0}$ sufficiently small such that $M(\epsilon_{0})<\frac{\epsilon}{2}$. We then choose the neighborhood $\operatorname{\mathcal{U}}$ of $\rho_{0}$ for $\epsilon^{\prime}=\epsilon_{0}$; see Eq. 3.6. Then for every $\rho\in\operatorname{\mathcal{U}}$, we have Eq. 3.6 and thus the above discussion implies that Eq. 3.7 holds. This establishes the claim and finishes the proof of part (4).

Proof of Part (5) To prove (5), we start with the same setup as in the proof of (2). We choose a finite family of $C^{1,2}$-local trivializations $\{\mathcal{O}_{i}\times F\}$ of $X_{F}/\widehat{\rho}_{0}(\Gamma)$ and let $\left\{\operatorname{\mathcal{W}}_{i}\times F\right\}$ be the local trivialization charts as earlier. As in the proof of (2), we get a family $\left\{g_{y}:y\in\operatorname{\mathcal{W}}_{i}\right\}$ of $C^{1,2}$ Riemannian metrics on $F$ parametrized by $\mathcal{W}_{i}$, which is homeomorphic to a ball in some $\operatorname{\mathbb{R}}^{\dim\operatorname{\mathcal{W}}_{i}}$. The set $\mathcal{W}_{i}\times F$ has a continuous foliation $\operatorname{\mathcal{H}}=\left\{\operatorname{\mathcal{H}}_{\xi^{\prime}}:\xi^{\prime}\in F\right\}$ whose leaves $\operatorname{\mathcal{H}}_{\xi^{\prime}}:=\{(y,\pi_{y}^{-1}\xi^{\prime}):y\in\mathcal{W}_{i}\}$ are $C^{1}$. As in Eq. 3.5, for every $\xi\in F$ and $y\in\operatorname{\mathcal{W}}_{i}$, we have

[TABLE]

where $\varphi_{\xi_{i}}(y)=\pi_{y}^{-1}(\xi_{i})$ and $\xi_{i}=\rho_{0}(\gamma_{j})\rho(\gamma_{j}^{-1})(\xi).$ When $\rho\to\rho_{0}$, $\xi_{i}$ converges to $\xi$ and hence, $\varphi_{\xi_{i}}(y)$ converges to $\varphi_{\xi}(y)$ locally uniformly on $\operatorname{\mathcal{W}}_{i}$. Then the diameters $\operatorname{diam}\{\varphi_{\xi_{i}}(y):i=1,\dots,q\}$ tend to 0 for every $y\in\mathcal{W}_{i}$. We are now in the setup of applying Proposition 2.9.

Fix $\epsilon>0$. Let $\delta$ be as in Proposition 2.9. There exists a neighborhood $\operatorname{\mathcal{U}}$ of $\rho_{0}$ such that if $\rho\in\operatorname{\mathcal{U}}$, then $\left(\sup_{y\in\operatorname{\mathcal{W}}_{i}}\operatorname{diam}\left\{\varphi_{\xi_{i}}(y):1\leq i\leq q\right\}\right)<\delta$. Then, by Proposition 2.9 for any $\rho\in\operatorname{\mathcal{U}}$ and $y\in\operatorname{\mathcal{W}}_{i}$,

[TABLE]

where $\eta_{y}=\pi_{y}(\widetilde{f}(y,\xi))$.

Now we explain how to get this for any point $(y,\xi)$. As $\widehat{\rho_{0}}(\Gamma)$ acts by isometries on $(X_{F},h)$, the above Grassmanian distance does not change under $\rho_{0}(\Gamma)$ action. As $X_{F}/\widehat{\rho_{0}}(\Gamma)$ is compact, we can choose finitely many local trivializations to cover it; as discussed in the beginning of the proof of part (2). As above, we now apply Proposition 2.9 for each of these finitely many local trivializations and conclude that there exists a neighborhood $\operatorname{\mathcal{U}}$ of $\rho_{0}$ for which Eq. 3.8 holds. ∎

4. Preliminaries II: General Results about Topological Actions

The main purpose of this section is proving the local semi-rigidity holds for general uniform lattices if it holds for torsion-free lattices. Readers may skip this section if you want to assume the lattices being torsion-free.

In this section, we will work in the very general framework of a group acting by homeomorphisms (on one occasion, by bi-Lipschitz homeomorphisms) on a compact metric space. We will apply these general results later on in the paper to derive two important applications: first, to discuss uniqueness properties of semi-conjugacies between nearby actions, and second, to promote semi-conjugacies from subgroups of finite index to the whole group. In particular, this will allow us to prove local uniqueness of the semi-conjugacies in our main theorem as well as bypass the torsion elements in lattices.

Before we get into the results, let us consider some elementary examples that illustrate some of the subtleties of working with semi-conjugacies. First, the semi-conjugacies (even conjugacies) between actions need not be unique. Moreover, one may think that if $\rho\to\rho_{0}$, then any family of semi-conjugacies $\left\{\phi_{\rho}\right\}$ must automatically converge to $\operatorname{id}$. However this also need not be the case.

Example 4.1**.**

Consider the action of $SO(n+1)$ on $S^{n}$. The action is conjugated to itself by both $\operatorname{id}$ and $-\operatorname{id}$. If we want a discrete group action, we can consider an embedding of a lattice into $SO(n+1)$ and consider the action of this lattice on $S^{n}$.

More generally we have,

Example 4.2**.**

Consider an action $\rho$ of a group $G$ on a manifold $M$ and suppose $\Gamma<G$ is a subgroup with a nontrivial centralizer $Z(\Gamma)<G$. Then if $\rho_{0}$ is the restriction of the action to $\Gamma$, then $\rho_{0}$ is self-conjugate both by $\rho(\sigma)$ for any $\sigma\in Z(\Gamma)$. However for $\sigma\neq 1$, the homeomorphism $\rho(\sigma)$ need not be close to the identity.

However, we will prove some results in Section 4.2 and Section 4.3 to control the behavior of semi-conjugacies under some additional assumptions.

4.1. Some Definitions

For this section, we assume that $(F,d_{F})$ is a compact metric space. We equip $C^{0}(F,F)$, the space of continuous functions from $F$ to itself, with the distance $d_{0}(f,g)=\sup_{x\in F}d_{F}(f(x),g(x))$. We will denote by $\operatorname{\mathcal{B}}_{r}(f)$ the open ball in $(C^{0}(F,F),d_{0})$ of radius $r$ around the point $f$. Our focus will be on the group $\operatorname{Homeo}(F)$ of homeomorphisms of $F$, equipped with the compact open topology, i.e. the topology generated by the basic open sets $\left\{f\in\operatorname{Homeo}(F):f(K)\subset U\right\}$ where $K\subset F$ is a compact set and $U\subset F$ is an open set.

Here $\Gamma$ will always denote a finitely generated group. We assume that $\Gamma$ acts on $F$ by homeomorphisms. A faithful $C^{0}$-action of $\Gamma$ on $F$ is an injective homomorphism $\rho:\Gamma\to\operatorname{Homeo}(F)$. The space $\operatorname{Hom}(\Gamma,\operatorname{Homeo}(F))$ is the space of faithful actions of $\Gamma$ on $F$ by homeomorphisms. We endow it with the compact open topology.

Definition 4.3**.**

Suppose $\rho,\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ are two actions of $\Gamma$. A continuous surjective map $\phi:F\to F$ is a $(\rho|_{H},\rho_{0}|_{H})$-semi-conjugacy for the group $H<\Gamma$ if

[TABLE]

If $H=\Gamma$, we will simplify the notation to $(\rho,\rho_{0})$ semi-conjugacy.

Definition 4.4**.**

Suppose $\lambda>1$.

1. (1)

We say that $f\in\operatorname{Homeo}(F)$ is $\lambda$-expanding on an open subset $U\subset F$ if: for all $x^{\prime},x^{\prime\prime}\in U$,

[TABLE] 2. (2)

If $\operatorname{\mathcal{U}}$ is an open cover of $F$, we say that $\rho:\Gamma\to\operatorname{Homeo}(F)$ is a $(\lambda,\operatorname{\mathcal{U}})$ uniformly expanding action on $F$ if: for each $x\in F$, there exists $U_{x}\in\operatorname{\mathcal{U}}$ and $g_{x}\in\Gamma$ such that $\rho(g_{x})$ is $\lambda$-expanding on $U_{x}$.

4.2. Uniqueness of Semi-Conjugacies

As noted in the examples above, semi-conjugacies and even conjugacies between actions are not always unique. However, under additional assumptions they become unique provided the actions are sufficiently close and the (semi-)conjugacy is sufficiently close to identity. We will provide such conditions in this subsection.

Lemma 4.5**.**

Suppose $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ is a $(\lambda,\operatorname{\mathcal{U}})$-uniformly expanding action where $\lambda>1$ and $\operatorname{\mathcal{U}}$ is an open cover of $F$. Then there exists $\epsilon>0$ such that: if $\rho:\Gamma\to\operatorname{Homeo}(F)$ is another action and $\phi,\phi^{\prime}:F\to F$ are $(\rho,\rho_{0})$ semi-conjugacies satisfying $d_{0}(\phi,\phi^{\prime})<\epsilon$, then $\phi=\phi^{\prime}$.

Proof.

By compactness of $F$, we can extract a finite sub-cover of $\operatorname{\mathcal{U}}$ such that $\rho_{0}$ is $(\lambda,\left\{U_{1},\dots,U_{k}\right\})$-uniformly expanding. Let $\epsilon$ be the Lebesgue number of the sub-cover $\left\{U_{1},\dots,U_{k}\right\}$ of $F$, i.e. if $d_{F}(x,y)<\epsilon$, then $x,y\in U_{i}$ for some $1\leq i\leq k$. Suppose $\phi,\phi^{\prime}$ are $(\rho,\rho_{0})$-semi-conjugacies such that $d_{0}(\phi^{\prime},\phi)<\epsilon$. It suffices to show that $\phi^{-1}(p)=(\phi^{\prime})^{-1}(p)$ for all $p\in F$.

To prove this fix $p\in F$. Let $x\in\phi^{-1}(p)$ and $y\in(\phi^{\prime})^{-1}(p)$, i.e. $\phi(x)=\phi^{\prime}(y)=p$. We will show that $y\in\phi^{-1}(p)$, i.e. $\phi(y)=\phi(x)$. Note that by symmetry it will imply that $x\in(\phi^{\prime})^{-1}(p)$ and finish the proof.

Suppose, on the contrary, that $\phi(x)\neq\phi(y)$. But $d_{F}(\phi(x),\phi(y))=d_{F}(\phi^{\prime}(y),\phi(y))<\epsilon$. Then there exists $U_{i}$ that contains $\phi(x)$ and $\phi(y)$. By the $(\lambda,\left\{U_{1},\dots,U_{k}\right\})$-uniform expansion assumption, there exists $\gamma\in\Gamma$, such that

[TABLE]

We let $\gamma_{1},\dots,\gamma_{k}$ be elements of $\Gamma$ such that $\gamma_{i}$ is $(\lambda,U_{i})$-expansive.

Then there exists $i_{1}\in\{1,\dots,k\}$ such that $\phi(x)$ and $\phi(y)$ belong to $U_{i_{1}}$. By the $(\lambda,U_{i_{1}})$- expansion assumption,

[TABLE]

If $d_{F}(\phi(\rho(\gamma_{i_{1}}x),\phi(\rho(\gamma_{i_{1}}y)))<\epsilon$, then there exists $i_{2}\in\{1,\dots,k\}$ such that $\phi(\rho(\gamma_{i_{1}}x)),\phi(\rho(\gamma_{i_{1}}y))\in U_{i_{2}}$. Applying the action by $\rho_{0}(\gamma_{i_{2}})$, we get

[TABLE]

If $d_{F}(\phi(\rho(\gamma_{i_{2}}\gamma_{i_{1}})x),\phi(\rho(\gamma_{i_{2}}\gamma_{i_{1}})y))<\epsilon$, we continue above process by induction. Since $\lambda>1$, there is $m\in\mathbb{N}$ such that

[TABLE]

but

[TABLE]

On the other hand,

[TABLE]

This contradicts with the assumption that $d_{0}(\phi,\phi^{\prime})<\epsilon$.

Thus, we must have $\phi(x)=\phi(y)$, which finishes the proof. ∎

4.3. Reduction to Normal Subgroups

We will now apply the criteria from the previous subsection to obtain semi-conjugacies between actions of a group assuming existence of semi-conjugacies for suitable normal subgroups.

Lemma 4.6**.**

Suppose $\Gamma_{0}<\Gamma$ is a normal subgroup and $\Gamma/\Gamma_{0}$ is finitely generated. Suppose that for any $\epsilon>0$ there is a neighborhood $U=U(\epsilon)$ of $\rho_{0}$ such that if $\rho\in U$, then either $\rho$ is not semi-conjugate to $\rho_{0}$ or there is exactly one $(\rho|_{\Gamma_{0}},\rho_{0}|_{\Gamma_{0}})$-semi-conjugacy in $\operatorname{\mathcal{B}}_{\epsilon}(\operatorname{id})$ that we denote by $\phi_{\rho}$. Then there exists a small enough neighborhood $U_{0}$ of $\rho_{0}$ such that for any $\rho\in U_{0}$ such that $\rho|_{\Gamma_{0}}$ is semi-conjugate to $\rho_{0}|_{\Gamma_{0}}$, then $\phi_{\rho}$ is a $(\rho,\rho_{0})$-semi-conjugacy (i.e. for the action of the whole group $\Gamma$).

Proof.

Fix a finite generating set $\left\{a_{1}\dots,a_{n}\right\}$ of $\Gamma/\Gamma_{0}$. Fix an $\epsilon>0$ and let $U$ be a neighborhood satisfying the assumption. For $1\leq i\leq n$, define $\phi_{i}:=\rho_{0}(a_{i}^{-1})\circ\phi_{\rho}\circ\rho(a_{i})$. It sufficies to show that we can find a smaller open subset $U_{0}\subset U$ such that for $\rho\in U_{0}$ for which $\rho|_{\Gamma_{0}}$ is semiconjugate to $\rho_{0}|_{\Gamma_{0}}$ by $\phi_{\rho}$, we have $\phi_{i}=\phi_{\rho}$ for $1\leq i\leq n$.

Since $\Gamma_{0}$ is normal in $\Gamma$, it is easy to verify that $\phi_{i}$ is a $(\rho|_{\Gamma_{0}},\rho_{0}|_{\Gamma_{0}})$-semi-conjugacy. We claim that whenever $\rho$ is sufficiently close to $\rho_{0}$, then for all $i=1,\dots,n$ we have,

[TABLE]

Indeed, for the first term we can choose $U_{0}$ sufficiently small so that $\max_{1\leq i\leq n}d(\rho_{0}(a_{i}^{-1}),\rho(a_{i}^{-1}))<\epsilon/2$. The second $\frac{\epsilon}{2}$ estimate can be achieved by further shrinking $U_{0}$ as follows. We note that the conjugation action of $\operatorname{Homeo}(F)$ on itself is continuous. Hence there exists $\delta\geq 0$ such that

[TABLE]

for every $\varphi\in B_{\delta}(\operatorname{id})$. Now we let $U_{0}=U(\delta)$ be the neighborhood of $\rho_{0}$ obtained by applying the assumption to the constant $\delta$. It follows that $d(\rho(a_{i})^{-1}\phi_{\rho}\rho(a_{i}),\operatorname{id})<\frac{\epsilon}{2}$.

Thus $\phi_{i}\in\operatorname{\mathcal{B}}_{\epsilon}(\operatorname{id})$ is a $(\rho|_{\Gamma_{0}},\rho_{0}|_{\Gamma_{0}})$-semi-conjugacy. Then by the hypothesis on $U_{0}$, $\phi_{i}=\phi_{\rho}$. ∎

Corollary 4.7**.**

Suppose $\Gamma_{0}<\Gamma$ is a normal subgroup, $\Gamma/\Gamma_{0}$ is finitely generated and $\rho_{0}:\Gamma\to\operatorname{Homeo}(F)$ is such that $\rho_{0}|_{\Gamma_{0}}$ is a $(\lambda,\operatorname{\mathcal{U}})$-uniformly expanding action on $F$ for some open cover $\operatorname{\mathcal{U}}$. Suppose that for any $\epsilon>0$ there is a neighborhood $U=U(\epsilon)$ of $\rho_{0}$ such that if $\rho\in U$ and $\rho|_{\Gamma_{0}}$ is semi-conjugate to $\rho_{0}|_{\Gamma_{0}}$, then there is a $(\rho|_{\Gamma_{0}},\rho_{0}|_{\Gamma_{0}})$-semi-conjugacy in $\operatorname{\mathcal{B}}_{\epsilon}(\operatorname{id})$. Then there exists $\epsilon_{0}>0$ and a neighborhood $U_{0}=U(\epsilon_{0})$ of $\rho_{0}$ such that: if $\rho\in U_{0}$ and $\rho|_{\Gamma_{0}}$ is semi-conjugate to $\rho_{0}|_{\Gamma_{0}}$, then $\rho$ is semi-conjugate to $\rho_{0}$. Moreover, there is a $(\rho,\rho_{0})$-semi-conjugacy $\phi_{\rho}$ such that $d_{0}(\phi_{\rho},\operatorname{id})<\epsilon_{0}$.

Proof.

Let $\epsilon$ be the constant obtained by applying Lemma 4.5 to the action $\rho_{0}|_{\Gamma_{0}}$ and the open cover $\operatorname{\mathcal{U}}$. Set $\epsilon_{0}:=\epsilon/2$ and $U_{0}:=U(\epsilon_{0})$. Suppose $\rho\in U_{0}$ is such that $\rho|_{\Gamma_{0}}$ is semi-conjugate to $\rho_{0}|_{\Gamma_{0}}$. By assumption, there is a semi-conjugacy $\phi_{\rho}$ such that $d_{0}(\phi_{\rho},\operatorname{id})<\epsilon_{0}$. If $\phi^{\prime}$ is another such semi-conjugacy in $\operatorname{\mathcal{B}}_{\epsilon_{0}}(\operatorname{id})$, then Lemma 4.5 implies that $\phi^{\prime}=\phi_{\rho}$. So $\operatorname{\mathcal{B}}_{\epsilon_{0}}(\operatorname{id})$ contains exactly one semi-conjugacy. Then Lemma 4.6 implies that $\phi_{\rho}$ is a $(\rho,\rho_{0})$-semi-conjugacy. ∎

4.4. Upgrading semi-conjugacy to conjugacy

Let $\operatorname{BiLip}(F)$ be the space of bi-Lipschitz homeomorphisms of $F$. The Lipschitz distance between $f\in\operatorname{BiLip}(F)$ and $\operatorname{id}$ is defined by $d_{\operatorname{Lip}}(f,\operatorname{id}):=\sup_{x\in F}\dfrac{d_{F}(f(x),f(y))}{d_{F}(x,y)}$. We consider the distance function $d_{L}$ defined for $f,g\in\operatorname{BiLip}(F)$ by

[TABLE]

If $f$ is $\lambda$-Lipschitz, then $d_{\operatorname{Lip}}(g\circ f^{-1},\operatorname{id})<\epsilon$ implies that $g$ is $\lambda\epsilon$-Lipschitz. Thus, if $f\in\operatorname{BiLip}(F)$ is $\lambda$-biLipschitz, then every point in the $\epsilon$-ball around $f$ (in $d_{L}$ metric) is $(\lambda\epsilon)$-biLipschitz.

Suppose $\Gamma$ is a finitely generated group and fix a generating set $S$. We will say that two actions $\rho,\rho_{0}:\Gamma\to\operatorname{BiLip}(F)$ are $\epsilon$-close if $\max_{s\in S}d_{L}(\rho(s),\rho_{0}(s))<\epsilon$. The action $\rho_{0}$ is Lipschitz locally rigid if there exists $\epsilon>0$ such that any action $\rho:\Gamma\to\operatorname{BiLip}(F)$ that is $\epsilon$-close to $\rho_{0}$, is $C^{0}$-conjugate to $\rho_{0}$.

We now prove a very general result about upgrading an equivariant $C^{0}$ semi-conjugacy to a conjugacy when the actions have a uniform expansion property.

Proposition 4.8**.**

Suppose $\rho_{0}:\Gamma\to\operatorname{BiLip}(F)\subset\operatorname{Homeo}(F)$ is a $(\lambda,\operatorname{\mathcal{U}})$ uniformly expanding action. Then there exists $\epsilon_{0}>0$ and a neighborhood $V_{0}$ of $\rho_{0}$ in $\operatorname{BiLip}(F)$ such that: if $\rho\in V_{0}$ and there is a $(\rho,\rho_{0})$ semi-conjugacy $\phi_{\rho}$ in $\operatorname{\mathcal{B}}_{\epsilon_{0}}(\operatorname{id})$, then $\phi_{\rho}$ is a homeomorphism (i.e. a conjugacy).

Proof.

By compactness of $F$, we may extract a finite subcover $\left\{U_{i}\right\}_{i=1}^{k}$ of $\operatorname{\mathcal{U}}$ with corresponding expanding group elements $\gamma_{1},\dots,\gamma_{k}$. Let $\epsilon$ be the Lebesgue number of this finite cover (i.e. every ball of radius $\epsilon$ lies entirely in at least one $U_{i}$). Set $\epsilon_{0}:=\frac{\epsilon}{2}$.

Fix some $\lambda^{\prime}$ such that $\lambda>\lambda^{\prime}>1$. Consider the neighborhood of $\rho_{0}$ in given by

[TABLE]

Then for any $\rho\in V_{0}$, $\rho(\gamma_{i})$ is $\lambda^{\prime}$-expanding on $U_{i}$ for $1\leq i\leq k$.

We will show that the conclusion of this proposition is satisfied by this $V_{0}$ and $\epsilon_{0}$. Suppose $\rho\in V_{0}$ and $\phi_{\rho}$ is a $(\rho,\rho_{0})$ semiconjugacy in $\operatorname{\mathcal{B}}_{\epsilon_{0}}(\operatorname{id})$. In order to prove this result, it suffices to show that $\phi_{\rho}$ is bijective.

We first show that $\phi_{\rho}$ is injective. Note that if $d_{F}(x,y)\geq\epsilon$, then $\phi_{\rho}(x)\neq\phi_{\rho}(y)$. Indeed, if $\phi_{\rho}(x)=\phi_{\rho}(y)$, then

[TABLE]

where the last inequality comes from $d_{0}(\phi_{\rho},\operatorname{id})<\epsilon_{0}$.

Suppose there exist $\xi,\zeta\in F$ such that $\phi_{\rho}(\xi)=\phi_{\rho}(\zeta)$. Then $d_{F}(\xi,\zeta)<\epsilon$ and both points lie in some $U_{i}$. Suppose

[TABLE]

As $\phi_{\rho}(\xi)=\phi_{\rho}(\zeta)$,

[TABLE]

However this is impossible as

[TABLE]

Continuing inductively as in the proof of Lemma 4.5, we conclude that for any $n\geq 1$, $\phi_{\rho}(\xi)\neq\phi_{\rho}(\zeta)$ whenever $\frac{\epsilon}{(\lambda^{\prime})^{n}}<d_{F}(\xi,\zeta)<\frac{\epsilon}{(\lambda^{\prime})^{n-1}}$. As $\lambda^{\prime}>1$, this implies that $\phi_{\rho}$ is injective. ∎

Corollary 4.9**.**

Suppose $\rho_{0}:\Gamma\to\operatorname{BiLip}(F)$ is a $(\lambda,\operatorname{\mathcal{U}})$ uniformly expanding action. Further assume that for every $\epsilon>0$ sufficiently small, there exists a neighborhood $V=V(\epsilon)$ of $\rho_{0}$ in $\operatorname{BiLip}(F)$ such that: if $\rho\in V$, then there exists a $(\rho,\rho_{0})$ semi-conjugacy in $B_{\epsilon}(\operatorname{id})$. Then $\rho_{0}$ is Lipschitz locally rigid.

Proof.

Suppose $\epsilon_{0}$ and $V_{0}$ are as in the previous proposition. Let $0<\epsilon<\epsilon_{0}$ and $V=V(\epsilon)$ be as in the hypothesis. Then $V\cap V_{0}$ is a neighborhood of $\rho_{0}$ such that: whenever $\rho\in V\cap V_{0}$, $\rho$ is semi-conjugate to $\rho_{0}$. Indeed, the hypothesis implies the existence of a $(\rho,\rho_{0})$ semi-conjugacy in $\operatorname{\mathcal{B}}_{\epsilon}(\operatorname{id})\subset\operatorname{\mathcal{B}}_{\epsilon_{0}}(\operatorname{id})$. The previous proposition then implies that $\phi_{\rho}$ is a conjugacy. This finishes the proof. ∎

4.5. A Generalized Denjoy Construction

We remark that there are many actions which are semiconjugate but not conjugate. This had been understood by Bowden and Mann in their work on boundary actions on spheres - cf. Proposition 4.1 of [BM22] which generalizes the Denjoy construction ([Den32]). Using their main argument, we provide a simple generalization that works for any countable group acting on a manifold with a dense orbit.

We will first need the following Proposition 4.10. It is a variant of [BM22, Proposition 4.1 ] and is a consequence of [dS22, Proposition 2.2]. We will first recall [dS22, Proposition]. Let $M^{n}_{n-1}$ denote the Sierpinski space of dimension $n-1$, i.e. $M^{n}_{n-1}:=S^{n}-\cup_{i\in\operatorname{\mathbb{N}}}U_{i}$ where $\left\{U_{i}:i\in\operatorname{\mathbb{N}}\right\}$ is a family of open balls dense in $S^{n}$ and $\left\{i\in\operatorname{\mathbb{N}}:\operatorname{diam}(U_{i})<\epsilon\right\}$ is finite for any $\epsilon>0$. If $A\subset S^{n}$ is finite, let $M^{n}_{n-1}/\sim_{A}$ be obtained by collapsing each $\partial U_{i}$ to a single point for which $U_{i}$ does not intersect $A$. Proposition 2.2 of [dS22] states that $M^{n}_{n-1}/\sim_{A}$ is homeomorphic to $S^{n}$ minus a set of $\leavevmode\nobreak\ \#A$ disjoint open tame balls. Here an open ball $U\subset S^{n}$ is tame if both $\overline{U}$ and $S^{n}-U$ are homeomorphic to closed balls.

Observe that Proposition 2.2 of [dS22] applies equally well in all dimensions $n$, including dimension $n=4$. This is because the proof of [dS22, Proposition 2.2] follows from the Decomposition Theorem (see [dS22, Proposition 1.11]) which in turn follows from Quinn’s annulus theorem and both of these results work in any dimension. Hence our Proposition 4.10 also works in all dimensions including $n=4$.

Proposition 4.10**.**

Given a countable dense subset $Z$ of an open $n$-disk $B^{n}$ there exists a surjective, continuous map $h:\overline{B^{n}}\to\overline{B^{n}}$ that is identity on the boundary $\partial(B^{n})$, injective on the complement of $h^{-1}(Z)$, and such that the pre-image of each point in $Z$ is a closed ball.

Proof.

Let $B^{n}\cup\left\{z_{0}\right\}$ be the one-point compactification of $B^{n}$ and thus is homeomorphic to $S^{n}$. We also enumerate $Z=\left\{z_{1},z_{2},\dots\right\}$.

We first observe that the proof of Proposition 4.1 of [BM22] applies to any countable set, and not just an orbit. Then we apply this to the countable set $Z^{\prime}=Z\cup\left\{z_{0}\right\}\subset S^{n}$ to obtain a continuous surjective map $f:S^{n}\to S^{n}$ which is injective on the complement of $f^{-1}(Z^{\prime})$ and such that $\left\{C_{i}:=f^{-1}(z_{i}):z_{i}\in Z^{\prime}\right\}$ is a family of disjoint closed balls. As shown in the proof of [BM22, Proposition 4.1], the space $S^{n}-\bigcup_{z^{\prime}\in Z^{\prime}}{\rm int}\left({f}^{-1}(z^{\prime})\right)$ is a Sierpinski (or Menger) space of dimension $n-1$, denoted by $M^{n}_{n-1}$.

Let $A=\left\{z_{0}\right\}$ and apply [dS22, Proposition 2.2]; see the remarks before the statement of this proposition. Consider the quotient map $q_{A}:M^{n}_{n-1}\to M^{n}_{n-1}/\sim_{A}$ where $M^{n}_{n-1}/\sim_{A}$ is obtained by collapsing each $\partial C_{i}$ to a single point for $i\geq 1$. By the proposition, $M^{n}_{n-1}/\sim_{A}$ is homeomorphic to the complement of a tame open ball in $S^{n}$, and hence is a closed ball $\overline{B^{n}}$ and $q_{A}$ is injective on $M^{n}_{n-1}-(\cup_{i\geq 1}\partial C_{i})$. Consider the map $h^{\prime}:M^{n}_{n-1}\cup(\cup_{i\geq 1}C_{i})\to M^{n}_{n-1}/\sim_{A}$ such that $h^{\prime}|_{M^{n}_{n-1}}=q_{A}$ and $h^{\prime}|_{C_{i}}$ collapses $C_{i}$ to the point $q_{A}(\partial C_{i})$. Now observe that both $M^{n}_{n-1}\cup(\cup_{i\geq 1}C_{i})$ and $M^{n}_{n-1}/\sim_{A}$ are homemorphic to $\overline{B^{n}}$. Then, by abuse of notation, we can think of $h^{\prime}$ as a map from $\overline{B^{n}}$ to itself that is continuous, surjective, is injective on the complement of $h^{\prime-1}(Z)$ and $(h^{\prime})^{-1}(z)$ is a closed ball for any $z\in Z$.

Consider $h^{\prime}|_{\partial\overline{B^{n}}}$ which is a homeomorphism as $Z\cap\partial\overline{B^{n}}=\varnothing$. Extend $(h^{\prime}|_{\partial\overline{B^{n}}})^{-1}$ to a homeomorphism $h^{\prime\prime}$ of $\overline{B^{n}}$, e.g. by coning it off. Then let $h:=h^{\prime}\circ h^{\prime\prime}$. ∎

This allows us to give a very general construction of non-conjugate perturbations.

Construction 4.11**.**

Consider any $C^{1}$-action $\rho_{0}$ of a countable group $\Gamma$ on a compact smooth manifold $M$. If $\rho_{0}(\Gamma)$ has a dense orbit, then there exists a $C^{0}$ action $\rho$ of $\Gamma$ on $M$ that is not $C^{0}$ conjugate to $\rho_{0}$. Moreover, $\rho$ can be chosen to be arbitrarily $C^{0}$ close to $\rho_{0}$.

Proof.

Suppose that $\Gamma\cdot x$ is a dense orbit. We will then form a new $C^{0}$-action $\rho$ by first expanding a portion of the orbit $\Gamma\cdot x$ into a small disjoint union of closed balls $\bigcup_{\gamma\in\Gamma}\overline{B}(\gamma x)$. As we will explain below, the radii of the closed balls $\overline{B}(\gamma x)$ can be chosen to decay in such a way that the resulting manifold $M^{\prime}$ is still homeomorphic to $M$.

We start by fixing a Riemannian metric on $M$ and choosing one closed ball $\overline{B}(x_{0},r_{0})\subset M$ for some $0<r_{0}<{\rm injrad}(M)$ and with $x_{0}\in\Gamma\cdot x$ whose boundary misses the orbit $\Gamma\cdot x$. We can do this since there are an uncountable number of choices for the radius $r_{0}$, but only a countable number of orbit points. Proceeding inductively, having chosen disjoint balls $\overline{B}(x_{0},r_{0}),\overline{B}(x_{1},r_{1}),\dots,\overline{B}(x_{n},r_{n})$ we can choose a closed ball $\overline{B}(x_{n+1},r_{n+1})$ with $0<r_{n+1}<{\rm injrad}(M)$ contained in the open set $M\setminus\bigcup_{i=0}^{n}\overline{B}(x_{i},r_{i})$ centered at some $x_{n+1}\in\Gamma\cdot x$ whose boundary misses $\Gamma\cdot x$. By induction we can find a countable number of balls such that $M\setminus\bigcup_{i\geq 0}\overline{B}(x_{i},r_{i})$ has no interior and contains no orbit point. Now for each closed ball $\overline{B}(x_{i},r_{i})$ we apply Proposition 4.10 to obtain a surjective continuous map $h_{i}:\overline{B}_{i}\to\overline{B}(x_{i},r_{i})$ from a smooth closed ball $\overline{B}_{i}$ such that the pre-image of each orbit point $\gamma x\in\overline{B}(x_{i},r_{i})$ is a closed ball $\overline{B}(\gamma x)$ and is injective on the complement of these disks. As the $\overline{B}(x_{i},r_{i})$ are all disjoint and $h_{i}$ restricts to a homeomorphism on $\overline{B}_{i}$, these maps can then be combined together to an overall surjective continuous map $h:M^{\prime}\to M$ where $M^{\prime}$ is formed by removing each $\overline{B}(x_{i},r_{i})$ from $M$ and gluing in \overline{B}_{i}=h_{i}^{-1}(\overline{B}(x_{i}{\raisebox{-2.15277pt}{|}_{\partial\overline{B}_{i}}},r_{i})) using h_{i}{\raisebox{-2.15277pt}{|}_{\partial\overline{B}_{i}}}. The map $h$ is injective on the complement of $h^{-1}(\Gamma\cdot x)$. Clearly $M^{\prime}$ is homeomorphic to $M$ because the map $f^{\prime}:M^{\prime}\to M$ defined by $f^{\prime}|_{B_{i}}=h_{i}$ and $f^{\prime}|_{M^{\prime}-\cup_{i}B_{i}}=\operatorname{id}$ is a homeomorphism.

Since the action of $\rho_{0}$ is $C^{1}$, it is locally linear around each orbit point. We can extend each $C^{1}$ diffeomorphism $\rho_{0}(\gamma^{\prime})$ to $M^{\prime}-\bigcup_{\gamma\in\Gamma}\overline{B}(\gamma x)\cong(M-\Gamma\cdot x)\bigcup_{\gamma\in\Gamma}\partial\overline{B}(\gamma x)$ by taking for each $\gamma$ and $v\in\partial\overline{B}(\gamma x)\cong S^{n-1}$ the limit of $\rho_{0}(\gamma^{\prime})z^{v}_{i}$ along a sequence of points $z^{v}_{i}$ which approach $\gamma x$ radially (with respect to some $C^{1}$ chart) along the direction $v$. (Here we are assuming that the $\epsilon_{\gamma}$ are chosen sufficiently small.) This sequence will necessarily tangentially approach $\gamma^{\prime}\gamma x$ along a radial direction $w$ which we identify with a unique point in $\partial\overline{B}(\gamma^{\prime}\gamma x)$. This will produce a homeomorphism $\partial\overline{B}(\gamma x)\to\partial\overline{B}(\gamma^{\prime}\gamma x)$. Lastly, we extend this homeomorphism on $\partial\overline{B}(\gamma x)$ to all of $\overline{B}(\gamma x,\epsilon_{\gamma})$ by coning to $\gamma x$ to produce $\rho(\gamma^{\prime}):\overline{B}(\gamma x)\to\overline{B}(\gamma^{\prime}\gamma x)$. The compatibility of these homeomorphisms follows from the fact that $\rho_{0}$ is a $C^{1}$ action on $M$, and thus we obtain an action $\rho$ on $M\cong M^{\prime}$. (See the proof of Proposition 4.1 of [BM22] for details: While they consider only actions on $S^{n}$, their arguments apply in our generality.) The desired semiconjugacy collapses each $\overline{B}(\gamma x,\epsilon_{\gamma})$ to $\gamma x$. Note that $\rho$ cannot have a dense orbit by construction, and hence cannot be $C^{0}$-conjugate to $\rho_{0}$. ∎

5. Preliminaries III: Symmetric Spaces and Furstenberg Boundaries

Here we review some well-known facts about symmetric spaces and their various boundaries. These can be found in [Ebe96, Section 2.7].

Let $X$ be a symmetric space of non-compact type without Euclidean factors and $\mathrm{Isom}(X)^{0}=G$ a semisimple Lie group. For any point $p\in X$, $K:={\rm Stab}_{G}(p)$ is a maximal compact subgroup of $G$. The orbit of the point $p$ induces a natural diffeomorphism of $X$ with $G/K$. Moreover, $X$ carries a left $G$-invariant metric of non-positive curvature. We denote by $\partial X$ the (smooth) geodesic boundary of $X$ with respect to any such metric. The diffeomorphism classes of $X$ and $\partial X$ are independent of the choice of such metric.

Definition 5.1**.**

A subgroup $Q<G$ is called a parabolic subgroup if there exists $\mathfrak{X}\in\partial X$ such that $Q=G_{\mathfrak{X}}:=\{g\in G:g\cdot\mathfrak{X}=\mathfrak{X}\}$.

We note that $G_{\mathfrak{X}}$ acts transitively on the symmetric space $X$.

Fix $o\in X$ and $\mathfrak{X}\in\partial X$. Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition of the Lie algebra $\mathfrak{g}$ determined by $o$. We define the (real) rank of $G$, $\operatorname{rk}(G)$, to be the dimension of any maximal abelian subalgebra of $\mathfrak{p}$. We fix a maximal abelian subalgebra of $\mathfrak{p}$ and label it $\mathfrak{a}$.

Note that $\mathfrak{p}$ can be identified with the tangent space to $X$ at $o$. Then let $v\in\mathfrak{p}$ be given by $v:=\gamma_{o\mathfrak{X}}^{\prime}(0)$ where $\gamma_{o\mathfrak{X}}$ is the geodesic ray from $o$ to $\mathfrak{X}$. Consider the homomorphism $T_{\mathfrak{X}}:G_{\mathfrak{X}}\to G$ defined by

[TABLE]

where this limit exists by [Ebe96, Proposition 2.17.3].

Let $Z(v):=\{w\in\mathfrak{g}:[v,w]=0\}$ be the centralizer of $v$ in the Lie algebra $\mathfrak{g}$. We then define the following subsets:

[TABLE]

We note that $Z_{\mathfrak{X}}$, $M_{\mathfrak{X}}$, and $N_{\mathfrak{X}}$ are subgroups of $G$, while $A_{\mathfrak{X}}$ is generally only a subset (unless $Z(v)\subset\mathfrak{p}$, i.e. $v$ is a regular vector). Setting $Q=G_{\mathfrak{X}}$, we will also use the notation $Z_{Q},M_{Q},N_{Q},A_{Q}$ to denote $Z_{\mathfrak{X}},M_{\mathfrak{X}},N_{\mathfrak{X}},A_{\mathfrak{X}}$ respectively.

In this section, we list some facts about the geometry associated to a parabolic subgroup $Q=G_{\mathfrak{X}}$.

5.1. Generalized Iwasawa Decomposition

From [Ebe96, Proposition 2.17.5(5)], we have the Generalized Iwasawa decomposition

[TABLE]

The indicated decomposition of elements of $G$ is unique.

By [Ebe96, Proposition 2.17.5(4)], there is an analogous decomposition of $Q$:

[TABLE]

Again, the indicated decomposition of elements of $Q$ is unique.

5.2. Parallel Sets

The reference for this subsection is [Ebe96, Pg 138]. By slight abuse of notation, let $\gamma_{o\mathfrak{X}}$ also denote the bi-infinite geodesic extension of the geodesic ray from $o$ to $\mathfrak{X}$.

The parallel set of $\gamma_{o\mathfrak{X}}$, denoted by $F(\gamma_{o\mathfrak{X}})$, is the union of all bi-infinite geodesics parallel to $\gamma_{o\mathfrak{X}}$. We will also use the notation $F(v)$ for the union of all geodesics parallel to the geodesic with initial vector $v$. In particular, $F(\gamma_{o\mathfrak{X}})=F(\gamma_{o\mathfrak{X}}^{\prime}(0))$. The set $F(\gamma_{o\mathfrak{X}})$ is a totally geodesic submanifold of $X$ and

[TABLE]

In fact, we also have $F(\gamma_{o\mathfrak{X}})=Z_{\mathfrak{X}}\cdot o.$

Let $E(\gamma_{o\mathfrak{X}})$ be the intersection of all maximal flats of $X$ that contain $\gamma_{o\mathfrak{X}}$. Define

[TABLE]

Then

[TABLE]

and $E(\gamma_{o\mathfrak{X}})$ a totally geodesic submanifold that is isometric to $\mathbb{R}^{d}$, where $1\leq d=\dim(E_{\mathfrak{X}})\leq{\rm rk}(G)$.

The set $F(\gamma_{o\mathfrak{X}})$ isometrically splits as $\mathbb{R}^{d}\times CS(\gamma_{o\mathfrak{X}})$ where $d=\dim E_{\mathfrak{X}}$ and $CS(\gamma_{o\mathfrak{X}})$ is a symmetric space of rank $({\rm rk}(G)-d)$ [Ebe96, Lemma 2.20.9].

The parallel sets produce a totally geodesic foliation of the symmetric space with isometric leaves. More precisely, $X$ is foliated by totally geodesic subspaces $\left\{F(\gamma_{q\mathfrak{X}}):q\in X\right\}$. The leaf space of this foliation can be naturally identified with $N_{\mathfrak{X}}$ (since $F(\gamma_{o\mathfrak{X}})=Z_{\mathfrak{X}}\cdot o$ and $G_{\mathfrak{X}}=N_{\mathfrak{X}}Z_{\mathfrak{X}}$). Given $o$ and $q$ in $X$, there is a unique isometry $H_{o,q}:F(\gamma_{o\mathfrak{X}})\to F(\gamma_{q\mathfrak{X}})$ such that for any sequence $(o_{n})$ of points in $F(\gamma_{o\mathfrak{X}})$ that tends to $\mathfrak{X}$, the distance $d(o_{n},H_{o,q}(o_{n}))$ tends to 0.

Using the above foliation, we can define a retraction map $retr:X\to F(\gamma_{o\mathfrak{X}})$ such that

[TABLE]

for every $q\in X$. This retraction map is distance non-expanding. Furthermore, $retr^{-1}(o)=N_{\mathfrak{X}}\cdot o$. In particular, the symmetric space $X$ is diffeomorphic to the product $F(\gamma_{o\mathfrak{X}})N_{\mathfrak{X}}$.

5.3. Weyl chamber $Q$-faces at infinity

The Weyl chamber face at infinity associated to $\mathfrak{X}$ is defined as $\mathcal{C}(\mathfrak{X})=\{\Upsilon\in\partial_{\infty}X:G_{\Upsilon}^{0}=G_{\mathfrak{X}}^{0}\}$.

We will say that two Weyl chamber $Q$-faces at infinity, $\mathcal{C}(\mathfrak{X})$ and $\mathcal{C}(\Upsilon)$, are of same type if $\mathcal{C}(\Upsilon)=g\mathcal{C}(\mathfrak{X})$ for some $g\in G$. Then $G$ acts transitively on the set of Weyl chamber $Q$-faces at infinity of the same type with the stabilizer of $\mathcal{C}(\mathfrak{X})$ being equal to $G_{\mathfrak{X}}=Q$. Thus the set of all Weyl chambers $Q$-faces at infinity of the same type as $\operatorname{\mathcal{C}}(\mathfrak{X})$ can be identified with $G/Q$ and inherits the structure of a smooth manifold. Moreover, $G/Q$ is diffeomorphic with $K/M_{Q}$ by the generalized Iwasawa decomposition. Using this identification, we may write the Weyl chamber $Q$-face at infinity $\operatorname{\mathcal{C}}(\mathfrak{X})$ as simply $\xi$ whenever $Q_{\mathfrak{X}}$ is the stabilizer of $\xi$ for some $\xi\in G/Q$.

5.4. Weyl chamber $Q$-faces based at a point

The Weyl chamber face associated to $\mathfrak{X}$ based at $o$ is defined as $\mathcal{C}_{o}(\mathfrak{X})=\exp\{tY:t\geq 0,Y(o)=\gamma_{o\Upsilon}^{\prime}(0),\Upsilon\in\mathcal{C}(\mathfrak{X})\}\cdot o$. Observe that $\mathcal{C}_{o}(\mathfrak{X})$ consists of the union of geodesic rays starting at $o$ which point to the corresponding Weyl chamber $Q$-face at infinity $\mathcal{C}(\mathfrak{X})$. The stabilizer of $\mathcal{\operatorname{\mathcal{C}}}_{o}(\mathfrak{X})$ in $G$ is $M_{\mathfrak{X}}=M_{Q}$.

Given a Weyl chamber $Q$-face at infinity $\xi\in G/Q$, we let $\operatorname{\mathcal{C}}_{o}(\xi)$ be the unique Weyl chamber $Q$-face base at $o$ that belongs to the equivalent class $\xi$. Note that if $\mathfrak{X}\in\partial_{\infty}X$ is such that $Q=G_{\mathfrak{X}}$, then $\operatorname{\mathcal{C}}_{o}(\xi)=\operatorname{\mathcal{C}}_{o}(\mathfrak{X})$.

5.5. Opposite Weyl chamber $Q$-faces

Given a Weyl chamber $Q$-face based at $o$, say $\operatorname{\mathcal{C}}_{o}(\xi)$, we let $\operatorname{\mathcal{C}}_{o}(\xi^{*})$ denote the opposite Weyl chamber $Q$-face based at $o$. Explicitly this face is given by $\operatorname{\mathcal{C}}_{o}(\xi^{*})=\exp\{tY:t\geq 0,Y=-\gamma_{o\Upsilon}^{\prime}(0),\Upsilon\in\mathcal{C}(\xi)\}\cdot o$.

Similarly, we will say that a Weyl chamber $Q$-face at infinity $\xi\in G/Q$ is an opposite Weyl chamber $Q$-face to $\eta\in G/Q$ if there exists a point $o$ for which $\operatorname{\mathcal{C}}_{o}(\xi)$ is opposite to $\operatorname{\mathcal{C}}_{o}(\eta)$.

5.6. The Weyl chamber $Q$-face bundle $G/M_{Q}$

Suppose $Q$ is a parabolic subgroup with compact subgroup $M_{Q}$ as in Section 5. The Weyl chamber $Q$-face bundle on $X$ is the is the space of Weyl chamber faces associated to $Q$ based at the (any) point $o\in X$. Observe that $G$ acts transitively on this space. In other words, it can be identified as $G\cdot\mathcal{C}_{o}(\mathfrak{X})\cong G/M_{Q}$.

We spend the rest of this section discussing the geometry of the space

[TABLE]

The natural map $p_{Q}:G/M_{Q}\to G/K$ gives $G/M_{Q}$ the structure of a smooth fiber bundle over $G/K$ with fiber $K/M_{Q}$. The fiber over the point $gK\in G/K$ is given by $F_{gK}=gK/M_{Q}$, and is identified with the space of all Weyl chamber $Q$-faces based at $gK$. We now explain the map $\pi_{gK}:F_{gK}\to K/M_{Q}$ that identifies each fiber with $K/M_{Q}$. For every $g\in G$, there is a unique decomposition $g=n_{Q}a_{Q}k$ coming from the generalized Iwasawa decomposition $G\cong N_{Q}A_{Q}K$; see Section 5.1. Then define the map $\pi_{gK}:gK/M_{Q}\to K/M_{Q}$ by

[TABLE]

Finally, the fiber bundle is globally trivial since $G/K$ is contractible. But the generalized Iwasawa decomposition (Section 5.1) gives an explicit description of a global trivialization. Notice that there is a natural smooth bundle isomorphism (and diffeomorphism) $\Phi:G/M_{Q}\to G/K\times K/M_{Q}\cong G/K\times G/Q$ arising from the smoothness of the (reverse) Iwasawa decomposition $G\cong N_{Q}A_{Q}K$ and the resulting diffeomorphism $G/K\cong N_{Q}A_{Q}$. In particular, if $g=n_{Q}a_{Q}k=k^{\prime}a^{\prime}_{Q}n^{\prime}_{Q}$ by the generalized Iwasawa decomposition, then

[TABLE]

Note that the map $g\mapsto k^{\prime}M_{Q}$ coming from the generalized Iwasawa decomposition is well-defined. This is because $M_{Q}A_{Q}N_{Q}=A_{Q}N_{Q}M_{Q}$, see Eq. 5.2. Hence the map $\Phi$ is well-defined. Further $\Phi$ is clearly a smooth function. Also observe that this definition of $\Phi$ agrees with the earlier definition Eq. 3.1 in Lemma 3.7.

5.7. Riemannian metric on $G/M_{Q}$

Fixing a point $o=eK\in X$, let us consider the corresponding Cartan decomposition

[TABLE]

where $\mathfrak{k}=T_{e}K$. Let us choose a maximal abelian subalgebra $\mathfrak{a}$ in $\mathfrak{p}$. Fix some choice of positive roots $\Lambda^{+}$ for $\mathfrak{a}$ and let $\Lambda^{++}$ denote the positive simple roots. Then fix a positive Weyl chamber $\mathfrak{a}^{+}\subset\mathfrak{p}$ (with respect to $\Lambda^{++}$) based at $o$.

Suppose $\alpha$ is any left $G$-invariant and $K$-bi-invariant metric on $G$. This induces a left $G$-invariant metric on $G/M_{Q}$. We will always equip $G/M_{Q}$ with such a metric.

We remark that for any $\alpha$ as above, $\mathfrak{p}=(T_{e}K)^{\perp}$. Indeed, $\alpha$ is an $\operatorname{Ad}(K)$ invariant metric on $\mathfrak{g}$ and $\operatorname{Ad}(K)$ preserves the Cartan decomposition. Moreover, there exists $k_{0}\in K$ such that $\operatorname{Ad}(k_{0})(\mathfrak{a}^{+})=-\mathfrak{a}^{+}$ (i.e. the opposition involution). Let $v\in\mathfrak{a}^{+}$ be the barycenter of the unit vectors in $\mathfrak{a}^{+}$. Then $\operatorname{Ad}(k_{0})v=-v$. Thus, for any $v_{1}\in\mathfrak{k}$,

[TABLE]

which implies that $v\in\mathfrak{k}^{\perp}$. Then, $\operatorname{Ad}(K)$-invariance of $\alpha$ implies that $\operatorname{Ad}(K)v\subset\mathfrak{k}^{\perp}$. Finally $\operatorname{Ad}(K)v$ spans $\mathfrak{p}$. The comparison of dimensions of $\mathfrak{p}$ and $\mathfrak{k}$ proves that $\mathfrak{p}=\mathfrak{k}^{\perp}$.

Furthermore, we also observe that such a Riemannian metric $\alpha$ also makes all simple factors of $T_{e}G$ orthogonal to each other. In fact, a Riemannian metric $\alpha$ as above is unique up to scaling on each simple component of the Lie algebra (see Section 2.3 of [Ebe96]).

Lemma 5.2**.**

Let $Q$ be any parabolic subgroup, and suppose $\alpha$ is a Riemannian metric on $G/M_{Q}$ as above. Setting $\operatorname{\mathcal{E}}_{g}=gA_{Q}N_{Q}\cdot M_{Q}\subset G/M_{Q}$, then:

1. (1)

each fiber $F_{g^{\prime}}$ is orthogonal to $\operatorname{\mathcal{E}}_{g}$, and 2. (2)

$\operatorname{\mathcal{E}}_{g}$* is isometric to the Riemannian globally symmetric space $(G/K,(p_{Q})_{*}\alpha)$.* 3. (3)

each fiber $F_{g}$ is totally geodesic.

Proof.

By left $G$-invariance and right $K$-invariance of $\alpha$, $dp_{Q}\alpha$ is a left $G$-invariant Riemannian metric on $G/K$, hence it endows $G/K$ with the structure of a Riemannian globally symmetric space. (Indeed, by Ad(K) invariance on $T_{e}(G/K)$, the metric $dp_{Q}\alpha$ differs from a fixed Riemannian symmetric metric only by re-scaling corresponding to the simple factors of $G$). Recall that $\operatorname{\mathcal{E}}_{g}$ is the image of a section of the bundle $p_{Q}:G/M_{Q}\to G/K$. Then left multiplication by $g^{-1}$ is an isometry between $\operatorname{\mathcal{E}}_{g}$ and $\operatorname{\mathcal{E}}_{e}:=A_{Q}N_{Q}\cdot M_{Q}$. By left $G$-invariance of $\alpha$, it is enough to show that:

• •

$T_{eM_{Q}}(K/M_{Q})$ is orthogonal to $T_{e}\operatorname{\mathcal{E}}_{e}$,

• •

$dp_{Q}:T_{eM_{Q}}(G/M_{Q})\to T_{eK}(G/K)$ is a Riemannian submersion,

• •

the restriction $dp_{Q}|_{T_{e}\operatorname{\mathcal{E}}_{e}}:T_{e}\operatorname{\mathcal{E}}_{e}\to T_{e}(G/K)$ is a Riemannian isometry.

Note that $T_{e}\operatorname{\mathcal{E}}_{e}$ is canonically identified with $\mathfrak{p}=T_{e}(G/K)$, where $\mathfrak{p}$ is as in Eq. 5.3. By definition of the metric $\alpha$, $(T_{eM_{Q}}(K/M_{Q}))^{\perp}$ is canonically identified with $T_{e}(G/K)$. This proves the first part. Next note that $dp_{Q}$ is clearly a submersion. Moreover,

[TABLE]

which is equal to $T_{e}\operatorname{\mathcal{E}}_{e}=T_{e}(G/K)$ and $(p_{Q})_{*}\alpha$ is an isometry between $\ker(dp_{Q})^{\perp}$ and $T_{e}(G/K)$. This proves both the first and the second parts.

For proving the third part, it is enough to show that $K$ is totally geodesic in $G$. But this is clear since the subgroup $K$ is invariant under the Cartan involution on $G$ induced by the Riemannian geodesic symmetry of $G/K$ at $eK$. ∎

5.8. Weyl Chamber flow on $G/M_{Q}$

Let $A$ be the connected subgroup of $G$ whose Lie algebra is $\mathfrak{a}$. Consider the group ${A^{\prime}_{Q}}<A$, the subgroup of $A$ consisting of elements commuting with $M_{Q}$. There is a natural action of ${A^{\prime}_{Q}}$ on $G/M_{Q}$ defined by right multiplication, i.e. $gM_{Q}\mapsto gaM_{Q}$ for every $a\in{A^{\prime}_{Q}}$ and $g\in G$. It is well-defined because ${A^{\prime}_{Q}}$ commutes with $M_{Q}$. This action is called the Weyl chamber flow associated to the parabolic subgroup $Q$.

Recall that $\mathfrak{a}^{+}$ is the positive Weyl chamber with respect to a choice of positive simple roots, see Section 5.7. We can intersect the Lie algebra of $A_{Q}^{\prime}$ with $\mathfrak{a}^{+}$ to obtain the positive Weyl chamber face $\mathfrak{a}^{+}_{Q}$. We define the positive Weyl chamber flow on $G/M_{Q}$ to be the restricted action of ${A^{\prime}_{Q}}^{+}:=\exp(\mathfrak{a}^{+}_{Q})\subset A_{Q}^{\prime}$.

5.9. Foliations on $G/M_{Q}$

Recall the notion of Weyl chamber $Q$-faces (based at a point and at infinity) as well as the notion of opposite Weyl chambers from Sections 5.3, 5.4 and 5.5.

Center stable and center unstable foliations

We will call two based Weyl chamber $Q$-faces, $\operatorname{\mathcal{C}}_{p}(\xi)$ and $\operatorname{\mathcal{C}}_{q}(\eta)$, asymptotic if their Hausdorff distance is bounded, or equivalently, if $\xi=\eta$. Then, an equivalence class of asymptotic based Weyl chamber $Q$-faces is a Weyl chamber $Q$-face at infinity.

For $\xi\in G/Q$, we define the center stable manifold in $G/M_{Q}$ as

[TABLE]

It is the set of all based Weyl chamber $Q$-faces asymptotic to $\operatorname{\mathcal{C}}_{o}(\xi)$. In particular, is a smooth manifold as it is the orbit of a closed subgroup of $G$, by definition. This coincides with the dynamical definition that the center stable manifold of a point $gM_{Q}$ in $G/M_{Q}$ consists of all points that maintain a bounded distance under the positive Weyl chamber flow to the positive Weyl chamber given by the $A_{Q}^{{}^{\prime}+}$ orbit of $gM_{Q}$.

The center unstable manifold is defined as:

[TABLE]

where $\xi^{*}$ is the Weyl chamber $Q$-face at infinity that is opposite to $\xi$; see Section 5.5.

Then $\left\{\operatorname{\mathcal{W}^{cs}}(\xi):\xi\in G/Q\right\}$ and $\left\{\operatorname{\mathcal{W}^{cu}}(\xi):\xi\in G/Q\right\}$ are two different continuous foliations of $G/M_{Q}$ with smooth leaves. The leaves of these foliations intersect along orbits of the Weyl chamber flow. Indeed, $\operatorname{\mathcal{W}^{cs}}(\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta)=\left\{\operatorname{\mathcal{C}}_{p}(\xi):p\in X,\operatorname{\mathcal{C}}_{p}(\xi^{*})\in\operatorname{\mathcal{W}^{cs}}(\eta^{*})\right\}$ is the set of Weyl chamber $Q$-faces that are forward asymptotic to $\xi$ and backward asymptotic to $\eta$. Under the map $p_{Q}:G/M_{Q}\to G/K$, this set is diffeomorphic to $F(\gamma)$, where $\gamma$ is a (any) geodesic with end-points $\gamma(+\infty)\in\xi$ and $\gamma(-\infty)\in\eta$. Thus the intersection is either empty or a smooth submanifold homeomorphic to $G_{Q}/K_{Q}\cong A_{Q}N_{Q}$; see Section 5.2.

Let $\Phi:G/M_{Q}\to G/K\times G/Q$ be the map defined as above in Section 5.6. Then $\Phi^{-1}(p,\xi)=\operatorname{\mathcal{C}}_{p}(\xi)$ and $\Phi^{-1}(X\times\xi)=\operatorname{\mathcal{W}^{cs}}(\xi)$ for any $\xi\in G/Q$.

Finer foliations

Each leaf $\operatorname{\mathcal{W}^{cs}}(\xi)$ admits finer foliations. Indeed, there is a natural homeomorphism

[TABLE]

where $F(\gamma_{p\mathfrak{X}})$ is identified with a center leaf in $\operatorname{\mathcal{W}^{cs}}(\xi)$, and the unipotent subgroup $N_{\mathfrak{X}}$ is identified with the strong unstable leaf of $(x,\xi)$. The foliation of $\operatorname{\mathcal{W}^{cs}}(\xi)$ by $\{F(\gamma_{q\mathfrak{X}}):q\in X\}$ is perpendicular to the foliation by orbits of $N_{\mathfrak{X}}$ with respect to the metric defined earlier on $G/M_{Q}$. (This follows from the fact that $K,A_{\mathfrak{X}}$ and $N_{\mathfrak{X}}$ are all perpendicular with respect to $\alpha$.) A similar foliation can be defined for each leaf of $\operatorname{\mathcal{W}^{cu}}(\xi)$.

5.10. Example

Consider the symmetric space $X=\operatorname{\mathbb{H}}^{2}\times\operatorname{\mathbb{H}}^{2}$ and let $G=\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})\times\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})$ be the connected component of the isometry group of $X$.

We consider the following standard subgroups of $\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})$: $A_{*}=\left\{\begin{bmatrix}e^{t}&0\\ 0&e^{-t}\end{bmatrix}:t\in\operatorname{\mathbb{R}}\right\}$, $N_{*}=\left\{\begin{bmatrix}1&s\\ 0&1\end{bmatrix}:s\in\operatorname{\mathbb{R}}\right\}$, $M_{*}=\left\{e\right\}$, and $P_{*}=M_{*}A_{*}N_{*}=\left\{\begin{bmatrix}e^{t}&s\\ 0&e^{-t}\end{bmatrix}:s,t\in\operatorname{\mathbb{R}}\right\}$.

Let $K\cong\mathrm{SO}(2)\times\mathrm{SO}(2)$ be the maximal compact subgroup of $G$ so that $X\cong G/K$. Consider the geodesic $\gamma(t)=(p,a_{t}\cdot p)$ where $p=eK_{0}\in\operatorname{\mathbb{H}}^{2}$ is a fixed base-point. Then $\gamma|_{[0,\infty)}$ is a singular geodesic ray and the parabolic subgroup $Q=\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})\times P_{2}$ is the stabilizer of $\mathfrak{X}=\gamma(\infty)$. Then

[TABLE]

Moreover $G/Q\cong\left\{[e]\right\}\times\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})/P_{*}\cong\left\{\mathfrak{X}\right\}\times\partial_{\infty}\operatorname{\mathbb{H}}^{2}\subset\partial_{\infty}\operatorname{\mathbb{H}}^{2}\times\partial_{\infty}\operatorname{\mathbb{H}}^{2}\subset\partial_{\infty}X$. Then the Weyl chamber bundle $G/M_{Q}\cong\operatorname{\mathbb{H}}^{2}\times\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})$, which is a smooth $S^{1}$-bundle over the symmetric space $X=\operatorname{\mathbb{H}}^{2}\times\operatorname{\mathbb{H}}^{2}$. The ${A^{\prime}_{Q}}$ action on $G/M_{Q}$ is trivial on the first factor $\operatorname{\mathbb{H}}^{2}$ and acts by right multiplication on the second factor $\operatorname{PSL}_{2}(\operatorname{\mathbb{R}})$. Finally, note that for the above geodesic ray, $\gamma$, we have $F(\gamma)\cong\operatorname{\mathbb{H}}^{2}\times\operatorname{\mathbb{R}}$.

5.11. Example

Consider the symmetric space $X=\mathrm{SL}_{3}(\operatorname{\mathbb{R}})/\mathrm{SO}(3)$ and let $G=\mathrm{SL}_{3}(\operatorname{\mathbb{R}})$. Consider the subgroups

[TABLE]

and P=MAN. Then $P$ is up to conjugacy the unique minimal parabolic, and there are two distinct conjugacy classes of proper non-minimal parabolics, given by $Q_{-}=\begin{pmatrix}*&*&*\\ *&*&*\\ 0&0&*\end{pmatrix}$ and $Q_{+}:=\begin{pmatrix}*&*&*\\ 0&*&*\\ 0&*&*\end{pmatrix}$ respectively. Note that the corresponding Furstenberg boundaries correspond to partial flag varieties, namely $G/Q_{-}\cong{\rm Gr}_{2}(\operatorname{\mathbb{R}}^{3})$ and $G/Q_{+}\cong\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{3})$. Observe that these two boundaries are diffeomorphic via the map from ${\rm Gr}_{2}(\operatorname{\mathbb{R}}^{3})$ to $\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{3})$ which takes a 2-plane $V$ to the equivalence class $[V^{\perp}]$. However, these two boundaries cannot be equivariantly diffeomorphic.

Consider geodesic $\gamma(t)=\begin{pmatrix}e^{t}&0&0\\ 0&e^{t}&0\\ 0&0&e^{-2t}\end{pmatrix}\cdot K$. This is a unit speed geodesic for a suitable choice of symmetric metric on $X$ (all of which differ only by scale). Then $\mathfrak{X}=\gamma(\infty)\in\partial_{\infty}X$ is stabilized by $Q$.

The subgroups are:

[TABLE]

Note that $A_{Q_{-}}$ consists of all matrices in $Z_{Q_{-}}$ that are symmetric and positive definite.

Then $G/Q_{-}\cong K/M_{Q_{-}}\cong SO(3)/O(2)$ is diffeomorphic to the quotient of $S^{2}$ by a degree two map, i.e. a copy of the two dimensional real projective space. This matches with our previous remark that $G/Q_{-}$, which is canonically isomorphic to ${\rm Gr}_{2}(\operatorname{\mathbb{R}}^{3})$ can be identified with $\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{3*})$.

Moreover, $G/M_{Q_{-}}=PK/M_{Q_{-}}\cong P\times{\rm Gr_{2}}(\operatorname{\mathbb{R}}^{3})$ has the structure of a smooth ${\rm Gr}_{2}(\operatorname{\mathbb{R}}^{3})$ bundle over $P\cong X$.

The above subgroups and spaces can be analogously determined for $Q_{+}$ and they are related by the duality between $\operatorname{\mathbb{R}}^{3}$ and $\operatorname{\mathbb{R}}^{3*}$.

More generally, consider the symmetric space $X=\mathrm{SL}_{d}(\operatorname{\mathbb{R}})/\mathrm{SO}(d)$. The set of positive simple roots is given by $\Delta=\left\{e_{i}^{*}-e_{i+1}^{*}:1\leq i\leq d-1\right\}$. For any $\Theta\subset\Delta$, we consider the Weyl chamber face $\mathfrak{a}_{\Theta}^{+}$ in the canonical flat $A$ given by the positive cone on the root vectors corresponding to the roots in $\Theta$. (This will be a Euclidean \left\lvert\vbox{\hbox{\displaystyle\Theta}}\right\rvert-simplex.) We can express the exponential action of $\mathfrak{a}^{+}_{\Theta}$ on $p\in X$ as $\mathcal{C}_{p}(\mathfrak{X})$ for some $\mathfrak{X}\in\partial_{\infty}X$. Define $Q=P_{\Theta}$ as the stabilizer in $G$ of $\mathfrak{X}$. (Any parabolic subgroup is conjugate to some such $P_{\Theta}$.) Then $G/P_{\Theta}$ is the partial flag variety consisting of the $G$-orbit of $\exp(\mathfrak{a}_{\Theta}^{+})$. In terms of matrices, the groups $P_{\Theta}$ will be conjugate to the set of matrices that vanish below the diagonal in the coordinates corresponding to $\Theta$. Then $M_{Q}$ will be the group isomorphic to $SO(q)$ for q=d-\left\lvert\vbox{\hbox{\displaystyle\Theta}}\right\rvert which acts by identity in the coordinates corresponding to $\Theta$. The group $A_{Q}$ will consist of the matrices in $P_{\Theta}$ that are symmetric and $A_{Q}^{\prime}$ will be the diagonal matrices that are $1$ off of the $\Theta$ coordinates.

6. Coarse Geometry of Symmetric Spaces

Let us fix a Riemannian symmetric space $X$ and let $G=\mathrm{Isom}(X)^{0}$.

We will call any isometric totally geodesic embedding of $\operatorname{\mathbb{R}}^{\operatorname{rk}(G)}$ into $X$ a (maximal) flat. The intersection of two flats is called a (singular) subflat. A half-flat $H$ is a subset of a flat such that $\partial H$ is a subflat of dimension $\operatorname{rk}(G)-1$. A subset of a flat is called a polytope if it is the intersection of finitely many half-flats.

We also define the parallel set of a singular subflat (compare with Section 5.2). Given a singular subflat $S$, the parallel set of the singular subflat $S$ is

[TABLE]

We call any $(L,C)$-quasi-isometric embedding of $\mathbb{R}^{\operatorname{rk}(G)}$ into $X$ an $(L,C)$-quasiflat. In particular, quasiflats are always assumed to be of dimension $\operatorname{rk}(G)$ unless stated otherwise.

Lemma 6.1**.**

[KL97, Theorem 1.2.5]**[EF97, Theorem 1.1]** For every $L\geq 1$ and $C\geq 0$ there exist $k\in\mathbb{N}$ and $R\geq 0$ such that if $F$ an $(L,C)$-quasiflat then $F$ is contained in an $R$-neighborhood of a union of $k$ flats in $X$. Moreover, if $L$ is sufficiently close to 1 (the ‘sufficiently close’ depending on type of the root system of $G$), then $k=1$.

Proof.

We refer to [KL97, Theorem 1.2.5] and the paragraph after it. ∎

Definition 6.2**.**

Let $A,B\subset X$. We say $A$ and $B$ coarsely intersect in a set $D$ if there is a constant $R=R(A,B)$ such that $d^{\operatorname{Haus}}(\operatorname{Nbhd}_{R}(A)\cap\operatorname{Nbhd}_{R}(B),D)<+\infty$. In this case, we say that $D$ is a coarse intersection of $A$ and $B$.

Note that the notion of coarse intersection is well-defined only up to finite Hausdorff distance. Hence we use the term ‘a coarse intersection’.

Fact: The coarse intersection of two flats in a symmetric space is a convex polytope, possibly unbounded; see for example [Esk98, Appendix B, Lemma B.1].

Let $\gamma_{pz}$ be a geodesic ray in $X$. Recall that $F(\gamma_{pz})$ is the parallel set of $\gamma_{pz}$, see Section 5.2. Let

[TABLE]

Note that $F*$ is in fact the unique minimal singular subflat that contains the geodesic extension of the ray $\gamma_{pz}$. Here, minimal means there is no singular subflat of strictly lower dimension that contains $\gamma_{pz}$. The symmetric subspace $F(\gamma_{pz})$ is then $F(F*)$, the parallel set of $F*$.

Lemma 6.3**.**

Let $q^{*}:F(\gamma_{pz})\to X$ be a $L$-biLipschitz embedding and let $F*$ be the intersection of all flats in $F(\gamma_{pz})$. Suppose there is a constant $C$ such that if $F$ is a flat in $F(\gamma_{pz})$, then there exists a flat $F^{\prime}$ in $X$ such that

[TABLE]

1. (1)

Let $F_{1}$ and $F_{2}$ be two flats such that $F_{1}\cap F_{2}=F*$. Then the coarse intersection of $F_{1}^{\prime}$ and $F_{2}^{\prime}$, which we denote by $\operatorname{\mathcal{S}}(F_{1}^{\prime},F_{2}^{\prime})$, is a singular subflat of dimension $\dim(F*)$. Furthermore, if the angle between $F_{1}$ and $F_{2}$ is bounded away from 0 in the following sense:

[TABLE]

for any $R\geq 0$. Then there is a choice of $\operatorname{\mathcal{S}}(F_{1}^{\prime},F_{2}^{\prime})$ such that

[TABLE]

where the constant $C_{0}$ is uniform and depends only on the symmetric space $X$ and the constants $C$ and $L$. 2. (2)

Let $F_{1},F_{2},F_{3}$ be flats such that $F_{i}\cap F_{j}=F*$, for $1\leq i\neq j\leq 3$. Then the coarse intersections $\operatorname{\mathcal{S}}(F_{1}^{\prime},F_{2}^{\prime})$ and $\operatorname{\mathcal{S}}(F_{1}^{\prime},F_{3}^{\prime})$ are parallel.

Proof.

(1) Let $D$ be the coarse intersection of $F_{1}^{\prime}$ and $F_{2}^{\prime}$. Let $\omega$ be a non-principal ultrafilter and let $(r_{n})$ be any sequence converging to infinity. Choose the sequence of base points as $(o_{n})=(o)$, where $o\in F*$. The map $q^{*}$ induces a biLipschitz embedding, denoted by $q^{**}$, of the asymptotic cone $Cone(F(\gamma_{pz}),\omega,r_{n},o_{n}))$ into the asymptotic cone $Cone(X,\omega,r_{n},q*(o_{n}))$. The asymptotic cone of the union $F_{1}\cup F_{2}$ is diffeomorphic to $F_{1}\cup F_{2}$. The map $q^{**}$ maps this set to the asymptotic cone of $F_{1}^{\prime}\cup F_{2}^{\prime}$. On the other hand the asymptotic cone of $F_{1}^{\prime}\cup F_{2}^{\prime}$ is a union of two flats along the asymptotic cone of $D$. The biLipschitz map $q^{**}$ maps $F*$ (more precisely, the asymptotic cone of $F*$) to the asymptotic cone of $D$. Hence the asymptotic cone of $D$ is isometric to $F*$. It follows that there is a choice of $\operatorname{\mathcal{S}}(F_{1}^{\prime},F-2^{\prime})$, the coarse intersection of $F_{1}^{\prime}$ and $F_{2}^{\prime}$, such that $\operatorname{\mathcal{S}}(F_{1}^{\prime},F_{2}^{\prime})$ is a singular subflat of dimension $\dim(F*)$.

For the second claim, let $D_{1}\subset F_{1}^{\prime}$ be the coarse intersection of $F_{1}^{\prime}$ and $F_{2}^{\prime}$. Since $q^{*}(F*)$ is in the $C$-neighborhood of both $F_{1}^{\prime}$ and $F_{2}^{\prime}$, we can choose the singular subflat $D_{1}$ to be in a $2C$-neighborhood of $F_{2}^{\prime}$. Since $q^{*}$ is $L$-bilipschitz, $q^{*}(\operatorname{Nbhd}_{4LC}(F_{i}))\supset\operatorname{Nbhd}_{2C}(F_{i}^{\prime})$. By assumption on the angle between flats $F_{1}$ and $F_{2}$, we get that

[TABLE]

It follows that the singular subflat $D_{1}$ is in a $4L^{2}C$-neighborhood of $q^{*}(F*)$.

(2) Let $D_{1}\subset F_{1}^{\prime}$ and $D_{2}\subset F_{1}^{\prime}$ be the coarse intersection of $F_{1}^{\prime}$ and $F_{2}^{\prime}$ and the coarse intersection of $F_{1}^{\prime}$ and $F_{3}^{\prime}$ respectively. By definition of coarse intersection, there exists $R>0$ such that

[TABLE]

and

[TABLE]

There exists $R^{\prime}$ such that

[TABLE]

and

[TABLE]

On the other hand, $q^{*}(F*)$ has finite Hausdorff distances to $\operatorname{Nbhd}_{R^{\prime}}(q^{*}(F_{1}))\cap\operatorname{Nbhd}_{R^{\prime}}(q^{*}(F_{2}))$ and $\operatorname{Nbhd}_{R^{\prime}}(q^{*}(F_{1}))\cap\operatorname{Nbhd}_{R^{\prime}}(q^{*}(F_{3}))$. Hence, $D_{1}$ and $D_{2}$ have finite Hausdorff distance. Since $D_{1}$ and $D_{2}$ are singular subflats of the same dimension, having finite Hausdorff distance implies that they are parallel. ∎

In the next result, we will use the notion of parallel set for a singular subflat from 6.1.

Proposition 6.4**.**

Let $q^{*}:F(\gamma_{pz})\to X$ be a $L$-biLipschitz embedding and let $F*$ be the intersection of all flats in $F(\gamma_{pz})$. We assume that there is a constant $C$ such that the image of any flat in $F(\gamma_{pz})$ has Hausdorff distance at most $C$ to a flat in $X$. Then there is a singular subflat $D$, of the same dimension as $\dim(F*)$, such that $q^{*}(F(\gamma_{pz}))$ is contained in a bounded neighborhood of $F(D)$, the parallel set of $D$.

Proof.

We first let $Y$ be a totally geodesic symmetric subspace such that $F(\gamma_{pz})$ isometrically splits as $Y\times F*$. Note that in this splitting, we identify $F*$ with $\{o\}\times F*$, where $o$ is a fixed point in $Y$.

We pick finitely many flats $\{l_{1},\dots,l_{k}\}$ in $Y$ passing through $o$ such that for any point $y\in Y$ and any flat $l$ containing $o$ and $y$, there is a flat $l_{i}$ in $Y$ such that $l\times F*$ and $l_{i}\times F*$ intersect in $F*$ at the an angle bounded away from 0 in the sense stated in Lemma 6.3. By Lemma 6.3, the images of these two flats are contained in bounded neighborhood of two flats that are parallel to a singular subflat $D$. Furthermore, $D$ has a bounded Hausdorff distance to $q^{*}(F*)$

Therefore, $q^{*}(F(\gamma_{pz}))$ is contained in a bounded neighborhood of $F(D)$. ∎

7. $C^{0}$ Local Semi-rigidity of Boundary Actions of Lattices

In this section, we study actions of higher rank lattices in semisimple Lie groups of the non-compact type $G$ on projective boundaries of $G$ and prove the main theorem. These are much better behaved than the actions of fundamental groups of general non-positively curved manifolds on boundary spheres, and rather resemble negatively curved manifolds. We attribute this to the shared good properties for quasi-geodesics and quasi-flats and the uniform transversality properties of weak stable and unstable manifolds.

We obtain the following local stability result which is a slightly more detailed version of Theorem 1.1:

Theorem 7.1**.**

Let $G$ be a connected linear semisimple Lie group without compact factors, $\Gamma$ be a uniform lattice in $G$, $Q$ be a parabolic subgroup of $G$ and $\rho_{0}:\Gamma\to\operatorname{Homeo}(G/Q)$ is the boundary action (i.e. action by left multiplication). Then there is a neighborhood $\operatorname{\mathcal{U}}_{0}$ of $\rho_{0}$ and a neighborhood $V_{0}$ of $\operatorname{id}$ (in the space of continuous maps on $G/Q$) such that: for every $\rho\in\operatorname{\mathcal{U}}_{0}$, there exists a unique $(\rho,\rho_{0})$-semiconjugacy $\phi_{\rho}\in V_{0}$. Moreover, $\phi_{\rho}$ converges to $\operatorname{id}$ uniformly as $\rho$ converges to $\rho_{0}$.

Remark 7.2**.**

We do not necessarily require that $G$ is linear, but rather that $G$ is semisimple without compact factors and $\Gamma<G$ has a torsion-free normal subgroup $\Gamma_{0}\vartriangleleft\Gamma$ such that $\Gamma/\Gamma_{0}$ is finitely generated and $\Gamma_{0}$ is isomorphic to a lattice in $\operatorname{Isom}(X)^{0}$ for the symmetric space $X$ of noncompact type associated to $G$. However, we state it in the form above for a simpler statement.

In Theorem 7.1, the uniqueness of $\phi_{\rho}$ in $V_{0}$ is an application of Lemma 4.5. Our goal in this section is to prove the existence of the semiconjugacies $\phi_{\rho}$ for all perturbations close to $\rho_{0}$, i.e. the $C^{0}$ local semi-rigidity of $\rho_{0}$. The construction of semi-conjugacies $\varphi_{\rho}$ will be done in Section 7.1. Then we will show in Section 7.2 that $\varphi_{\rho}$ is continuous and that it converges to the identity uniformly as $\rho$ converges to $\rho_{0}$.

Initial reductions: We will now explain how in Theorem 7.1, we can always reduce to the case that $G$ is a connected center-free semisimple Lie group without compact factors and $\Gamma$ is a torsion-free uniform lattice in $G$.

Since $G$ is linear, Selberg’s lemma implies there is a normal torsion-free finite index subgroup $\Gamma_{0}\vartriangleleft\Gamma$. By Corollary 4.7, it suffices to prove that the action $\rho|_{\Gamma_{0}}$ of $\Gamma_{0}$ is $C^{0}$-locally semi-rigid and that a semi-conjugacy is sufficiently close to $\operatorname{id}$ is unique.

Let $Z(G)$ be the center of $G$ and $h:G\to G/Z(G)$. Then $\Gamma_{0}$ is isomorphic to $h(\Gamma_{0})$ where $h(\Gamma_{0})$ is a torsion-free uniform lattice in $h(G)$. Moreover $h(G)$ is a connected semisimple group with trivial center and without compact factors. Finally, $h(Q)$ is a parabolic subgroup of $h(G)$ and $G/Q$ is homeomorphic to $h(G)/h(Q)$.

Setup: Henceforth in this section, we will work with the following assumptions and notations. Let $G$ be a connected semisimple Lie group with trivial center and without compact factors. Let $X$ be the Riemannian symmetric space attached to $G$, i.e. $G=\mathrm{Isom}(X)^{0}$. Let $Q<G$ be a parabolic subgroup. We fix a torsion-free uniform lattice $\Gamma$ in $G$.

• •

Let $\mathfrak{X}\in\partial_{\infty}X$ be a point such that $Q=G_{\mathfrak{X}}$.

• •

Fix a basepoint $p\in X$ and let $\gamma_{p\mathfrak{X}}$ be the geodesic in $X$ that is asymptotic to $\mathfrak{X}$. We will denote by $F(\gamma_{p\mathfrak{X}})$ its parallel set (see Section 5.2).

• •

The space $X_{G/Q}:=G/M_{Q}$ is the Weyl chamber $Q$-face bundle equipped with a $G$-invariant Riemannian metric, see Section 5.7. Recall that $p:X_{G/Q}\to X$ is a fiber bundle with base $X$ and fibers homeomorphic to $K/M_{Q}\cong G/Q$. The map $\Phi:X_{G/Q}\to X\times G/Q$ is the diffeomorphism introduced in Section 5.6.

• •

The action $\rho_{0}:\Gamma\to\operatorname{Homeo}(G/Q)$ will always denote the algebraic action of $\Gamma$ on $G/Q$ by left multiplication. Then $\rho_{0}$ is a standard action with associated data $(X_{G/Q},p,\Phi,\{\pi_{x}\},\{g_{x}\},\ell)$ where $\ell\geq 1$. See Section 3.1 for the definition of standard action, Remark 3.8 for the explanation of associated data in this case, and Section 3.2.2 for a proof of this fact.

• •

Let $\operatorname{\mathcal{W}^{cs}},\operatorname{\mathcal{W}^{cu}}$ be the center-stable and center-unstable manifolds in $G/M_{Q}$ for the Weyl chamber flow, as in Section 5.9.

• •

If $\rho:\Gamma\to\operatorname{Homeo}(G/P)$, then $\widehat{\rho}:\Gamma\to\operatorname{Homeo}(X\times G/Q)$ is the induced action (as in Definition 3.4).

7.1. Construction of equivariant maps

The construction proceeds in several steps. As the whole argument is rather complicated, let us give an outline here. For simplicity, suppose the parabolic $Q$ is minimal. Then, eventually, we want to produce a quasi-flat that comes about as the intersection of unperturbed center-unstable with the “perturbed center stable”. This gives us a perturbation of the center foliation of the unperturbed action which is given by flats. The hope then is that these actually are quasi-flats. Then we can apply the work of Kleiner Leeb and Eskin Farb on quasi-flats which gives us shadowing of the “perturbed center leaves” in Proposition 7.10 and allows to define a $\rho-\rho_{0}$ equivariant map on the boundary in Proposition 7.18. To get there we first get local bi-Lipschitz control of the “perturbed center foliation” in Proposition 7.3 (the Bowden-Mann Lemma). From this we then derive global bi-Lipschitz in Proposition 7.7. From a dynamical point of view, this is the analogue of orbit equivalence of the perturbation of an Anosov flow, i.e. structural stability.

We now begin the construction. We first apply the generalized lemma about standard actions to our setup of perturbing the standard action of a uniform lattice on the Furstenberg boundary.

Proposition 7.3**.**

If $\rho:\Gamma\to\operatorname{Homeo}(F)$ is sufficiently close to $\rho_{0}$, then there is a continuous map $\widetilde{f}:X\times G/Q\to X_{G/Q}$ satisfying:

1. (1)

$p(\widetilde{f}(x,\xi))=x$* and $\widetilde{f}(\gamma\cdot x,\rho(\gamma)\xi)=\widehat{\rho}_{0}(\gamma)\widetilde{f}(x,\xi)$* 2. (2)

the map $\widetilde{f}(\cdot,\xi):X\to\widetilde{f}(X\times\{\xi\})$ is a $C^{1}$ diffeomorphism for every $\xi\in G/Q$. Moreover, for any $(x_{0},\xi_{0})\in X\times F$, $(\rho,y,\xi)\mapsto D_{y}\widetilde{f}(y,\xi)$ is continuous in all three variables at $(\rho_{0},x_{0},\xi_{0})$. 3. (3)

for every $\epsilon>0$ and for every $x\in X$, there exists a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(x,\epsilon)$ of $\rho_{0}$ such that

[TABLE] 4. (4)

for every $\epsilon,R>0$, there is a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon,R)$ of $\rho_{0}$ such that: if $\rho\in\operatorname{\mathcal{U}}$, $(x,\xi)\in X\times G/Q$ and $\eta:=\pi_{x}(\widetilde{f}(x,\xi))\in G/Q$, then

[TABLE]

where $d_{F_{y}}$ is the distance on the fiber $F_{y}$ (induced by the Riemannian metric $g_{y}$). 5. (5)

We consider the Riemannian metric on $X_{G/Q}$ constructed in Section 5.7. For any $\epsilon>0$, there exists a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon)$ of $\rho_{0}$ such that

[TABLE]

where $\eta_{x}:=\pi_{x}(\widetilde{f}(x,\xi))$.

Proof.

This is essentially an application Proposition 3.9 to the standard action $\rho_{0}$ of $\Gamma$ on $G/Q$. But deriving this result from Proposition 3.9 requires a few observations we will now remark on. Note that the maps $\pi_{x}:F_{x}\to F$ are in fact smooth and each fiber $F_{x}$ is totally geodesic in $G/M_{Q}$. Let $g_{x}$ denote the induced Riemannian metric on $F_{x}$. If $x\in X$ and $\gamma\in\Gamma$, then the map $\gamma_{x}:(F_{x},g_{x})\to(F_{\gamma x},g_{\gamma x})$ is an isometry (so more than just affine).

The smoothness of $\pi_{x}$ implies that the Hölder assumption in Proposition 3.9 part (4) is satisfied. Since the action $\widehat{\rho_{0}}$ is by isometries on $G/M_{Q}$, note that we can make the following assumption when applying part (4): we can assume that $x$ lies in a fundamental domain of $X/\Gamma$. Moreover, $F$ is also compact. Then the neighborhood $\operatorname{\mathcal{U}}$ of $\rho_{0}$ in the conclusion of part (4) can be chosen in such a way that it does not depend on $x$ or $\xi$.

Finally, part (5) of Proposition 3.9 applies because $\widehat{\rho_{0}}$ acts by isometries on $G/M_{Q}$; see Section 5.7.

We also remark that in the last claim of Proposition 7.3, since $\rho_{0}(\Gamma)\backslash X_{G/Q}$ is a closed manifold, the claim still holds for any lifted metric on $X_{G/Q}$ from a Riemannian metric on $\rho_{0}(\Gamma)\backslash X_{G/Q}$. ∎

For the rest of this section, $\widetilde{f}:X\times G/Q\to X_{G/Q}$ will always denote the map obtained in Proposition 7.3. In the next proposition, we will control the images of $\widetilde{X}\times\left\{\xi\right\}$ under $\widetilde{f}$. But first we make a definition.

Definition 7.4**.**

For any $\xi,\eta\in G/Q$, we define

[TABLE]

Note that $\operatorname{\mathcal{Q}}_{\xi,\eta}$ may be an empty set for some choices of $\xi$ and $\eta$.

Remark 7.5**.**

Note that $\widetilde{f}$ depends on $\rho:\Gamma\to\operatorname{Homeo}(G/P)$. Thus $\operatorname{\mathcal{Q}}_{\xi,\eta}\equiv\operatorname{\mathcal{Q}}_{\xi,\eta}(\rho)$. But we suppress $\rho$ for the sake of brevity in this subsection. Further, if $\rho=\rho_{0}$, then

[TABLE]

Thus, informally speaking, one should think of $\operatorname{\mathcal{Q}}_{\xi,\eta}\equiv\operatorname{\mathcal{Q}}_{\xi,\eta}(\rho)$ as ‘perturbed center’ manifolds, where the perturbation comes from perturbing the action $\rho_{0}$ to $\rho$.

7.1.1. $\operatorname{\mathcal{Q}}_{\xi,\eta}$** is contained in the neighborhood of a center manifold $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$**

The main result of this subsection is Proposition 7.10 which shows that each non-empty ‘perturbed center’ manifold $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is contained in a bounded neighborhood of a center manifold $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$. Along the way, we will establish several properties of the ‘perturbed center manifolds’ $\operatorname{\mathcal{Q}}_{\xi,\eta}$.

In the following lemma, $d^{\operatorname{Gr}}_{v}$ is the Grassmanian distance between linear subspaces of $T_{v}(G/M_{Q})$, induced by the Riemannian metric on $G/M_{Q}$.

Lemma 7.6**.**

For any $\epsilon>0$, there exists a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon)$ of $\rho_{0}$ such that: for any $\rho\in\operatorname{\mathcal{U}}$, if $\xi,\eta\in G/Q$ and $x\in X$ are such that $\widetilde{f}(x,\xi)\in\mathcal{Q}_{\xi,\eta}$, then

[TABLE]

where $\zeta_{0}=\pi_{x}(\widetilde{f}(x,\xi))$.

Proof.

Fix any $\epsilon^{\prime}>0$ and let $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon^{\prime})$ be the neighborhood of $\rho_{0}$ given by Proposition 7.3 part (5). Suppose $\operatorname{\mathcal{Q}}_{\xi,\eta}\neq\emptyset$, $x\in X$ and $\zeta_{0}=\pi_{x}(\widetilde{f}(x,\xi))$. By Proposition 7.3 part (5), the tangent spaces $T_{\widetilde{f}(x,\xi)}\widetilde{f}(X\times\{\xi\})$ and $T_{\widetilde{f}(x,\xi)}\operatorname{\mathcal{W}^{cs}}(\zeta_{0})$ are $\epsilon^{\prime}$-close in the Grassmannian distance $d^{\operatorname{Gr}}_{\widetilde{f}(x,\xi)}$. On the other hand, for every $\widetilde{f}(x,\xi)\in\mathcal{Q}_{\xi,\eta}$,

[TABLE]

Since $\widetilde{f}(x,\xi)\in\operatorname{\mathcal{Q}}_{\xi,\eta}$, $\widetilde{f}(x,\xi)\in\operatorname{\mathcal{W}^{cu}}(\eta)$. Thus $\widetilde{f}(x,\xi)\in\operatorname{\mathcal{W}^{cs}}(\pi_{x}(\widetilde{f}(x,\xi))\cap\operatorname{\mathcal{W}^{cu}}(\eta)=\operatorname{\mathcal{W}^{cs}}(\zeta_{0})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$. Since $\operatorname{\mathcal{W}^{cs}}(\zeta_{0})$ and $\widetilde{f}(X\times\{\xi\})$ form angles at least $\frac{\pi}{2}-\epsilon^{\prime}$ with $\operatorname{\mathcal{W}^{cu}}(\eta)$, there exists $\epsilon=\epsilon(\epsilon^{\prime})$ such that

[TABLE]

Furthermore $\epsilon$ tends to 0 when $\epsilon^{\prime}$ tends to 0. We then obtain the lemma by choosing $\epsilon^{\prime}$ sufficiently small. ∎

Next, we will show that each connected component of $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is biLipschitz diffeomorphic to the parallel set $F(\gamma_{p\mathfrak{X}})$.

Proposition 7.7**.**

*Let $L>1$. Then there is a neighborhood $\mathcal{U}$ of $\rho_{0}$ such that: if $\rho\in\mathcal{U}$ and $\xi,\eta\in G/Q$ are such that $\mathcal{Q}_{\xi,\eta}\neq\varnothing$, then for each connected component $\operatorname{\mathcal{Q}}$ of $\operatorname{\mathcal{Q}}_{\xi,\eta}$ there is a map $\alpha_{\xi,\eta}:F(\gamma_{p\mathfrak{X}})\to\operatorname{\mathcal{W}^{cu}}(\eta)$ which is an $L$-biLipschitz diffeomorphism onto $\operatorname{\mathcal{Q}}$, where the distance on $\mathcal{Q}$ is the restriction of the distance on $\operatorname{\mathcal{W}^{cu}}(\eta)$. *

Proof.

Let us fix the connected component $\operatorname{\mathcal{Q}}$ of $\operatorname{\mathcal{Q}}_{\xi,\eta}$.

Let $\xi=gQ\in G/Q$ and let $\gamma_{p\mathfrak{X}^{\prime}}$ be the geodesic passing asymptotic to $\mathfrak{X}^{\prime}=g\mathfrak{X}$. We can identify $\operatorname{\mathcal{W}^{cu}}(\xi)$ with $F(\gamma_{p\mathfrak{X}^{\prime}})(gN_{Q}g^{-1})$ by first identifying $\operatorname{\mathcal{W}^{cu}}(\xi)$ with $X$, and then identifying $X$ with $F(\gamma_{p\mathfrak{X}^{\prime}})(gN_{Q}g^{-1})$. The overall identification can be thought of as the global product structure on a center unstable manifold in terms of center manifolds and strong unstable manifolds.

The center-unstable leaf $\operatorname{\mathcal{W}^{cu}}(\eta)$ is foliated by the family of symmetric subspaces $\{F(\gamma_{p{\mathfrak{X}}}):p\in X\}$. Each subspace in the family is identified with a center manifold of a point $\operatorname{\mathcal{C}}_{q}(\eta)\in\operatorname{\mathcal{W}^{cu}}(\eta)\subset X_{G/Q}$. The projection along strong unstable leaves maps every center leaf isometrically to a fixed center leaf.

Consider the map $retr:\operatorname{\mathcal{W}^{cu}}(\eta)\to F(\gamma_{p{\mathfrak{X}}^{\prime}})$ which is obtained by projection along strong unstable leaves. Then set $q_{\xi,\eta}:=retr|_{\operatorname{\mathcal{Q}}}$, the restriction of $retr$ to $\operatorname{\mathcal{Q}}\subset\operatorname{\mathcal{W}^{cu}}(\eta)$.

By Lemma 7.6, the tangent spaces of $\mathcal{Q}$ are $\epsilon$-close to $\operatorname{\mathcal{W}^{cs}}(\zeta)\cap\operatorname{\mathcal{W}^{cu}}(\eta)$ where $\zeta=\pi_{x}(\widetilde{f}(x,\xi))$. Hence, the map $q_{\xi,\eta}$ is locally $L$-biLipschitz if $\epsilon$ is small enough. That is, for every $\widetilde{f}(x,\xi)\in\operatorname{\mathcal{Q}}$, there is a neighborhood of $\widetilde{f}(x,\xi)$ such that $q_{\xi,\eta}$ is $L$-biLipschitz. Then note that $q_{\xi,\eta}$ is globally $L$-Lipschitz.

We now claim that $q_{\xi,\eta}$ is a diffeomorphism. This is an application of Lemma B.2 from the Appendix. For applying the Lemma, we first need to fix a suitable foliation chart. Note that the center unstable manifold $\operatorname{\mathcal{W}^{cu}}(\eta)$ has a global product structure in terms of center manifolds and strong unstable manifolds. This provides two mutually orthogonal foliations on $\operatorname{\mathcal{W}^{cu}}(\eta)$ and $\operatorname{\mathcal{W}^{cu}}(\eta)$ is homeomorphic to $\operatorname{\mathbb{R}}^{\dim(X)}$. Finally $\operatorname{\mathcal{Q}}$ is a complete connected $C^{1}$ sub-manifold of $\operatorname{\mathcal{W}^{cu}}(\eta)$ with tangent spaces which are uniformly close to the center distribution (see Lemma 7.6). Thus, we can apply Lemma B.2 and prove that $q_{\xi,\eta}$ is a diffeomorphism.

Then we define $\alpha_{\xi,\eta}:F(\gamma_{p\mathfrak{X}})\to\operatorname{\mathcal{W}^{cu}}(\eta)$ by setting $\alpha_{\xi,\eta}:=q_{\xi,\eta}^{-1}$. Thus $\alpha_{\xi,\eta}$ is a diffeomeorphism with image $\operatorname{\mathcal{Q}}$. As $\alpha_{\xi,\eta}=q_{\xi,\eta}^{-1}$ is locally $L$-Lipschitz, this implies that $\alpha_{\xi,\eta}$ is globally $L$-Lipschitz. Since $q_{\xi,\eta}=\alpha_{\xi,\eta}^{-1}$ is also globally $L$-Lipschitz, this implies that $\alpha_{\xi,\eta}$ is globally $L$-biLipschitz. ∎

Next, we show that each connected component $\operatorname{\mathcal{Q}}$ of $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is contained in a uniformly bounded neighborhood of some ‘center’ leaf $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$.

Lemma 7.8**.**

Let $\rho$ be sufficiently close to $\rho_{0}$ and suppose $\xi,\eta\in G/Q$ are such that $\mathcal{Q}_{\xi,\eta}$ is non-empty. Then for each connected component $\operatorname{\mathcal{Q}}$ of $\mathcal{Q}_{\xi,\eta}$ there is a unique $\zeta\in G/Q$ such that $\operatorname{\mathcal{Q}}$ is contained in a uniform bounded neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$.

Proof.

By Proposition 7.7, each connected component $\operatorname{\mathcal{Q}}$ of $\mathcal{Q}_{\xi,\eta}$ is the bi-Lipschitz image of $F(\gamma_{p\mathfrak{X}})$ in $\operatorname{\mathcal{W}^{cu}}(\eta)$, which is isometric to the symmetric space $X$. By Proposition 6.4, there is a singular subflat $D$ such that $\mathcal{Q}$ is in a bounded neighborhood of the parallel set of $D$. As a subset of $\operatorname{\mathcal{W}^{cu}}(\eta)$, the parallel set of $D$ is the center manifold of some $\zeta\in G/Q$.

For the uniqueness of $\zeta$, suppose that $\zeta^{\prime}\in G/Q$ such that $\mathcal{Q}$ is contained in a bounded neighborhood of $\operatorname{\mathcal{W}^{cs}}(\zeta^{\prime})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$. Let $F$ be a maximal flat in $F(\gamma_{pz})$ that maps to $q_{\xi,\eta}^{-1}(F)$ and has finite Hausdorff distance to a flat $F^{\prime}$ in $\operatorname{\mathcal{W}^{cu}}(\eta)$. On the other hand, if there is a flat contained in a bounded neighborhood of a center manifold then the flat is itself contained in the center manifold. Thus, as the flat $F^{\prime}$ is contained in a bounded neighborhood of two centers $\operatorname{\mathcal{W}^{cs}}(\zeta)\cap\operatorname{\mathcal{W}^{cu}}(\eta)$ and $\operatorname{\mathcal{W}^{cs}}(\zeta^{\prime})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$, the two centers has non-empty intersection, hence coincide. Therefore, $\zeta=\zeta^{\prime}$. ∎

Note that we do not know whether $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is connected. So a priori, the previous lemma does not imply that $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is contained in a uniformly bounded neighborhood of the ‘center’ leaf $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$. We prove this in Proposition 7.10. But first, we need the following lemma which is a consequence of Proposition 7.3.

Lemma 7.9**.**

For every $R,D>0$, there exists a neighborhood $\mathcal{U}^{\prime}$ of $\rho_{0}$ such that: suppose $\rho\in\mathcal{U}^{\prime}$ and $\xi,\eta\in G/Q$ are such that $\mathcal{Q}_{\xi,\eta}$ is non-empty. Then if $\widetilde{f}(x,\xi)\in\mathcal{Q}_{\xi,\eta}$ and $\zeta:=\pi_{x}(\widetilde{f}(x,\xi))$, we get

[TABLE]

Proof.

By Proposition 7.3 part (4), for every $\epsilon,R>0$, there exists a neighborhood $\operatorname{\mathcal{U}}$ of $\rho_{0}$ such that for every $\rho\in\operatorname{\mathcal{U}}$,

[TABLE]

Note that $B_{R}(\widetilde{f}(x,\xi))\cap\widetilde{f}(X\times\xi)\subset\widetilde{f}(B_{R}(x),\xi)$. It follows that

[TABLE]

Thus,

[TABLE]

Now given any $R,D>0$, the lemma follows immediately by choosing $\epsilon=D$. ∎

In the next proposition, the key idea is that any two components of $\mathcal{Q}_{\xi,\eta}$ “fellow-travel” and each lies in a uniform neighborhood of some center leaf which must therefore be the same.

Proposition 7.10**.**

Let $\rho$ be sufficiently close to $\rho_{0}$ and suppose $\xi,\eta\in G/Q$ be such that $\mathcal{Q}_{\xi,\eta}$ is non-empty. Then there is a unique $\zeta\in G/Q$ such that $\mathcal{Q}_{\xi,\eta}$ is contained in a uniformly bounded neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$.

Proof.

Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be two connected components of $\mathcal{Q}_{\xi,\eta}$. By Lemma 7.9, there exists $D_{0}$ such that each of $\mathcal{Q}_{i}$ is contained in a $D_{0}$-neighborhood of a center leaf. We note that $D_{0}$ depends only on the neighborhood of $\rho_{0}$ and the symmetric space $X$. We would like to prove the two center leaves coincide.

We let $\zeta_{1},\zeta_{2}\in G/Q$ such that $\mathcal{Q}_{i}$ is contained in $D_{0}$-neighborhood of $\operatorname{\mathcal{W}^{cs}}(\zeta_{i})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$, for $i=1,2$. Suppose for that $\zeta_{1}\neq\zeta_{2}$. We let $y\in\operatorname{\mathcal{W}^{cu}}(\eta)$ such that $y$ has distance at most $\frac{D_{0}}{2}$ to both $\operatorname{\mathcal{W}^{cs}}(\zeta_{1})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$ and $\operatorname{\mathcal{W}^{cs}}(\zeta_{2})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$ but $\operatorname{Nbhd}_{10D_{0}}(B_{R}(y)\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{i})\cap\operatorname{\mathcal{W}^{cu}}(\eta))$ does not contain $B_{R}(y)\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{3-i})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$, for $i=1,2$, and where the constant $R$ is from Lemma 7.9. Furthermore, any intersection of a $L$-biLipschitz flat in $\mathcal{Q}_{i}$ with $B_{R}(y)$ is not contained in a $8D_{0}$-neighborhood of $B_{R}(y)\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{3-i})\cap\operatorname{\mathcal{W}^{cu}}(\eta)$.

We assume that $\rho$ is sufficiently close to $\rho_{0}$ so that $D<D_{0}$ and $R>20D_{0}$. By Lemma 7.9, $\mathcal{Q}_{\xi,\eta}\cap B_{R}(y)$ is contained in a $D$-neighborhood of a center leaf. But this contradicts the last paragraph. ∎

7.1.2. The $\zeta$ in Proposition 7.10 is independent of $\eta$

By Proposition 7.10, each $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is contained in the bounded neighborhood of a center manifold $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$. In this subsection, we will show that given $\xi$, this $\zeta$ is independent of $\eta$ (provided $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is non-empty). This is the content of Lemma 7.17, the main result of this subsection. Then, an application of this lemma is Proposition 7.18 where we construct a $(\rho,\rho_{0})$ equivariant map on $G/Q$.

But first, we need some preparatory lemmas. Recall the notion of opposite points in $G/Q$ from Section 5.5.

Lemma 7.11**.**

If $\alpha,\beta$ in $G/Q$, then the set of elements opposite to both $\alpha$ and $\beta$ is open dense in $G/Q$.

Proof.

The set of points opposite to $\alpha$ corresponds to an open dense Bruhat cell. The same holds for $\beta$. Hence the desired set is an intersection of open dense sets. ∎

Lemma 7.12**.**

Suppose $\zeta_{1}\neq\zeta_{2}\in G/Q$ and $\eta_{1},\eta_{2}\in G/Q$ are both opposite to $\zeta_{1}$. Then there exists a sequence $(\gamma_{n})$ in $\Gamma$ such that

[TABLE]

while

[TABLE]

Proof.

We will first prove this statement for $G$ in place of $\Gamma$.

We recall that every element $\eta$ of $G/Q$ corresponds to an equivalence class of asymptotic based Weyl chambers $Q$-faces. It follows that $\eta$ can be identified with the geodesic boundary of a based Weyl chamber $Q$-face in the equivalence class, and thus can be treated as a closed subset in the geodesic boundary at infinity of $X$.

We let $F*$ be the minimal singular subflat with $\eta_{1}$ and $\zeta_{1}$ contained in its geodesic boundary at infinity(i.e. the singular subflat of minimal dimension containing $\eta_{1}$ and $\zeta_{1}$ in its boundary).

Let $g\in G$ be an element whose axis is the parallel set of $F*$ (here, by axis of $g$, we mean the $\left\{x\in X:d_{X}(x,g\cdot x)=\inf_{y\in X}d_{X}(y,g\cdot y)\right\}$). By switching to $g^{-1}$ if necessary, we may assume that $\zeta_{1}$ is the source and $\eta_{1}$ is the sink of $g$. Under iteration, $g$ contracts to $\eta_{1}$ the (open dense) set of elements of $G/Q$ which are opposite to $\zeta_{1}$.

In particular, $\eta_{2}$ converges to $\eta_{1}$ under iterations. Moreover, under iterations, the only element of $G/Q$ converging to $\zeta_{1}$ is $\zeta_{1}$ itself. Hence, $\zeta_{2}$ converges to a point distinct from $\zeta_{1}$ under iterations of $g$.

To obtain the claim for $\Gamma$, we approximate iterations of $g$ by elements of the lattice. Indeed, writing $g^{n}=c_{n}\gamma_{n}$, where $c_{n}$ lies in a compact fundamental domain, and passing to a subsequence, we can assume that the elements $c_{n}$ converge to a fixed element $c$. Hence the result follows using $\gamma_{n}$ instead of $g^{n}$. ∎

By pulling back the ‘perturbed center stable’ leaves $\operatorname{\mathcal{Q}}_{\xi,\eta}$, we now introduce a foliation $\operatorname{\mathcal{F}}(\xi,\eta)$ of $X\times G/Q$. This is the content of the next two lemmas. First however let us define $\operatorname{\mathcal{F}}(\xi,\eta)$. Let

[TABLE]

Definition 7.13**.**

For $(\xi,\eta)\in\mathcal{V}_{0}$, we let

[TABLE]

We note that each $\mathcal{F}(\xi,\eta)$ is a $C^{1}$ manifold. Furthermore, the disjoint union of all such $\mathcal{F}(\xi,\eta)$ is $X\times G/Q$. The set of leaves is identified with an open subset $\mathcal{V}_{0}\subset G/Q\times G/Q$. We note that the intersection of the unstable foliation of $X_{G/Q}$ with $\widetilde{f}(X\times\{\xi\})$ gives rise to a foliation of the $C^{1}$ manifold $\widetilde{f}(X\times\{\xi\})$. By pulling this back by $\widetilde{f}$ we obtain a foliation of $X\times\{\xi\}$ whose leaves are

[TABLE]

Then, for any $\gamma\in\Gamma$, $\left\{\widehat{\rho}(\gamma)\mathcal{F}(\xi,\eta):\eta\in G/Q,(\xi,\eta)\in\mathcal{V}_{0}\right\}$ is a foliation of $X\times\left\{\rho(\gamma)\xi\right\}$ as $\widetilde{f}$ is $(\widehat{\rho},\widehat{\rho}_{0})$-equivariant.

We will now prove the continuity of $\operatorname{\mathcal{Q}}_{\xi,\eta}$ and $\operatorname{\mathcal{F}}(\xi,\eta)$ using Lemma B.3.

Lemma 7.14**.**

The map $\operatorname{\mathcal{V}}_{0}\ni(\xi,\eta)\mapsto\operatorname{\mathcal{Q}}_{\xi,\eta}$ is continuous with respect to the pointed Hausdorff topology.

Proof.

Continuity in $\eta$ is straightforward as the unstable leaves produce a continuous foliation of $\widetilde{f}(X\times\left\{\xi\right\})$ for any fixed $\xi$. For proving continuity in $\xi$, fix $(\xi_{0},\eta)\in\operatorname{\mathcal{U}}$ and consider a sequence $(\xi_{n},\eta)$ in $\operatorname{\mathcal{U}}$ converging to $(\xi_{0},\eta)$. Apply Lemma B.3 with $M_{n}=\widetilde{f}(X\times\left\{\xi_{n}\right\})$ for $n\geq 0$, and $N=\operatorname{\mathcal{W}^{cu}}(\eta)$. ∎

Lemma 7.15**.**

The map $\operatorname{\mathcal{V}}_{0}\ni(\xi,\eta)\mapsto\mathcal{F}(\xi,\eta)$ is continuous with respect to the pointed Hausdorff topology.

Proof.

Continuity in $\eta$ is immediate from the fact that $\{\mathcal{F}(\xi,\eta):\eta\in G/Q\text{ such that }\mathcal{F}(\xi,\eta)\neq\varnothing\}$ is a continuous foliation of $X\times\{\xi\}$.

For proving continuity in $\xi$, let $(\xi_{n})$ be a sequence converging to $\xi_{0}\in G/Q$. Note that since $\widetilde{f}$ covers $\operatorname{id}$ on $X$,

[TABLE]

Since $\xi_{n}\to\xi_{0}$, Lemma 7.14 implies that $\operatorname{\mathcal{Q}}_{\xi_{n},\eta}\to\operatorname{\mathcal{Q}}_{\xi_{0},\eta}$ in the pointed Hausdorff topology. Then $p(\operatorname{\mathcal{Q}}_{\xi_{n},\eta})\to p(\operatorname{\mathcal{Q}}_{\xi_{0},\eta})$ in the pointed Hausdorff topology. This finishes the proof. ∎

Remark 7.16**.**

We point out that the last two lemmas do not follow from a general statement about the continuity of leaves of continuous foliations of the leaves of a smooth foliation.

Indeed, Bonatti and Franks [BF04, Theorem 1.1] even provide a Hölder continuous distribution $E^{\prime}$ on $\mathbb{R}^{2}$ tangent to a continuum $(\mathscr{F}_{s})_{s\in(0,1)}$ of pairwise distinct foliations. Then consider the Hölder continuous distribution $E$ on $\mathbb{R}^{3}$ that consists of a copy of $E^{\prime}$ on each $\left\{t\right\}\times\mathbb{R}^{2}$. Now we can choose a discontinuous map $\sigma:\mathbb{R}\to(0,1)$ and consider for each $t\in\mathbb{R}$ the Bonatti-Franks foliation $\mathscr{F}_{\sigma(t)}$ on $\left\{t\right\}\times\mathbb{R}^{2}$. Now the union of these foliations of $\mathbb{R}^{2}$ does not form even a $C^{0}$ foliation of $\mathbb{R}^{3}$. In particular, a sequence of leaves $\ell_{t}$ through the point $(t,0,0)$ need not converge to the leaf $\ell_{0}$ through $(0,0,0)$.

Now we are in a position to prove the key lemma of this subsection. Before starting the proof, we remark that the argument below is a higher rank analogue of the argument in [BM22, Proposition 3.8]. However, it is considerably more complicated because the leaves have higher dimension and because the $\rho_{0}$-action is no longer a convergence group action.

Lemma 7.17**.**

Let $\xi,\eta_{1},\eta_{2}\in G/Q$ be such that $\operatorname{\mathcal{Q}}_{\xi,\eta_{1}}$ and $\operatorname{\mathcal{Q}}_{\xi,\eta_{2}}$ are non-empty. For $i\in\left\{1,2\right\}$, let $\zeta_{i}\in G/Q$ be the unique point (obtained by Proposition 7.10) such that $\mathcal{Q}_{\xi,\eta_{i}}$ is contained in a uniform neighborhood of $\operatorname{\mathcal{W}^{cs}}(\zeta_{i})\cap\operatorname{\mathcal{W}^{cu}}(\eta_{i})$. Then $\zeta_{1}=\zeta_{2}$.

Proof.

Assume the lemma holds in the special case when $\eta_{1}$ and $\eta_{2}$ are both opposite to $\zeta_{1}$. Let us first derive the lemma for $\eta_{2}$ possibly not opposite to $\zeta_{1}$ from the special case. In that case, note that $\left\{\eta:\operatorname{\mathcal{Q}}_{\xi,\eta}\neq\varnothing\right\}$ is an open set. Also note that $\eta_{1}$ is always opposite to $\zeta_{1}$ by definition of $\zeta_{1}$. Then, by Lemma 7.11, we may choose $\eta_{3}$ such that $\eta_{3}$ is opposite to both $\zeta_{1}$ and $\zeta_{2}$ and $\operatorname{\mathcal{Q}}_{\xi,\eta_{3}}\neq\varnothing$. Let $\zeta_{3}$ be such that $\operatorname{\mathcal{Q}}_{\xi,\eta_{3}}$ is contained in a uniform neighborhood of $\operatorname{\mathcal{W}^{cs}}(\zeta_{3})\cap\operatorname{\mathcal{W}^{cu}}(\eta_{3})$. Then, applying the special case to $\eta_{i}$ and $\eta_{3}$, we obtain that $\zeta_{i}=\zeta_{3}$ where $i\in\left\{1,2\right\}$. Thus it suffices to prove the special case.

Now suppose that we are in the special case, i.e. $\eta_{1},\eta_{2}$ are both opposite to $\zeta_{1}$. Then by Lemma 7.12, there is a sequence $(\gamma_{n})$ such that $\rho_{0}(\gamma_{n})\zeta_{1}$ and $\rho_{0}(\gamma_{n})\zeta_{2}$ converge to two distinct points in $G/Q$, while $\rho_{0}(\gamma_{n})\eta_{1}$ and $\rho_{0}(\gamma_{n})\eta_{2}$ converge to the same point.

We introduce the following notation: if $\eta\in G/Q$ such that $\operatorname{\mathcal{Q}}_{\xi,\eta}$ is non-empty, let $\zeta_{\xi,\eta}\in G/Q$ be the point such that $\mathcal{Q}_{\xi,\eta}$ is contained in a uniform neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{\xi,\eta})$. Note that $\zeta_{\xi,\eta}$ is well-defined by Proposition 7.10 and that $\zeta_{i}=\zeta_{\xi,\eta_{i}}$ for $i=1,2$ where $\zeta_{i}$ is as in the statement of the lemma.

Since $G/Q$ is compact, up to passing to a subsequence, we assume that $\xi_{0}:=\lim_{n\to\infty}\rho(\gamma_{n})\xi$ exists. Let $\eta_{0}:=\lim_{n\to\infty}\rho_{0}(\gamma_{n})\eta_{1}$. We will obtain a contradiction by showing that there are two distinct leaves in $X\times\{\xi_{0}\}$ belonging to $\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{0}))$.

We let $x_{0}\in X$ be a point contained in the minimal singular subflat whose geodesic boundary contains $\eta_{0}$ and $\lim_{n}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}$. Let $C$ be the uniform constant such that whenever $\mathcal{Q}_{\xi^{\prime},\eta^{\prime}}\neq\emptyset$, $\operatorname{\mathcal{Q}}_{\xi^{\prime},\eta^{\prime}}$ is contained in a $C$-neighborhood $\operatorname{\mathcal{W}^{cs}}(\zeta_{\xi^{\prime},\eta^{\prime}})\cap\operatorname{\mathcal{W}^{cu}}(\eta^{\prime})$. Let $D$ be the ball of radius $7C$ around $x_{0}$.

Claim**.**

There exists a sequence $(\eta_{n})_{n\geq 3}$ with $\eta_{n}\in G/Q$ such that:

1. (1)

$\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{n}))\cap(X\times\{\rho(\gamma_{n})\xi\})$* has non-empty intersection with $D\times\{\rho(\gamma_{n})\xi\}$,* 2. (2)

$\lim_{n\to\infty}\zeta_{\rho(\gamma_{n})\xi,\eta_{n}}$* is different from $\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}$,* 3. (3)

the sequence $(\eta_{n})$ converges to $\eta_{0}=\lim_{n\to\infty}\rho_{0}(\gamma_{n})\eta_{1}$.

We will postpone the proof of this claim until after finishing the proof of Lemma 7.17.

For the latter, we proceed by contradiction. By Lemma 7.15, the sequence of $C^{1}$ manifolds $\mathcal{F}(\rho(\gamma_{n})\xi,\eta_{n})=\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{n}))\cap(X\times\{\rho(\gamma_{n})\xi\})$ converges to $\mathcal{F}(\xi_{0},\eta_{0})=\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{0}))\cap(X\times\{\xi_{0}\})$.

We remark that, since $\widetilde{f}$ covers the identity $\operatorname{id}:X\to X$,

[TABLE]

By the choice of $\gamma_{n}$ and Lemma 7.8, $\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\widetilde{f}(X\times\{\xi_{0}\})$ is contained in a bounded neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}})$. In other words, $p\left(\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\widetilde{f}(X\times\{\xi_{0}\})\right)$ is contained in a bounded neighborhood of $p\left(\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}})\right)$.

On the other hand, $\operatorname{\mathcal{W}^{cu}}(\eta_{n})\cap\widetilde{f}(X\times\{\rho(\gamma_{n})\xi\})$ is in a bounded neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta_{n})\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{\rho(\gamma_{n})\xi,\eta_{n}})$. Set $\Upsilon:=\lim_{n\to\infty}\zeta_{\rho(\gamma_{n})\xi,\eta_{n}}$. Then the $\widetilde{f}$ image of the limit manifold is contained in a bounded neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\Upsilon)$. Thus, the manifold $\mathcal{F}(\xi_{0},\eta_{0})=\lim_{n}\mathcal{F}(\rho(\gamma_{n})\xi,\eta_{n})$ is contained in a bounded neighborhood of $p\left(\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\Upsilon)\right)\times\{\xi_{0}\}$. Therefore, $p(\mathcal{F}(\xi_{0},\eta_{0}))$ is contained in a bounded neighborhood of $p\left(\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\Upsilon)\right)$. Since $\Upsilon\neq\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}$, and each fiber $F_{x}\cong G/Q$ is compact, this contradicts Proposition 7.10. ∎

Proof of Claim..

For every $n$, we let $U_{n}$ be the neighborhood of $\eta_{0}$ defined as follows

[TABLE]

Recall that $(\gamma_{n})$ is a sequence as in the proof of the above lemma (i.e. $\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{1}\neq\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{2}$ while $\lim_{n\to\infty}\rho_{0}(\gamma_{n})\eta_{1}=\lim_{n\to\infty}\rho_{0}(\gamma_{n})\eta_{2}$). We let $\sigma_{n}$ be the shortest geodesic segment (with respect to a fixed Riemannian metric on $G/Q$) connecting $\rho_{0}(\gamma_{n})\eta_{1}$ and $\rho_{0}(\gamma_{n})\eta_{2}$. As $\rho_{0}(\gamma_{n})\eta_{1}$ and $\rho_{0}(\gamma_{n})\eta_{2}$ converge to the same point $\eta_{0}$, the segment $\sigma_{n}$ converges to $\eta_{0}$. Thus, for $n$ sufficiently large, $\sigma_{n}$ is contained entirely in $U_{n}$. We are going to pick $\eta_{n}$ belonging to $\sigma_{n}$, thus satisfying item (3). We note that $\widetilde{f}|_{X\times\{\rho(\gamma_{n})\xi\}}$ is a homeomorphism to its image. Thus $\{\mathcal{F}(\rho(\gamma_{n})\xi,\nu)=\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\nu))\cap(X\times\{\rho(\gamma_{n})\xi\}):\nu\in\sigma_{n}\}$ is a continuous family of $C^{1}$ submanifolds of $X\times\{\rho(\gamma_{n})\xi\}$. We pick $\eta_{n}\in\sigma_{n}$ to be the first point (in the direction from $\rho_{0}(\gamma_{n})\eta_{1}$ to $\rho_{0}(\gamma_{n})\eta_{2}$) such that $\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{n}))\cap(X\times\{\rho(\gamma_{n})\xi\})$ does not intersect $B_{5C}(x_{0})\times\{\rho(\gamma_{n})\xi\}$. If there is no such point in $\sigma_{n}$, then we can pick $\eta_{n}=\rho_{0}(\gamma_{n})\eta_{2}$.

Observe that with this choice of $\eta_{n}$ parts (1) and (3) of the claim follow immediately because of our choices of $D$. For the second item of the claim, in the case $\eta_{n}=\rho_{0}(\gamma_{n})\eta_{2}$, it is immediate. Thus we only need to verify the case that $\eta_{n}$ is the first point in $\sigma_{n}$ such that $\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{n}))\cap(X\times\{\rho(\gamma_{n})\xi\})$ does not intersect $B_{5C}(x_{0})\times\{\rho(\gamma_{n})\xi\}$. Recall that the manifold $\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\eta_{n}))\cap(X\times\{\rho(\gamma_{n})\xi\})$ is in a $C$-neighborhood of $p(\operatorname{\mathcal{W}^{cu}}(\eta_{n})\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{\rho(\gamma_{n})\xi,\eta_{n}}))\times\{\rho(\gamma_{n})\xi\}$ and the manifold $\widetilde{f}^{-1}(\operatorname{\mathcal{W}^{cu}}(\rho_{0}(\gamma_{n})\eta_{1}))\cap(X\times\{\rho(\gamma_{n})\xi\})$ is in a $C$-neighborhood of $p(\operatorname{\mathcal{W}^{cu}}(\rho_{0}(\gamma_{n})\eta_{1})\cap\operatorname{\mathcal{W}^{cs}}(\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}))\times\{\rho(\gamma_{n})\xi\}$. By the triangle inequality of the Hausdorff pseudo-distance,

[TABLE]

for every $n$. It follows that their limits are distinct. The limit of $p(\operatorname{\mathcal{W}^{cu}}(\eta_{n})\cap\operatorname{\mathcal{W}^{cs}}(\zeta_{\xi,\eta_{n}}))$ is the manifold $p(\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\lim_{n}\zeta_{\rho(\gamma_{n})\xi,\eta_{n}})$. The limit of $p(\operatorname{\mathcal{W}^{cu}}(\rho_{0}(\gamma_{n})\eta_{1})\cap\operatorname{\mathcal{W}^{cs}}(\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}))$ is the manifold $p(\operatorname{\mathcal{W}^{cu}}(\eta_{0})\cap\operatorname{\mathcal{W}^{cs}}(\lim_{n}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}))$. As the two limit manifolds are different, $\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}\neq\lim_{n\to\infty}\rho_{0}(\gamma_{n})\zeta_{\xi,\eta_{1}}$. This completes the proof of the claim. ∎

Using the previous lemma, we can now define a continuous $(\rho,\rho_{0})$-equivariant map whenever $\rho$ is sufficiently close to $\rho_{0}$.

Proposition 7.18**.**

Suppose $\rho$ is sufficiently close to $\rho_{0}$. Then there is a $(\rho,\rho_{0})$-equivariant map $\varphi_{\rho}:G/Q\to G/Q$.

Proof.

The map $\varphi_{\rho}$ is defined as follows. For every $\xi\in G/Q$, let $\eta\in G/Q$ be such that $\mathcal{Q}_{\xi,\eta}$ is not empty. By Proposition 7.10, there is a unique $\zeta\in G/Q$ such that $\mathcal{Q}_{\xi,\eta}$ is contained in a bounded neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$. We then define $\varphi_{\rho}(\xi)=\zeta$. By Lemma 7.17, this map is well-defined as $\zeta$ does not depend on the choice of $\eta$.

Since $\widetilde{f}$ is $(\widehat{\rho},\widehat{\rho}_{0})$-equivariant,

[TABLE]

Thus, if $\mathcal{Q}_{\xi,\eta}$ is contained in a uniform neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta)$ then $\mathcal{Q}_{\rho(\gamma)\xi,\rho_{0}(\gamma)\eta}$ is contained in a uniform neighborhood of $\widehat{\rho}_{0}(\gamma)(\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\zeta))=\operatorname{\mathcal{W}^{cu}}(\rho_{0}(\gamma)\eta)\cap\operatorname{\mathcal{W}^{cs}}(\rho_{0}(\gamma)\zeta)$. It follows that $\varphi_{\rho}(\rho(\gamma)\xi)=\rho_{0}(\gamma)\varphi_{\rho}(\xi)$. ∎

7.2. Local semirigidity and continuity of $\varphi_{\rho}$

Proof of Theorem 7.1.

We let $\varphi_{\rho|_{\Gamma_{0}}}:G/Q\to G/Q$ be the map obtained from Proposition 7.18. We will show in Lemma 7.19 and Lemma 7.20 that $\varphi_{\rho|_{\Gamma_{0}}}$ is continuous and tends to $\operatorname{id}$ when $\rho\to\rho_{0}$.

The surjectivity of $\varphi_{\rho|_{\Gamma_{0}}}$ then follows from minimality of the ${\rho_{0}|_{\Gamma_{0}}}$-action, the $({\rho|_{\Gamma_{0}}},{\rho_{0}|_{\Gamma_{0}}})$-equivariance and the continuity of $\varphi_{\rho|_{\Gamma_{0}}}$. Since ${\rho_{0}|_{\Gamma_{0}}}$ is a uniformly expanding action, we can now apply Lemma 4.5 to prove the uniqueness of $\varphi_{\rho|_{\Gamma_{0}}}$ semi-conjugacy provided it is close enough to $\operatorname{id}$ and $\rho$ is close enough to $\rho_{0}$. ∎

For the next two lemmas, to simplify notations, we assume that the lattice $\Gamma$ is torsion-free.

Lemma 7.19**.**

Suppose $\rho$ is sufficiently close to $\rho_{0}$. Then the map $\varphi_{\rho}$ is continuous.

Proof.

Let $\xi_{0}\in G/Q$. We pick $\eta\in G/Q$ such that $(\xi_{0},\eta)\in\mathcal{U}$, or equivalently $\mathcal{Q}_{\xi_{0},\eta}\neq\varnothing$. If $(\xi_{n})$ is a sequence converging to $\xi_{0}$ then $(\xi_{n},\eta)\in\mathcal{U}$ for sufficiently large $n$ as $\mathcal{U}$ is an open set. By Lemma 7.14, the sequence $\mathcal{Q}_{\xi_{n},\eta}$ converges to $\mathcal{Q}_{\xi_{0},\eta}$ in the pointed Hausdorff topology. Let $x_{0}\in\mathcal{Q}_{\xi_{0},\eta}$. Then for every $\epsilon>0$ and for every $R>0$, there exists $n_{0}$ such that for every $n>n_{0}$,

[TABLE]

On the other hand, by Proposition 7.10, there is $C>0$ such that $\mathcal{Q}_{\xi_{n},\eta}$ is contained in the $C$-neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\varphi(\xi_{n}))$, and $\mathcal{Q}_{\xi_{0},\eta}$ is contained in the $C$-neighborhood of $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\varphi(\xi_{0}))$. In $\mathcal{Q}_{\xi_{n},\eta}$ and $\mathcal{Q}_{\xi_{0},\eta}$, we pick two biLipschitz flats $BF_{n}$ and $BF_{0}$ whose intersections with $B(x_{0},R)$ are $\epsilon$-close. By Lemma 6.1 and Proposition 7.7, when $\rho$ is close enough to $\rho_{0}$ so that the constant $L$ given there is close enough to $1$, the biLipschitz flat $BF_{n}$ is $C$-close to a flat $F_{n}$ in $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\varphi(\xi_{n}))$, and the biLipschitz flat $BF_{0}$ is $C$-close to a flat $F_{0}$ in $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\varphi(\xi_{0}))$. Thus, intersections of flats $F_{n}$ and $F_{0}$ with the ball $B(x_{0},R)$ are $(C+\epsilon)$-Hausdorff close. In other words, center manifolds $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\varphi(\xi_{n}))$ and $\operatorname{\mathcal{W}^{cu}}(\eta)\cap\operatorname{\mathcal{W}^{cs}}(\varphi(\xi_{0}))$ contain flats that are uniformly close on large balls for sufficiently large $n$. As the backward endpoints are $\eta$ in each case, it follows that the center manifolds are close in the pointed Hausdorff topology. Therefore, $\varphi(\xi_{n})$ converges to $\varphi(\xi_{0})$. This shows the continuity of $\varphi$. ∎

Lemma 7.20**.**

The map $\varphi_{\rho}$ tends to $\operatorname{id}$ uniformly as $\rho\to\rho_{0}$.

Proof.

We will use the subscript $\rho$ for $\widetilde{f}$ and $\mathcal{Q}$ to indicate the dependence on the action $\rho$. In particular, we will use the notations $\widetilde{f}_{\rho}$ and $\mathcal{Q}_{\rho,\xi,\eta}=\widetilde{f}_{\rho}(X\times\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta)$. Note that $\mathcal{Q}_{\rho_{0},\xi,\eta}=\operatorname{\mathcal{W}^{cs}}(\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta)$ and $\varphi_{\rho_{0}}=\operatorname{id}$. Then we have point-wise convergence of $\varphi_{\rho_{n}}$ to $\varphi_{\rho_{0}}$ for any sequence $\rho_{n}\to\rho_{0}$. Indeed, if $\rho_{n}\to\rho_{0}$, then Lemma B.3 implies that $\varphi_{\rho_{n}}(\xi)\to\varphi_{\rho_{0}}(\xi)=\xi$. We will prove that the convergence is uniform in $\rho$, i.e. for any $\epsilon>0$, there exists a neighborhood of $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon)$ of $\rho_{0}$ such that $\sup_{\xi\in G/Q}d(\varphi_{\rho}(\xi),\xi)<\epsilon$ for any $\rho\in U$.

We let $D$ be the chosen fundamental domain in the construction of $\widetilde{f}_{\rho}$ in Proposition 7.3 and Proposition 3.9. Let $x_{0}$ be a point in $D$. Let $d_{G/Q}$ be any Riemannian metric on $G/Q$. Since $G/Q$ is compact, the choice of a Riemannian metric on $G/Q$ will not affect the argument.

For every $\xi\in G/Q$, pick $\eta_{x_{0},\xi}\in G/Q$ such that the Weyl chamber $Q$-face $\mathcal{C}_{x_{0}}(\eta_{x_{0},\xi})$ based at $x_{0}$, is opposite to the Weyl chamber $Q$-face $\mathcal{C}_{x_{0}}(\xi)$ based at $x_{0}$. Observe that $\operatorname{\mathcal{W}^{cs}}(\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})\neq\varnothing$.

Claim**.**

There exist constants $\epsilon^{\prime},d^{\prime}>0$ such that for every $\epsilon<\epsilon^{\prime}$, there exists a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon)$ of $\rho_{0}$ such that: for every $\rho\in\operatorname{\mathcal{U}}$, the set $\mathcal{Q}_{\rho,\xi,\eta_{x_{0},\xi}}\neq\varnothing$ and contains a point of distance at most $d^{\prime}\epsilon$ from $p^{-1}(x_{0})\cap\operatorname{\mathcal{W}^{cs}}(\xi)$. Furthermore, the constants $\epsilon^{\prime}$, $d^{\prime}$ and the neighborhood $\operatorname{\mathcal{U}}$ are independent of $x_{0}$ and $\xi$.

Proof of Claim.

Set $\eta:=\eta_{x_{0},\xi}$. Let $\overline{x_{0}}\in\operatorname{\mathcal{W}^{cs}}(\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta)$ be such that $p(\overline{x_{0}})=x_{0}$. Let $m=\dim(X)$ and $k=\dim(G/Q)$. Pick a smooth local trivialization $\Psi_{\overline{x_{0}}}:U_{\overline{x_{0}}}\to(-1,1)^{m}\times(-1,1)^{k}$ where $U_{\overline{x_{0}}}$ is a neighborhood of $\overline{x_{0}}$. We can assume that $\Psi_{\overline{x_{0}}}(x_{0})=0^{m+k}$.

In this local trivialization, $\operatorname{\mathcal{W}^{cs}}(\xi)$ and $\operatorname{\mathcal{W}^{cu}}(\eta)$ are graphs $G_{1}$ and $G_{2}$ of two smooth functions defined on $(-1,1)^{m}$. Moreover, $G_{1}$ and $G_{2}$ intersect transversely along the center manifold, whose dimension is $m-k$. Let $\pi_{k}:(-1,1)^{m}\to(-1,1)^{k}$ be the projection to the first $k$ coordinates. Then we can choose smooth fiber-preserving coordinate change such that

[TABLE]

and

[TABLE]

Then $G_{1}\cap G_{2}=\left\{(0^{k},w,0^{k}):w\in(-1,1)^{m-k}\right\}$.

Observe that for any $\epsilon_{1}$ sufficiently small, there exist $\epsilon_{1}^{\prime}$ such that

[TABLE]

Now suppose $G_{3}$ is the graph of a continuous function defined on $(-1,1)^{m}$ and suppose that $G_{3}$ is $\epsilon_{2}$ close to $G_{1}$ for some $\epsilon_{2}>0$. In particular, this means that $G_{3}=\left\{(v,-\pi_{k}(v)+q(v)):v\in(-1,1)^{m}\right\}$ where $q:(-1,1)^{m}\to(-1,1)^{k}$ is a continuous function for which \left\lVert\vbox{\hbox{\displaystyle q(v)}}\right\rVert<\epsilon_{2}. We claim that there exists $\delta<\epsilon_{1}$ such that, whenever $\epsilon_{2}<\delta$ , then $G_{3}\cap G_{2}\neq\varnothing$ and $G_{3}\cap G_{2}$ contains a point that is $\epsilon_{1}^{\prime}$ close to the image of $\overline{x_{0}}$ in this trivialization.

To prove this claim, note that $G_{3}\cap G_{2}\neq\emptyset$ if and only if $q(v)=2\pi_{k}(v)$ for some $v\in(-1,1)^{m}$. Suppose this claim is false. Then there is a map $h:(-1,1)^{m}\to S^{k-1}(1)$ to the unit sphere $S^{k-1}(1)$ given by

[TABLE]

Let us restrict $h$ to the subset $S:=S^{k-1}(4\epsilon_{2})\times 0^{m-k}\subset(-1,1)^{m}$, which is homeomorphic with $S^{k-1}$. When $\epsilon_{2}$ is sufficiently small, $h|_{S}$ is close, and thus homotopic, to the $\frac{1}{4\epsilon_{2}}$-homothety $S^{k-1}(4\epsilon_{2})\to S^{k-1}(1)$.

Thus $h|_{S}$ has degree one. This is impossible as $h$ extends to a map of the $4\epsilon_{2}$-ball in $\operatorname{\mathbb{R}}^{k}$ to $S^{k-1}$ and hence must have degree zero. This shows that $G_{3}\cap G_{2}\neq\varnothing$.

Moreover, the above arguments further show that in fact, there exists $(w,0^{m-k})\in(-1,1)^{m}$ with \left\lVert\vbox{\hbox{\displaystyle w}}\right\rVert<4\epsilon_{2} for which $q(w,0)=2w$. In particular, this implies that $G_{3}\cap G_{2}$ contains a point at distance at most $4\epsilon_{2}$ to $0^{m+k}$.

Now we will explain how this above discussion proves the claim. Consider a sufficiently small neighborhood $\operatorname{\mathcal{U}}^{\prime}$ of $\rho$. For $\rho\in\operatorname{\mathcal{U}}^{\prime}$, set $G_{3}=\Psi_{\overline{x_{0}}}(\widetilde{f}_{\rho}(X\times\xi))$. Set $\epsilon^{\prime}:=\delta$. Given any $\epsilon<\epsilon^{\prime}$, set $\epsilon_{2}:=\epsilon$. Then choose a neighborhood $\operatorname{\mathcal{U}}=\operatorname{\mathcal{U}}(\epsilon)\subset\operatorname{\mathcal{U}}^{\prime}$ of $\rho_{0}$ such that $G_{3}$ is $\epsilon_{2}$ close to $G_{1}$. Then $\Psi_{\overline{x_{0}}}(\operatorname{\mathcal{Q}}_{\rho,\xi,\eta_{x_{0},\xi}}\cap\>U_{\overline{x_{0}}})=G_{3}\cap G_{2}$ is non-empty and contains a point at a distance at most $4\epsilon_{2}$ from $0=\Psi_{\overline{x_{0}}}(\overline{x_{0}})$. Then $\operatorname{\mathcal{Q}}_{\rho,\xi,\eta_{x_{0},\xi}}$ is non-empty and contains a point within a distance $d^{\prime}\epsilon_{2}$ from $\overline{x_{0}}$ where $\epsilon=\epsilon_{2}<\delta(=\epsilon^{\prime})$ and $d^{\prime}$ is a constant that depends on the chart $\Psi_{x_{0}}$. Thus the first part of the claim holds. But as $G$ acts transitively on $X_{G/Q}$ by isometries, we can pre-compose this fixed chart $\Psi_{\overline{x_{0}}}$ with appropriate isometries and assume that the constants $\epsilon^{\prime},d^{\prime}$ and the neighborhood $\operatorname{\mathcal{U}}$ are independent of $x_{0}$ and $\xi$. ∎

Now we use the Claim to prove the lemma. First fix $\epsilon^{\prime},d^{\prime}$ as in the Claim above.

Proposition 7.3 part (3) implies that for every $\epsilon>0$, there is an open neighborhood $U=U(x_{0},\xi)$ of $\rho_{0}$ such that for every $\rho\in U$, we have

[TABLE]

Choose $\epsilon<\epsilon^{\prime}$ small enough so that $d_{G/Q}(\pi_{x_{0}}(\widetilde{f}_{\rho}(x_{0},\xi),\xi)<\epsilon$ implies that $\operatorname{\mathcal{W}^{cs}}(\xi)$ and $\operatorname{\mathcal{W}^{cs}}(\pi_{x_{0}}(\widetilde{f}(x_{0},\xi)))$ are sufficiently close on a ball. More precisely, for every $R>0$ and $\epsilon>0$, there exists a neighborhood $U$ of $\rho_{0}$ such that for every $\rho\in U$,

[TABLE]

On the other hand, Proposition 7.3 part (4) implies that for any $\epsilon,R>0$, there is a neighborhood $U$ of $\rho_{0}$ such that for every $\rho\in U$,

[TABLE]

Thus triangle inequality implies that for any $\epsilon,R>0$, there exists a neighborhood $U$ of $\rho_{0}$ such that for any $\rho\in U$,

[TABLE]

It follows that $\widetilde{f}_{\rho}(\overline{B_{R}(x_{0})},\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})$ is contained in a $2\epsilon$-neighborhood of $\operatorname{\mathcal{W}^{cs}}(\xi)\cap\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi}))$. On the other hand, by Proposition 7.10, $\widetilde{f}_{\rho}(X\times\{\xi\})\cap\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})$ is contained in a $C$-neighborhood of $\operatorname{\mathcal{W}^{cs}}(\varphi_{\rho}(\xi))\cap\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})$, where $C$ is independent of $\rho$ in a sufficiently small neighborhood of $\rho_{0}$.

We fix $R>4C$ and $\epsilon<\epsilon^{\prime}$. Let $U$ be the neighborhood of $\rho_{0}$ as in Eq. 7.1. By further shrinking $U$ if necessary, we can assume that $L$ in Proposition 7.7 is sufficiently close to 1. We identify $\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})$ with the symmetric space $X$. All considered sets and points are subsets and points of $\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})$. Thus they can be considered as subsets and points of $X$. By the Claim above there exists $x_{0}^{\prime}\in\mathcal{Q}_{\rho,\xi,\eta_{x_{0},\xi}}$ of distance at most $d^{\prime}\epsilon$ from $x_{0}$. Thus $x_{0}$ is in a $(C+d^{\prime}\epsilon)$-neighborhood of the center manifold $\operatorname{\mathcal{W}^{cs}}(\varphi_{\rho}(\xi))\cap\operatorname{\mathcal{W}^{cu}}(\eta_{x_{0},\xi})$.

We denote by $\gamma_{x_{0},\xi}$ the geodesic ray from $x_{0}$ in the center direction of the Weyl chamber $Q$-face $\xi$ based at $x_{0}$. We let $y_{R}\in\gamma_{x_{0},\xi}$ be a point of distance $\frac{R}{2}$ from $x_{0}$. Let $z_{R}\in\mathcal{Q}_{\rho,\xi,\eta_{x_{0},\xi}}$ that is at most $d^{\prime}\epsilon$ away from $y_{R}$. We let $BF\subset\mathcal{Q}_{\rho,\xi,\eta_{x_{0},\xi}}$ be an $L$-biLipschitz flat containing $z_{R}$. Let $F$ be the unique flat in $X$ that is at most $C$-Hausdorff distance from $BF$, see Lemma 6.1. The flat $F$ is asymptotic to $\eta_{x_{0},\xi}$ and $\varphi_{\rho}(\xi)$ as opposite Weyl chamber $Q$-faces.

Let $w_{R}\in F$ be a point of distance at most $C$ from $z_{R}$. Recall that $x_{0}^{\prime}\in\mathcal{Q}_{\rho,\xi,\eta_{x_{0},\xi}}$ is at most $d^{\prime}\epsilon$ far away from $x_{0}$. Let $x_{0}^{\prime\prime}\in F$ be a point of distance at most $C$ from $x_{0}^{\prime}$. We denote by $\gamma_{x_{0},\eta_{x_{0},\xi}}$ and $\gamma_{x_{0}^{\prime\prime},\eta_{x_{0},\xi}}$ the geodesic rays in the center directions of Weyl chamber $Q$-faces $\eta_{x_{0},\xi}$ based at $x_{0}$ and $x_{0}^{\prime\prime}$, respectively. We note that the rays $\gamma_{x_{0},\eta_{x_{0},\xi}}$ and $\gamma_{x_{0}^{\prime\prime},\eta_{x_{0},\xi}}$ are at Hausdorff distance at most $C+d^{\prime}\epsilon$. Let $\gamma_{w_{R},\eta_{x_{0},\xi}}$ be the geodesic ray initiating from $w_{R}$ in the center direction of Weyl chamber $Q$-face $\eta_{x_{0},\xi}$. The two geodesic rays $\gamma_{y_{R},\eta_{x_{0},\xi}}$ and $\gamma_{w_{R},\eta_{x_{0},\xi}}$ are $(C+d^{\prime}\epsilon)$-Hausdorff close. It follows that, in the flat $F$, the two parallel geodesic rays $\gamma_{w_{R},\eta_{x_{0},\xi}}$ and $\gamma_{x_{0}^{\prime\prime},\eta_{x_{0},\xi}}$ have distance at most $2(C+d^{\prime}\epsilon)$. Hence, the geodesic segment $[x_{0}^{\prime\prime},w_{R}]$ is contained in a $2(C+d^{\prime}\epsilon)$-neighborhood of geodesic rays initiating from $x_{0}^{\prime\prime}$ in the center direction of $\varphi_{\rho}(\xi)$. Therefore, the intersections of the Weyl chamber $Q$-faces $\xi$ and $\varphi_{\rho}(\xi)$ based at $x_{0}$ with the ball radius $\frac{R}{3}$ are $4(C+d^{\prime}\epsilon)$-Hausdorff close. We note that $R$ is independent of $\xi$. As $R>4C$ is arbitrary, we can let $R\to\infty$ when the neighborhood $U$ of $\rho_{0}$ shrinks further. Thus as $\rho\to\rho_{0}$, the semiconjugacy $\varphi_{\rho}$ converges uniformly to $\varphi_{\rho_{0}}=\text{id}$. ∎

8. Applications of the Main Theorem: Differentiable and Lipschitz Local Rigidity

In this section, we give quick applications of our main result to local rigidity of projective actions of uniform lattices in higher rank in both the Lipschitz and $C^{1}$-topology. This yields well-known results by Kanai [Kan96] and Katok-Spatzier [KS97] for the $C^{1}$-topology and a recent one by Kapovich-Kim-Lee [KKL22] in the Lipschitz topology. The applications are deduced from Theorem 7.1 and the fact that the $\rho_{0}$ action of $\Gamma$ on $G/Q$ by left multiplication is a uniformly expanding action.

We first prove Corollary 1.2 which we restate here from the introduction. This recovers the result of [KKL22] for Lipschitz neighborhoods of the standard action.

Corollary 1.2.

Let $\Gamma<G$ be a uniform lattice in a connected linear semisimple Lie group $G$ of higher rank and without compact factors. Let $Q$ be a parabolic subgroup of $G$, and let $\rho_{0}$ be the action of $\Gamma$ on $G/Q$ by left translation. Then there is a neighborhood $U$ of $\rho_{0}$ in Lipschitz topology such that every $\rho\in U$ is $C^{0}$-conjugate to $\rho_{0}$.

Proof.

By Theorem 7.1, there is a neighborhood $U$ of $\rho_{0}$ such that every $\rho\in U$, there exists a semi-conjugacy $\varphi_{\rho}$. Proposition 4.8 shows that $\varphi_{\rho}$ is a conjugacy. ∎

If we restrict to $C^{1}$ neigborhoods of the standard action, we obtain the following corollary, which was proved by Katok and Spatzier [KS97, Theorem 17] and by Kanai under additional assumptions [Kan96]. Similar assertions have been conjectured for semisimple Lie groups of rank 1 but remain unresolved.

Corollary 1.3.

Let $\Gamma<G$ be a uniform lattice in a connected linear semisimple Lie group $G$ of higher rank and without compact factors. Let $Q$ be a parabolic subgroup of $G$, and let $\rho_{0}$ be the action of $\Gamma$ on $G/Q$ by left translation. Then there is a neighborhood $U$ of $\rho_{0}$ of smooth actions that are close to $\rho_{0}$ in $C^{1}$-topology such that every $\rho\in U$ is $C^{\infty}$-conjugate to $\rho_{0}$.

Proof.

By Theorem 7.1, there is such a neighborhood $U$ of $\rho_{0}$ such that every $\rho\in U$ is semi-conjugate to $\rho_{0}$. We claim that the semi-conjugacy maps are actually conjugacies. Indeed, this follows immediately from Proposition 4.8.

Now apply [GS18, Theorem 1.2] to the $C^{0}$ conjugacies to conclude that the conjugacies are $C^{\infty}$. ∎

9. Actions on the Geodesic Boundary

While in rank one symmetric spaces, geodesic boundaries coincide with Furstenberg boundaries, the situation in higher rank is different. The geodesic boundary of a higher rank symmetric space $X$ is diffeomorphic to a sphere $S^{\dim(X)-1}$, and always has dimension strictly larger than the dimension of any Furstenberg boundary. Our Theorem 1.1 shows that local semi-rigidity hold for standard actions on Furstenberg boundaries. However, we now prove Theorem 1.4 which shows that local semi-rigidity fails for standard actions on geodesic boundaries. First let us recall the statement.

Theorem 1.4.

Let $\Gamma$ be a lattice for a higher rank symmetric space of noncompact type $X$, not necessarily uniform. Then the standard action of $\Gamma$ on $\partial_{\infty}X$ is not locally semi-rigid among $C^{k}$ actions for any $k\in[0,\infty]$.

Before we start to construct perturbed actions that are not semi-conjugate to the standard ones, let us explain the structure of geodesic boundaries of higher rank symmetric spaces.

The geodesic boundary $\partial_{\infty}X$ of $X$ has the structure of a spherical building. It is the union of closed Weyl chamber simplices of dimension $\operatorname{rk}(X)-1$ where $\operatorname{rk}(X)$ is the rank of the symmetric space. The set of Weyl chamber simplices can be identified with the Furstenberg boundary $G/P$, where $G$ is a semisimple Lie group associated to $X$ and $P$ is a minimal parabolic subgroup of $G$. On $\partial_{\infty}X$ there is a visual topology and a Tits metric. All Weyl chamber simplices are isometric with respect to the Tits metric. We fix a model Weyl chamber $W$. Then there is a continuous map with respect to visual topology $\phi:\partial_{\infty}X\to W$ such that $\phi$ is an isometry with respect to the Tits metric on each Weyl chamber simplex. If we identify $W$ with a Weyl chamber in $\partial_{\infty}X$ and identify $\partial_{\infty}X=\cup_{g\in G}gW$, then the map $\phi$ coincides with translating points in $\partial_{\infty}X$ into $W$ by elements of $G$. Indeed, $G$ acts transitively on the Weyl chambers with stabilizer $P$, and $P$ itself acts by the identity on $W$.

On $G/P$, we let $\mu$ be a Haar measure. The action of $G$ on $G/P$ preserves the measure class of $\mu$. We thus have the following smooth Radon-Nikodym derivative cocycle $c:G\times G/P\to\mathbb{R}$, defined by

[TABLE]

Since $G$ does not preserve a probability measure on $G/P$, this cocycle is not trivial. Note also that for every $\alpha\in[0,+\infty)$, $c^{\alpha}$ is also a smooth cocycle.

Construction of perturbed actions.

We construct new $G$-actions with the following properties:

1. (1)

each $g\in G$ maps each closed Weyl chamber simplex to a closed Weyl chamber simplex by a homeomorphism ($C^{k}$-diffeomorphism resp.), 2. (2)

the induced action on $G/P$ is the standard action $\rho_{0}$ of $G$ on $G/P$.

We thus only need to define how $G$ acts on the model Weyl chamber simplex $W$. For this, first we choose a center $w\in\operatorname{int}(W)$, and pick a continuous homeomorphism $\tau:W\to\mathbb{R}^{\operatorname{rk}(X)-1}\cup\partial_{\infty}\mathbb{R}^{\operatorname{rk}(X)-1}$ with $\tau(w)=0$. For every $\delta\neq 0$, we let $D_{\delta}:\mathbb{R}^{\operatorname{rk}(X)-1}\to\mathbb{R}^{\operatorname{rk}(X)-1}$ be the linear dilation by factor $\delta$. It is clear that $D_{\delta}$ extends to a homeomorphism on $\mathbb{R}^{\operatorname{rk}(X)-1}\cup\partial_{\infty}\mathbb{R}^{\operatorname{rk}(X)-1}$ that fixes $\partial_{\infty}\mathbb{R}^{\operatorname{rk}(X)-1}$ pointwise. By pulling back by $\tau$, i.e. the map $\tau^{*}D_{\delta}=\tau^{-1}\circ D_{\delta}\circ\tau$, we obtain a homeomorphism on $W$ that fixes $w$ and $\partial W$ pointwise. Since points on $\partial W$ are fixed, $\tau^{*}D_{\delta}$ extends to a homeomorphism, still denoted by $\tau^{*}D_{\delta}$, from $W\cup\partial W$ into itself by fixing every point in $\partial W$ and $w$.

Now let $g\in G$ and $\mathcal{C}\subset\partial_{\infty}X$ be a Weyl chamber. We define the homeomorphism $\kappa_{g}^{\alpha}:\mathcal{C}\to g\mathcal{C}$ such that

[TABLE]

where $\mathcal{C}$ here is regarded as an element of $G/P$. In other words, $\kappa_{g}^{\alpha}$ from $\mathcal{C}$ to $g\mathcal{C}$ is just the pulling back of $\tau^{*}D_{c(g,\mathcal{C})^{\alpha}}$ by the model map $\phi$.

Explicitly, for $g\in G$, the map $\kappa_{g}^{\alpha}:\partial_{\infty}X\to\partial_{\infty}X$ defined as follows: if $x\in hW$ for an $h\in G$, then

[TABLE]

To extend $\kappa_{g}^{\alpha}$ to all of $\partial_{\infty}X$, note that $\kappa_{g}^{\alpha}$ agrees with the standard action on Weyl chamber faces. Indeed, the map $\tau^{*}D_{c(g,hW)^{\alpha}}$ is the identity on the Weyl chamber faces. This matters since the cocycle $c(g,hW)$ is not constant on the star of a face. From this it follows easily that $\kappa_{g}^{\alpha}$ is a homeomorphism.

For any fixed $k\in[0,\infty]$, we observe that, for a good choice of $\tau$, the map $\tau^{*}D_{\delta}$ is even $C^{k}$ and agrees with the standard action on $\partial W$ up to order $k$. Hence the action $g\mapsto\kappa_{g}^{\alpha}$ is also $C^{k}$.

Remark 9.1**.**

We can use flows other than the $D_{\delta}$ on the model Weyl chamber to define actions of $G$ on $G/P$ as long as the flows are the identity on the Weyl chamber faces. We suspect that such flows could lead to more semi-conjugacy classes.

Proof of Theorem 1.4.

Fix $k\in[0,\infty]$. The map $\rho_{\alpha}:g\mapsto\kappa_{g}^{\alpha}\in\operatorname{Diff}^{k}(\partial_{\infty}X)$ constructed above defines an action of $G$ on $\partial_{\infty}X$ by $C^{k}$-diffeomorphisms. Furthermore, when $\alpha$ varies from 0 to $+\infty$, we obtain a continuous deformation of the standard action. The actions of $\Gamma$ are just the restrictions of $G$-actions to $\Gamma$.

We prove that for $0\neq\alpha$, then the action $\rho_{\alpha}$ is not semi-conjugate to $\rho_{0}$. Suppose, for the sake of contradiction, that there exists a continuous and surjective map $\varphi:\partial_{\infty}X\to\partial_{\infty}X$ such that $\rho_{0}\circ\varphi=\varphi\circ\rho_{\alpha}$. We first need the following lemma.

Lemma 9.2**.**

Let $\mathcal{C}$ be a Weyl chamber such that there exists $\gamma\in\Gamma$ stabilizing $\mathcal{C}$ with $c(\gamma,C)\neq 1$. Then $\phi(\varphi(\mathcal{C}))=\{\phi(\varphi(c))\}$ where $c$ is the center of $\mathcal{C}$.

Proof.

Let $c\in\mathcal{C}$ be the center, namely the point such that $\phi(c)=w\in W$. For all $\alpha\in[0,\infty)$, by construction $\rho_{\alpha}(\gamma)$ stabilizes $\mathcal{C}$ since it has the same quotient action on $G/P$ as $\rho_{0}$. Moreover $c$ and $\partial\mathcal{C}$ are pointwise fixed by both $\rho_{\alpha}(\gamma)$ and $\rho_{0}(\gamma)$. Hence the same conclusions hold for $\rho_{\alpha}(\gamma^{n})$ for $n\in\mathbb{Z}$.

For any other point $x\in\operatorname{int}(\mathcal{C})$, after possibly swapping $\gamma$ with $\gamma^{-1}$, we have $\lim_{n\to\infty}\rho_{\alpha}(\gamma^{n})(x)=c$. Hence by continuity $\varphi(c)=\lim_{n\to\infty}\varphi(\rho_{\alpha}(\gamma^{n})x)=\lim_{n\to\infty}\rho_{0}(\gamma^{n})\varphi(x)$. It follows that,

[TABLE]

for every $x\in\operatorname{int}(\mathcal{C})$. By continuity of $\phi\circ\varphi$, $\phi(\varphi(\mathcal{C}))=\{\phi(\varphi(c))\}$. ∎

We continue the proof of Theorem 1.4. Let $\mathcal{C}$ be a Weyl chamber as in Lemma 9.2. Such $\mathcal{C}$ and associated $\gamma$ always exist by work of Prasad and Raghunathan [PR72, Theorem 2.8]. Then for every $\delta\in\Gamma$, $\delta\cdot\mathcal{C}$ also satisfies the assumption of Lemma 9.2. By the minimality of $\Gamma$-action on $G/P$, Lemma 9.2 holds for every Weyl chamber.

Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be two adjacent Weyl chambers and let $c_{1}$ and $c_{2}$ be their centers. We have that $\phi(\varphi(\mathcal{C}_{1}))=\{\phi(\varphi(c_{1}))\}$ and $\phi(\varphi(\mathcal{C}_{2}))=\{\phi(\varphi(c_{2}))\}$. Since $\varphi(\mathcal{C}_{1})$ and $\varphi(\mathcal{C}_{2})$ have non-empty intersection, the point $\phi(\varphi(c_{1}))$ coincides with $\phi(\varphi(c_{2}))$.

Note that for any pair of Weyl chambers, there exists a chain (gallery) of Weyl chambers from one to the other such that any two consecutive chambers in the chain are adjacent. It follows that there exists $z\in W$ such that $\phi(\varphi(\mathcal{C}))=\{z\}$ for every Weyl chamber $\mathcal{C}$. Since $\varphi$ is surjective, we obtain $\phi(\partial_{\infty}X)=\phi(\cup_{\mathcal{C}}\varphi(\mathcal{C}))=\{z\}$, a contradiction. Therefore, the action $\rho_{\alpha}$ is not semiconjugate to $\rho_{0}$. ∎

Appendix A Barycenters and Estimates

For the rest of this section, let us fix a compact, connected $C^{k}$ manifold $F$ where $k\geq 3$. We will study how the barycenter maps on $F$ vary as we vary measures and Riemannian metrics on $F$ along a parametrized family.

A.1. Two lemmas from differential geometry

The goal of this section is to establish two basic lemmas that will be used in the next section for estimating derivatives of the barycenter map. In this section, assume that $F$ is equipped with a Riemannian metric $g$ with associated norm \left\lVert\vbox{\hbox{\displaystyle\cdot}}\right\rVert. We will denote by $\nabla$ the Riemannian connection on $F$ and by $D$ the covariant derivative operator, namely if $z\in F$ and $X,Y$ are smooth vector fields on $F$, then $D_{z}Y(X)=(\nabla_{X}Y)(z)$. Moreover, if $f:F\to\operatorname{\mathbb{R}}$ is a smooth function, then $\nabla f$ is the vector field dual to the 1-form $Df$, i.e. for any $v\in T_{z}F$, $g(\nabla_{z}f,v)=D_{z}f(v)$.

Lemma A.1**.**

Suppose the sectional curvatures of $F$ are bounded by $-a^{2}\leq\kappa_{F}\leq b^{2}$ for $a,b\geq 0$. For any point $z\in B(x_{0},\frac{\pi}{2b})$, we have

[TABLE]

where $I$ is the identity and $T$ is a $(1,1)$-tensor with \left\lVert\vbox{\hbox{\displaystyle T}}\right\rVert\leq\max\left\{\frac{a^{2}}{3},\frac{b^{2}}{2}\right\}d(x_{0},z)^{2}.

Proof.

The $(1,1)$-form $D_{x=x_{0}}d(x,z)\otimes\nabla_{x=x_{0}}d(x,z)$ has norm at most $1$ since one term is the gradient of the distance function, and the other is its dual. It is equal to $1$ precisely in the radial direction at $x_{0}$ pointing away from $z$, and vanishes on the orthogonal subspace.

Since the points $z$ belong to a convex ball centered at $x_{0}$. Then the second fundamental form $D_{x=x_{0}}\nabla_{x}d(x,z)$ of the sphere $S(z,d(x_{0},z))$ at the point $x_{0}$ is positive semidefinite, and vanishes only in the radial direction $\nabla_{x=x_{0}}d(x,z)$.

Indeed, the $(1,1)$-tensor $d(x_{0},z)D_{x=x_{0}}\nabla_{x}d(x,z)$ is zero in the radial direction and in the tangent directions it is $d(x_{0},z)\cdot II(x_{0})$ where $II(x_{0}):T_{x_{0}}S(z,d(x_{0},z))\to T_{x_{0}}S(z,d(x_{0},z))$ is the second fundamental form of the sphere $S(z,d(x_{0},z))$ at the point $x_{0}$.

If the sectional curvatures are bounded above by $b^{2}$ for $b\geq 0$, then by comparison geometry ([Pet06, Theorem 27]), we may estimate the second fundamental form $II(y(t))$ of $S(z,t)$ at the point $y(t)$ by,

[TABLE]

where the inequality is as positive definite forms and $II(y(t))$ is positive definite for $t<\frac{\pi}{2b}$. We have the expansions

[TABLE]

Moreover, for all $t>0$, $at\coth(at)-1<\frac{a^{2}}{3}t^{2}$ and for all $0<t<\frac{\pi}{2b}$, we have $bt\cot(bt)-1>-\frac{b^{2}}{2}t^{2}$. Choosing $t=d(x_{0},z)$ we thus have the expansion,

[TABLE]

where \left\lVert\vbox{\hbox{\displaystyle T^{\prime}}}\right\rVert\leq\max\left\{\frac{b^{2}}{2},\frac{a^{2}}{3}\right\}d(x_{0},z)^{2}. Combining these two terms establishes the lemma. ∎

If $x,z\in F$ are connected by a unique minimal geodesic, we denote $\lVert^{x}_{z}:T_{z}F\to T_{x}F$ the parallel translation along this geodesic. This applies when $z\in B(x,\frac{\pi}{2b})$ where $b$ is an upper bound for sectional curvature of $F$.

Lemma A.2**.**

Suppose the sectional curvatures of $F$ are bounded by $-a^{2}\leq\kappa_{F}\leq b^{2}$ for $a,b\geq 0$. For any point $x_{0}\in B(z_{0},\frac{\pi}{2b})$,

[TABLE]

where $S$ is a $(1,1)$ tensor with $||S||\leq 2\max\left\{a^{2},b^{2}\right\}d(x_{0},z_{0})^{2}$.

Proof.

We observe that if $d(x,z)<\pi/2b$, then

[TABLE]

This implies that

[TABLE]

Since $x_{0}\in B(z_{0},\pi/2b)$, Lemma A.1 and the previous equation implies that:

[TABLE]

where \left\lVert\vbox{\hbox{\displaystyle T}}\right\rVert\leq\max\left\{\frac{a^{2}}{3},\frac{b^{2}}{2}\right\}d(z_{0},x_{0})^{2}. Then

[TABLE]

where

[TABLE]

In order to finish the proof of this lemma, it suffices to show that:

Claim**.**

\left\lVert\vbox{\hbox{\displaystyle T_{1}}}\right\rVert\leq\max\left\{a^{2},b^{2}\right\}d(x_{0},z_{0})^{2}.

We will now prove this claim. Note that it suffices to show that

[TABLE]

Let $t_{0}:=d(x_{0},z_{0})$. For $0\leq t\leq t_{0}$, let $S(x_{0},t):=\{z:d(x_{0},z)=t\}$. Then, if $z\in S(x_{0},t)$, then $\nabla_{z}d(x_{0},z)$ is the outward pointing unit normal to $S(x_{0},t)$ at $z$ while $\nabla_{x=x_{0}}d(x,z)$ is the inward pointing unit normal to $S(x_{0},t)$ at $z$, parallel translated to $x_{0}$ along the geodesic joining $x_{0}$ and $z$.

Let $v\in T_{z_{0}}F$ and let $\gamma_{v}:(-\varepsilon,\varepsilon)\to F$ be a curve on $S(x_{0},t_{0})$ such that $\gamma_{v}(0)=z_{0}$ and $\dot{\gamma_{v}}(0)=v$. If $v\in(T_{z_{0}}S(x_{0},t_{0}))^{\perp}$, then $T_{1}(v)=\lVert^{x_{0}}_{z_{0}}D_{\dot{\gamma}(t_{0})}(-\dot{\gamma})(t_{0})+D_{\dot{\gamma}(0)}\dot{\gamma}(0)=0$. So we now assume that $v\in T_{z_{0}}S(x_{0},t_{0})$ and consider the geodesic variation given by $p_{v}(s,t):=\exp_{x_{0}}(t\cdot\alpha_{v}(s))$ where $\alpha_{v}(s):=\frac{1}{t_{0}}\exp_{x_{0}}^{-1}(\gamma_{v}(s))$. Then

[TABLE]

Since the map $p_{v}$ is smooth, we can interchange the order of the derivatives above to get that

[TABLE]

where J_{v}(t):=\frac{\partial}{\partial s}\big{|}_{s=0}p_{v}(s,t) is a Jacobi field along the geodesic $c(t):=p_{v}(0,t)$ associated to the geodesic variation $p_{v}(s,.)$ [dC92, section 2, Proposition 2.4]. Also note that $J_{v}(t_{0})=v$. Thus

[TABLE]

However

[TABLE]

where $M:=\sup_{x\in F}||R_{x}||\cdot\sup_{t\in[0,t_{0}]}||J_{v}(t)||$. Note that $\sup_{x\in F}||R_{x}||\leq\max\left\{a^{2},b^{2}\right\}$. Also, by generalized Rauch Comparison (Corollary 2.19 of [Sak84]) we have for $t\leq\frac{\pi}{2b}$, \left\lVert\vbox{\hbox{\displaystyle J_{v}(t)}}\right\rVert\leq\left\lVert\vbox{\hbox{\displaystyle J^{b}(t)}}\right\rVert\leq 1 where $J^{b}(t)$ is any Jacobi field in constant curvature $b^{2}\geq 0$ with conditions $J^{b}(0)=0$ and \left\lVert\vbox{\hbox{\displaystyle J^{b}(t_{0})}}\right\rVert=1=\left\lVert\vbox{\hbox{\displaystyle J_{v}(t_{0})}}\right\rVert. Thus $M\leq\max\left\{a^{2},b^{2}\right\}$.

This establishes the Claim and hence the lemma. ∎

A.2. Derivatives of the barycenter map

The goal of this section is to establish Proposition A.10. Recall that for this entire Appendix, we have fixed a compact, connected $C^{1}$ (hence smooth) manifold $F$.

In the first part of this section (Lemma A.3 to Lemma A.6), we will assume the following standing assumptions.

Assumptions A: Let $a,b\geq 0$ and $n\in\operatorname{\mathbb{N}}$ be fixed constants. Suppose that:

1. (1)

$F$* is equipped with a $C^{2}$ Riemannian metric $g$ whose sectional curvatures are bounded between $-a^{2}\leq\kappa_{F}\leq b^{2}$.* 2. (2)

$\mu=\sum_{i=1}^{n}w_{i}\delta_{z_{i}}$* is a probability measure on $F$ where $0\leq w_{i}\leq 1$ are the weights and $\left\{z_{1},\dots,z_{n}\right\}\subset F$ is the support of $\mu$.* 3. (3)

We have $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<\min\left\{\frac{\pi}{4b},\frac{1}{2}\operatorname{InjRad}_{g}(F)\right\}$, where $\operatorname{InjRad}_{g}(F)$ denotes the injectivity radius of $(F,g)$.

*We remark that any open ball of radius $\min\left\{\frac{\pi}{2b},\frac{1}{2}\operatorname{InjRad}_{g}(F)\right\}$ is strongly geodesically convex (cf. [Cha06, Theorem IX.6.1]). *

Notation for Assumptions A: We standardize the following notation for Assumptions A. We will denote by $\nabla$ the Riemannian connection on $F$ and by \left\lVert\vbox{\hbox{\displaystyle\cdot}}\right\rVert the norm associated to the metric $g$. We will denote by $D$ the covariant derivative operator, namely if $z\in F$, $X$ is a smooth vector field, and $Y$ is a vector field on $F$, then $D_{z}Y(X)=(\nabla_{X}Y)(z)$. Moreover, if $f:F\to\operatorname{\mathbb{R}}$ is a smooth function, then $\nabla f$ is the vector field dual to the 1-form $Df$.

Under Assumptions A, the function $G$ defined in Eq. 2.1) becomes

[TABLE]

We denote by $\operatorname{bar}(\mu)$ denotes the barycenter of $\mu$ (which is defined as the unique point in $F$ such that $G(\operatorname{bar}(\mu),w,z)=0$; see Section 2 for details). This exists and is well defined (see Section 2 or [Cha06, Proposition IX.7.1]).

In what follows, we define a $(1,1)$-tensor $Q$ by

[TABLE]

Lemma A.3**.**

Under Assumptions A**, $\max_{1\leq i\leq n}d(\operatorname{bar}(\mu),z_{i})\leq 2\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}$, and**

[TABLE]

Proof.

By Remark 2.1, $\max_{i}d(\operatorname{bar}(\mu),z_{i})\leq 2\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}$. The second statement of the lemma follows from Lemma A.1 by letting $x_{0}=\operatorname{bar}(\mu)$, $z=z_{j}$ and since $\sum_{j}w_{j}=1$ (which implies that each summand is close to identity). ∎

Noting that $Q$ is a symmetric tensor, we immediately obtain the following from Lemma A.3.

Corollary A.4**.**

Under Assumptions A**, if $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<\frac{1}{3\max\left\{a,b\right\}}$, then \left\lVert\vbox{\hbox{\displaystyle Q-I}}\right\rVert\leq\frac{1}{4} and $Q$ is a strictly positive definite tensor and thus invertible. Moreover, \left\lVert\vbox{\hbox{\displaystyle Q^{-1}}}\right\rVert\leq\frac{4}{3}<2.**

Remark A.5**.**

We allow $\max\left\{a,b\right\}=0$ in the above corollary. In this case $Q=I$.

Lemma A.6** (Derivative of the barycenter with respect to the measure).**

Suppose Assumptions A**. If $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<\frac{1}{3\max\left\{a,b\right\}}$, then the partial derivative of the map $(w_{1},\dots,w_{n},z_{1},\dots,z_{n})\mapsto\operatorname{bar}(\mu)$ in the weight $w_{i}$ is**

[TABLE]

and the partial derivative in $z_{i}$ is

[TABLE]

Proof.

We observe that $0=G(\operatorname{bar}(\mu),w,z)$ and differentiating this we obtain,

[TABLE]

We have already seen from (2.2) that

[TABLE]

Then $D_{x=\operatorname{bar}(\mu)}G(x,w,z)=Q$ (cf. Eq. A.2). Since $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<\frac{1}{3\max\left\{a,b\right\}}$, Corollary A.4 implies that $Q$ is strictly positive definite and hence invertible. Moreover, from Eq. A.1,

[TABLE]

Then we can solve Eq. A.3 to obtain:

[TABLE]

The case of $D_{z_{i}}\operatorname{bar}(\mu)$ is similar. By differentiating $0=G(\operatorname{bar}(\mu),w,z)$ we obtain

[TABLE]

For the second term we observe that

[TABLE]

As $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<\frac{\pi}{4b}$, Lemma A.2 implies: $D_{z_{i}}G(x,w,z)|_{x=\operatorname{bar}(\mu)}=-\sum_{i=1}^{n}w_{i}\lVert^{\operatorname{bar}(\mu)}_{z_{i}}(I+S)$ (where $S$ is as in Lemma A.2). Thus, by similar reasoning as in the proof of Corollary A.4, we see that: there exists $r^{\prime}$ sufficiently small (depending only on the constants $a$ and $b$) such that $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<r^{\prime}$ implies $D_{z_{i}}G(x,w,z)$ is invertible. Then solving for $D_{z_{i}}\operatorname{bar}(\mu)$ from Eq. A.4 completes the result. ∎

In the next two lemmas, we will consider families of metrics $g_{s}$ on $F$ jointly satisfying Assumptions A**. Recall the definition of $C^{1,2}$ family of metrics from Section 2.2.**

We precisely formulate the:

**Assumptions B:

Let $\epsilon_{0}>0$, $a,b\geq 0$ and $m,n\in\operatorname{\mathbb{N}}$ be fixed constants. Suppose that:**

1. (1)

$B(0,\epsilon_{0})\ni s\mapsto g_{s}$** is a family of $C^{1,2}$ Riemannian metrics on $F$ where each $g_{s}$ is a $C^{2}$ Riemannian metric with sectional curvature $\kappa_{s}\in[-a^{2},b^{2}]$.** 2. (2)

for each $s$, there is a probability measure $\mu(s):=\sum_{i=1}^{n}w_{i}(s)\delta_{z_{i}(s)}$ where the weights $w_{i}(s)$ and the points $z_{i}(s)$ depend on the parameter $s\in B(0,\epsilon_{0})$ in a $C^{1}$ way. 3. (3)

We have $\operatorname{diam}_{s}\left\{z_{1}(s),\dots,z_{n}(s)\right\}<\min\left\{\frac{\pi}{4b},\frac{1}{2}\operatorname{InjRad}_{g_{s}}(F),\frac{1}{3\max\left\{a,b\right\}}\right\}$, where $\operatorname{InjRad}_{g_{s}}(F)$ denotes the injectivity radius of $(F,g_{s})$. (We interpret $\frac{\pi}{4b}=\infty$ when $b=0$ and $\frac{1}{3\max\left\{a,b\right\}}=\infty$ when $a=b=0$.)

Notations under Assumptions B: We also standardize the following notation for use henceforth. For each Riemannian metric $g_{s}$, we denote the corresponding Riemannian connection by $\nabla^{s}$, distance function by $d_{s}$, diameter measured in $d_{s}$ by $\text{diam}_{s}$, barycenter map by $\operatorname{bar}_{s}$, and the covariant derivative operator by $D^{s}_{x}$ (i.e. if $X,Y$ are smooth vector fields on $F$ and $x\in F$, then $D^{s}_{x}Y(X)=\nabla^{s}_{X}Y(x)$). Moreover, let $\nabla^{s}_{x}f$ be the gradient of a smooth function $f:F\to\operatorname{\mathbb{R}}$ for the metric $g_{s}$ (i.e. $\nabla^{s}_{x}f(v)=D_{v}f(x)$ for any $v\in T_{x}F$). Finally, $D_{s}$ is the usual covariant derivative operator on $B(0,\epsilon_{0})\subset\operatorname{\mathbb{R}}^{m}$.

Finally, for each $s$, let $Q_{s}$ denote the $(1,1)$ tensor $Q$ (cf. Eq. A.2) that corresponds to the distance function $d_{s}$, i.e.

[TABLE]

where $x_{s}:=\operatorname{bar}_{s}(\mu(s))$.

In the next three lemmas, we will compute the derivatives of the map $s\mapsto\operatorname{bar}_{s}(\mu(s))$ under Assumptions B as above.

Lemma A.7**.**

Suppose Assumptions B part (1)** holds. Then the map $E_{x}:T_{x}F\times B(s_{0},\epsilon)\to F$ defined by $E_{x}(v,s)=\exp^{g_{s}}_{x}(v)$ satisfies:**

1. (a)

it is $C^{1}$ in $x$ and $v$ and the first order derivatives (in $x$ and $v$) are $C^{1}$ in $s$, 2. (b)

$D_{s}E_{x}(v,s)$** is differentiable with respect to $v$ and the mixed partial derivatives with respect to $s$ and $v$ are equal (i.e. $D_{v}D_{s}E_{x}(v,s)=D_{s}D_{v}E_{x}(v,s)$).**

Moreover, for all $z$ and $x$ with $d_{s}(x,z)$ sufficiently small, $d_{s}(x,z)^{2}$ is $C^{2}$ in $x$ and $z$, $C^{1}$ in $s$ and $\nabla_{x}^{s}(d_{s}(x,z))^{2}$ is $C^{1}$ in $s$.

Proof.

By assumption, the Christoffel symbols for $g_{s}$ are $C^{1}$ in $x$ and $s$ and have derivatives up to order one which are also $C^{1}$ in $s$. Regularity of solutions of ODEs (the geodesic equations) implies that $E_{x}(v,s)$ is $C^{1}$ in $x$ and $v$ and its derivatives up to order $1$ are $C^{1}$ functions of $s$. This proves part (a). Part (b) follows from [Rud76, Theorem 9.41, page 236], because $E_{x}(v,s)$ is $C^{1}$ in $(v,s)$ and $D_{v}E_{x}(v,s)$ is $C^{1}$ in $s$.

By the Implicit Function Theorem, there is a map $\alpha_{x}:B(s_{0},\epsilon)\to T_{x}F$ with the same regularity as $E_{x}$ (i.e. $\alpha_{x}(s)$ is $C^{1}$ in both $x$ and $s$) such that $E_{x}(\alpha_{x}(s),s)=z$. However, observe that

[TABLE]

Hence the stated regularity of $d_{s}(x,z)^{2}$ follows. ∎

Corollary A.8**.**

Suppose Assumptions B part (1)** holds and $s\mapsto z(s)$ is $C^{1}$ as $s$ varies in $B(0,\epsilon_{0})$. Then $\nabla_{x}^{s}d_{s}(x,z(s))^{2}$ varies $C^{1}$ in both $s$ and $x$. **

Proof.

We will prove this on $B(s_{0},\epsilon)$ and let $g_{0}:=g_{s_{0}}$. Let $E_{x}(v,s)$ be as above and define $F_{x}(v,s):=(\exp^{g_{0}}_{x})^{-1}\circ E_{x}(v,s)-(\exp^{g_{0}}_{x})^{-1}(z(s))$. Then $F_{x}(v,s)$ is $C^{1}$ in both $x$ and $s$ and further,

[TABLE]

This implies that $D_{v}F_{x}(v,s)$ is invertible whenever \left\lVert\vbox{\hbox{\displaystyle v}}\right\rVert_{g_{0}} sufficiently small. Then, by Implicit Function theorem, there exists a function $\beta_{x}:B(s_{0},\epsilon)\to F$ which is $C^{1}$ in both $x$ and $s$ such that $F_{x}(\beta_{x}(s),s)=0$. Note that $F_{x}(v,s)=0$ if and only if $E_{x}(v,s)=z(s)$. Thus $E_{x}(\beta_{x}(s),s)=z(s).$ Then

[TABLE]

which finishes the proof as $\beta_{x}$ is $C^{1}$ in $s$ and $x$. ∎

Lemma A.9** (Derivative of the barycenter with respect to the metric).**

Suppose Assumptions B** hold. Assume that $w_{i}(s)=w_{i}$ and $z_{i}(s)=z_{i}$ are constant functions for each $i\in\left\{1,\dots,n\right\}$ and $\mu(s)\equiv\mu=\sum_{i=1}^{n}w_{i}\delta_{z_{i}}$. Then the derivative of the map $s\mapsto\operatorname{bar}_{s}(\mu)$ with respect to the parameter $s$ at $s=s_{0}$ is:**

[TABLE]

where $x_{0}:=\operatorname{bar}_{s_{0}}(\mu)$.

Proof.

For every $s$, we define a function $G_{s}(x,w,z)=G(x,w,z,s)$ for the distance function $d_{s}$ as in equation A.1. As in Lemma A.6 we begin by differentiating the equation $0=G(\operatorname{bar}_{s}(\mu),w,z,s)$ to obtain

[TABLE]

From (2.2) we have

[TABLE]

Note that $D_{x=\operatorname{bar}_{s_{0}}(\mu)}^{s_{0}}G(x,w,z,s_{0})=Q_{s_{0}}$ and since $\operatorname{diam}\left\{z_{1},\dots,z_{n}\right\}<\frac{1}{3\max\left\{a,b\right\}}$, Corollary A.4 implies that $Q_{s_{0}}$ is a strictly positive definite (hence invertible) tensor.

Moreover, we compute,

[TABLE]

We may now solve for $D_{s=s_{0}}\operatorname{bar}_{s}(\mu)$ to obtain the result stated. ∎

We now prove Proposition 2.8.

Proposition A.10**.**

Suppose Assumptions B** hold and let $r(s):=\operatorname{diam}_{(F,d_{s})}\left\{z_{1}(s),\dots,z_{n}(s)\right\}$. Then $s\mapsto\operatorname{bar}_{s}(\mu(s))$ is $C^{1}$ and for any $s_{0}\in B(0,\epsilon_{0})$,**

[TABLE]

where

[TABLE]

where \left\lVert\vbox{\hbox{\displaystyle g_{s}}}\right\rVert_{C^{1,2}} represents the bounds of derivatives up to order one in $s$ at $s\in B(0,\epsilon_{0})$ of derivatives up to order two of components of $g_{s}$ with respect to $g_{0}$-exponential coordinates.

Remark A.11**.**

In the above lemma (and its proof below), whenever we have a vector (or a tensor) decorated by $s$, then the corresponding norm \left\lVert\vbox{\hbox{\displaystyle\cdot}}\right\rVert should be understood as being taken with respect to the Riemannian metric $g_{s}$. For example, \left\lVert\vbox{\hbox{\displaystyle T_{s}}}\right\rVert, \left\lVert\vbox{\hbox{\displaystyle D_{s}w_{i}(s)}}\right\rVert,\left\lVert\vbox{\hbox{\displaystyle D_{s}z_{i}(s)}}\right\rVert, etc. are all norms taken with respect to the metric $g_{s}$. We avoid the more precise but cumbersome notation \left\lVert\vbox{\hbox{\displaystyle\cdot}}\right\rVert_{s}.

Proof.

We first note that the function $L(s)$ is well-defined. Indeed both $w_{i}(s)$ and $z_{i}(s)$ vary $C^{1}$ in $s$ and $(s,v)\to g_{s}(v,v)$ is at least $C^{2}$. Moreover Lemma A.7 and Corollary A.8 implies that whenever $d_{s}(x,z_{i}(s))$ is sufficiently small, $d_{s}(x,z_{i}(s))^{2}$ is $C^{2}$ in $x$ and $C^{1}$ in $s$ and its gradient $\nabla^{s}_{x}d_{s}(x,z_{i}(s))^{2}$ is $C^{1}$ in $s$. By the Implicit Function Theorem, we obtain that $s\mapsto\operatorname{bar}_{s}(\mu(s))$ is $C^{1}$ in $s$.

Fix any $s_{0}\in B(0,\epsilon_{0})$ and let $\mu:=\mu(s_{0})$, $w_{i}:=w_{i}(s_{0})$, $z_{i}:=z_{i}(s_{0})$ and $x_{0}:=\operatorname{bar}_{s_{0}}(\mu)$. Considering the map $(t,s)\mapsto\operatorname{bar}_{t}(\mu(s))$, and differentiating along the diagonal we obtain,

[TABLE]

We will evaluate the two terms on the right hand side of this equation using Lemmas A.9 and A.6 respectively.

Claim 1**.**

\left\lVert\vbox{\hbox{\displaystyle D_{s=s_{0}}\operatorname{bar}_{s}(\mu)}}\right\rVert\leq 32L(s_{0})r(s_{0}).**

Proof of Claim 1: To prove this, we use the equation for $D_{s}\operatorname{bar}_{s}(\mu)$ from Lemma A.9. Since $r(s_{0})<\frac{1}{3\max\left\{a,b\right\}}$, Corollary A.4 implies that \left\lVert\vbox{\hbox{\displaystyle Q_{s_{0}}^{-1}}}\right\rVert<2. Since \left\lVert\vbox{\hbox{\displaystyle Q_{s_{0}}^{-1}}}\right\rVert<2 and $\sum{w_{i}}=1$, it suffices to show that

[TABLE]

as

[TABLE]

In order to prove Eq. A.7, it suffices to show that

[TABLE]

where

[TABLE]

By Lemma A.7, $\alpha:B(s_{0},\epsilon)\to F$ is a $C^{1}$ map such that $E(\alpha(s),s)=z_{i}$ for all $s\in B(s_{0},\epsilon)$. Here $E(v,s)\equiv E_{x_{0}}(v,s)=\exp^{g_{s}}_{x_{0}}(v)$ is the exponential map in the $g_{s}$ metric, as defined in Lemma A.7. Then

[TABLE]

as 0=D_{s}E(\alpha(s),s)=D_{v=\alpha(s)}E(v,s)\circ D_{s}\alpha(s)+D_{s}E(v,s){\raisebox{-2.15277pt}{|}_{v=\alpha(s)}}.

For all $(v,s)$ in a sufficiently small neighbourhood of $(0,s_{0})\in T_{x_{0}}F\times B(s_{0},\epsilon)$ (whose diameter only depends on the curvature bounds $a$ and $b$ of $F$), $D_{v=0}E(v,s)=I:T_{x_{0}}F\to T_{x_{0}}F$ and $(D_{v}E(v,s))$ is uniformly bounded away from [math].

Since $g_{s}$ satisfies Assumptions B** part (1), Lemma A.7 implies that $D_{s=s_{0}}E(v,s)$ is differentiable at $v=0$, that is,**

[TABLE]

Since $D_{s=s_{0}}E(0,s)=0$,

[TABLE]

for some function with $\lim_{v\to 0}h(v)=0$. Since \left\lVert\vbox{\hbox{\displaystyle D_{v=0}D_{s=s_{0}}E(v,s)}}\right\rVert\leq\left\lVert\vbox{\hbox{\displaystyle g_{s_{0}}}}\right\rVert_{1,2}\leq L(s_{0}),

[TABLE]

whenever \left\lVert\vbox{\hbox{\displaystyle v}}\right\rVert is sufficiently close to [math]. Since at $v=\alpha(s_{0})$, we have \left\lVert\vbox{\hbox{\displaystyle\alpha(s_{0})}}\right\rVert=d_{s_{0}}(x,z_{i})<2r(s_{0}), this implies that

[TABLE]

By Theorem II.7.1 of ****[Cha06]****, $D_{v=\alpha(s_{0})}E(v,s_{0})(w)=\frac{1}{t_{0}}J(t_{0})$ where t_{0}=\left\lVert\vbox{\hbox{\displaystyle\alpha(s_{0})}}\right\rVert and $J$ is the Jacobi field along t\to E(t\frac{\alpha(s_{0})}{\left\lVert\vbox{\hbox{\displaystyle\alpha(s_{0})}}\right\rVert},s_{0}) such that $J(0)=0$ and $J^{\prime}(0)=w$. By Rauch comparison theorem, \left\lVert\vbox{\hbox{\displaystyle J(t)}}\right\rVert\geq\frac{\left\lVert\vbox{\hbox{\displaystyle w}}\right\rVert}{a}\sin(at). Since $\frac{\sin(t)}{t}$ is monotonically decreasing on $[0,\pi]$ and $t_{0}<\frac{1}{3a}<\frac{\pi}{2a}$, we have \left\lVert\vbox{\hbox{\displaystyle D_{v}E(v,s_{0})(w)}}\right\rVert\geq\frac{2}{\pi}\left\lVert\vbox{\hbox{\displaystyle w}}\right\rVert. Then, \left\lVert\vbox{\hbox{\displaystyle(D_{v=\alpha(s_{0})}E(v,s_{0}))^{-1}}}\right\rVert\leq\pi/2<2.

Thus

[TABLE]

which finishes the proof of Claim 1. ∎

We will now finish the proof of Proposition A.10 by establishing:

Claim 2**.**

\left\lVert\vbox{\hbox{\displaystyle D_{s=s_{0}}\operatorname{bar}{s{0}}(\mu(s))-Q_{s_{0}}^{-1}\circ\sum_{i=1}^{n}w_{i}\lVert_{z_{i}}^{\operatorname{bar}{s{0}}(\mu)}D_{s=s_{0}}z_{i}(s)}}\right\rVert\leq 16\max\left\{1,a^{2},b^{2}\right\}L(s_{0})r(s_{0}).**

Proof of Claim 2:** We first expand the term $D_{s=s_{0}}\operatorname{bar}_{s_{0}}(\mu(s))$ as,**

[TABLE]

We will now prove this claim by first showing that

[TABLE]

and then showing that

[TABLE]

We first prove the claimed Eq. A.9. The first part of Lemma A.6 implies that

[TABLE]

Note that $\nabla_{x=\operatorname{bar}_{s_{0}}(\mu)}d_{s_{0}}(x,z_{i})$ has norm at most 1. Since $r(s_{0})<\frac{1}{3\max\left\{a,b\right\}}$, Corollary A.4 implies that \left\lVert\vbox{\hbox{\displaystyle Q_{s_{0}}^{-1}}}\right\rVert\leq 2. Then

[TABLE]

The claimed Eq. A.9 then follows as

[TABLE]

In order to prove claimed Eq. A.10, first fix $i\in\left\{1,\dots,n\right\}$. Recalling that $x_{0}=\operatorname{bar}_{s_{0}}(\mu)$, we get

[TABLE]

We will now show that

[TABLE]

Indeed, the second part of Lemma A.6 gives the formula for $D_{z_{i}}\operatorname{bar}_{s_{0}}(\mu)$ and we have:

[TABLE]

Since $r(s_{0})<\frac{1}{3\max\left\{a,b\right\}}$, Corollary A.4 implies that \left\lVert\vbox{\hbox{\displaystyle Q_{s_{0}}^{-1}}}\right\rVert\leq 2. Then the above expression is at most

[TABLE]

As $d_{s_{0}}(x_{0},z_{i})<2r(s_{0})<\pi/2b$, Lemma A.2 implies that the term over the underbrace is equal to $-\lVert^{x_{0}}_{z_{i}}(I+S_{i})$ where $S_{i}$ is a $(1,1)$-tensor such that

[TABLE]

Hence

[TABLE]

As $\sum_{i=1}^{n}w_{i}=1$ and $r(s_{0})\leq 1$,

[TABLE]

This establishes Eq. A.10 and finishes the proof of Claim 2. ∎

Now Claims 1 and 2, taken together with Eq. A.6, finish the proof of Proposition A.10. ∎

Let $d^{\operatorname{Gr}}$ be the Grassmannian distance. The following is straightforward from the definition of $d^{\operatorname{Gr}}$.

Lemma A.12**.**

Suppose $A,C:\operatorname{\mathbb{R}}^{m}\to\operatorname{\mathbb{R}}^{p}$ and $B:\operatorname{\mathbb{R}}^{p}\to\operatorname{\mathbb{R}}^{p}$ are linear maps and $W=\operatorname{Im}(\operatorname{id}_{\operatorname{\mathbb{R}}^{m}},A)$ and $W^{\prime}=\operatorname{Im}(\operatorname{id}_{\operatorname{\mathbb{R}}^{m}},C+B\circ A)$. Then d^{\operatorname{Gr}}(W,W^{\prime})\leq\left\lVert\vbox{\hbox{\displaystyle C}}\right\rVert+\left\lVert\vbox{\hbox{\displaystyle B-I}}\right\rVert\left\lVert\vbox{\hbox{\displaystyle A}}\right\rVert.

We now prove Proposition 2.9.

Corollary A.13**.**

Assume that Assumptions B** hold. Suppose that there is a continuous foliation $\operatorname{\mathcal{H}}$ on $B(0,\epsilon_{0})\times F$ such that:**

1. (a)

$\operatorname{\mathcal{H}}$** has $C^{1}$ leaves which transversely vary continuously in the leafwise $C^{1}$-topology,** 2. (b)

for each $\xi\in F$, there is a $C^{1}$ function $\varphi_{\xi}:B(0,\epsilon_{0})\to F$ satisfying $\varphi_{\xi}(0)=\xi$ and the foliation $\operatorname{\mathcal{H}}$ is given by

[TABLE] 3. (c)

the tangent distribution of $\operatorname{\mathcal{H}}$ is denoted by $\operatorname{\mathcal{D}}$.

Fix $\xi_{1},\dots,\xi_{n}\in F$, smooth weight functions $w_{i}:B(0,\epsilon_{0})\to[0,1]$ and let $G$ denote the graph of the map

[TABLE]

Then for every $\epsilon>0$ and every $c\in[0,1)$, there exists $\delta=\delta(\epsilon,c)>0$ such that for every $s_{0}\in B(0,c\epsilon_{0})$, if $r(s_{o})<\delta$ then

[TABLE]

Moreover, the derivatives $D_{s=s_{0}}\operatorname{bar}_{s}(\mu(s))$ and distributions $T_{(s_{0},\operatorname{bar}_{s_{0}}\mu(s_{0}))}G$ vary continuously in $s_{0},\xi_{1},\dots,\xi_{n}$.

Proof.

Let $z_{i}(s):=\varphi_{\xi_{i}}(s)$ and $x_{0}:=\operatorname{bar}_{s_{0}}(\mu(s_{0}))$. Now

[TABLE]

We will first explain why the last moreover statement in the corollary follows directly from our previous results. Note that by the assumption on the foliation $\operatorname{\mathcal{H}}$, $\varphi_{\xi_{i}}$ is $C^{1}$ in $s$ and $D_{s}z_{i}(s)=D_{s}\varphi_{\xi_{i}}(s)$ is continuous in both $\xi_{i}$ and $s$ for each $i=1,\dots,n$.

By the formula for $D_{s=s_{0}}\operatorname{bar}_{s}(\mu(s))$ given in Eq. A.6 and Eq. A.8 above,

[TABLE]

The first term is continuous in $s_{0}$ and $\xi_{i}$ by Lemma A.9 and Lemma A.7 (for the continuity of $Q_{s}$, see Equation A.5 and Lemma A.7). The continuity of the second and the third term in $s_{0}$ and $\xi_{i}$ follows from Lemma A.6 and the above mentioned regularity of $D_{s}\phi_{\xi_{i}}$.

Thus $T_{(s_{0},x_{0})}G$ varies continuously in $s_{0},\xi_{1},\dots,\xi_{n}$.

We now prove the remaining statements. For that, we can fix $\xi_{1},\dots,\xi_{n}\in F$ and let $s$ vary in $B(0,\epsilon_{0})$. By Proposition A.10,

[TABLE]

where \left\lVert\vbox{\hbox{\displaystyle R_{s_{0}}}}\right\rVert_{s_{0}}\leq 32\max\left\{a^{2},b^{2}\right\}L(s_{0})r(s_{0}) for any $s_{0}\in B(0,\epsilon_{0})$.

Let

[TABLE]

Note that $W(s)$ is continuous in $s$. Then M^{\prime}:=\sup_{s_{0}\in B(0,\epsilon_{0})}\max_{1\leq i\leq n}\left\lVert\vbox{\hbox{\displaystyle D_{s=s_{0}}\varphi_{\xi_{i}}(s)}}\right\rVert exists. Then the above Lemma A.12 implies that

[TABLE]

for a constant $M$ that is independent of $s_{0}$, when we assume that $r(s_{0})<1$.

We will now show that $d^{Gr}(W(s_{0}),\operatorname{\mathcal{D}}(s_{0},x_{0}))<\epsilon$ whenever $r(s_{0})$ is sufficiently small, depending on $\epsilon$. Define

[TABLE]

where $\lVert^{x}_{z_{i}(s)}$ is the parallel transports along $\{\{s\}\times F:s\in B(0,\epsilon_{0})\}$, with respect to the metrics $g_{s}$.

First note that $W(s_{0})=\sum w_{i}(s_{0})\operatorname{\mathcal{D}}_{i}(s_{0},x_{0})$. Fix any $\epsilon>0$ and $c\in[0,1)$. We now claim that there exists $\delta^{\prime}=\delta^{\prime}(\epsilon,c)>0$ such that if $s_{0}\in B(0,c\epsilon_{0})$ and $r(s_{0})<\delta^{\prime}$, then

[TABLE]

Indeed, this follows from the uniform continuity of the distribution $\operatorname{\mathcal{D}}$ and uniform continuity of the parallel translation maps $\lVert^{(s,x)}_{(s,y)}$ on $\overline{B(0,c\epsilon_{0})}\times F$. More precisely, uniform continuity of $\operatorname{\mathcal{D}}$ and the parallel translation implies the existence of $\delta^{\prime}=\delta^{\prime}(\epsilon,c)$ such that: if $s\in B(0,c\epsilon_{0})$, $d_{s}(x,y)<\delta^{\prime}$ and $v\in T_{y}F$ then

[TABLE]

(here $d^{TF}$ is the distance in the tangent bundle of $F$ induced by $g_{s}$). Now observe that $(u,u^{\prime})\in\operatorname{\mathcal{D}}_{i}(s_{0},x_{0})$ if and only if $u^{\prime}=\lVert^{x_{0}}_{z_{i}(s_{0})}v^{\prime}$ where $v^{\prime}\in\operatorname{Im}(D_{s=s_{0}}\varphi_{\xi_{i}}(s))$. It follows that $(u,v^{\prime})\in\operatorname{\mathcal{D}}(s_{0},z_{i}(s_{0}))$. Indeed we have

[TABLE]

This finishes the proof of the claim that if $r(s_{0})<\delta^{\prime}$, then $d(\operatorname{\mathcal{D}}_{i}(s_{0},x_{0}),\operatorname{\mathcal{D}}(s_{0},x_{0}))<\epsilon$.

Now suppose $\delta^{\prime}=\delta^{\prime}(\epsilon,c)$ as above. We will now show that: if $r(s_{0})<\delta^{\prime}$, then

[TABLE]

Indeed, if $v\in\operatorname{\mathcal{D}}(s_{0},x_{0})$, then there exists $v_{i}\in\operatorname{\mathcal{D}}_{i}$ such that \left\lVert\vbox{\hbox{\displaystyle v-v_{i}}}\right\rVert<\epsilon. Then $\sum w_{i}(s_{0})v_{i}\in W(s_{0})$ and \left\lVert\vbox{\hbox{\displaystyle v-\sum w_{i}(s_{0})v_{i}}}\right\rVert\leq\sum w_{i}(s_{0})\left\lVert\vbox{\hbox{\displaystyle v-v_{i}}}\right\rVert<\epsilon. Conversely, if $v_{i}^{\prime}\in\operatorname{\mathcal{D}}_{i}$, then there exist $y_{i}^{\prime}\in\operatorname{\mathcal{D}}$ such that \left\lVert\vbox{\hbox{\displaystyle y_{i}-v_{i}^{\prime}}}\right\rVert<\epsilon. Then $\sum w_{i}y_{i}\in\operatorname{\mathcal{D}}$ and \left\lVert\vbox{\hbox{\displaystyle\sum v_{i}^{\prime}-\sum y_{i}}}\right\rVert<\epsilon.

By Section A.2 and Eq. A.12, for any $\epsilon>0$ and $c\in[0,1)$, there exists $\delta=\min\left\{\frac{\epsilon}{M},1,\delta^{\prime}(\epsilon,c)\right\}$ such that: if $s_{0}\in B(0,c\epsilon_{0})$ and $r(s_{0})<\delta$, then

[TABLE]

∎

Appendix B Some Lemmas about Foliations

A manifold $V$ is said to be uniformly transverse to a foliation $\mathcal{H}$ of a Riemannian manifold if there exists $0<\theta\leq\frac{\pi}{2}$ such that at every point $p\in V$, the angle of $V$ with the leaf of $\mathcal{H}$ through $p$ is at least $\theta$.

Lemma B.1**.**

For $0\leq k\leq n$, consider the product foliation $\mathcal{H}=\{\mathbb{R}^{k}\times\{y\}:y\in\mathbb{R}^{n-k}\}$ of $\mathbb{R}^{n}$. Let $V$ be a complete connected $C^{1}$-submanifold of $\mathbb{R}^{n}$ such that $V$ is uniformly transverse to the foliation $\mathcal{H}$. Then the intersection of $V$ and any leaf of $\mathcal{H}$ is either empty or path connected.

We note that the uniform transversality assumption is required as simple counterexamples show (e.g. the surface of revolution for the function $e^{x}$). However, if $V$ has dimension $n-k$, the lemma holds under just the transversality assumption.

Proof.

We first set up some preliminaries before starting the proof. Let $H$ be a leaf of the foliation $\operatorname{\mathcal{H}}$ and define $f:\mathbb{R}^{n}\to[0,\infty)$ as $f(x):=d(x,H)^{2}$. Note that $\nabla_{x}f$ is well-defined whenever $f(x)>0$. Then, if $x\in V$ and $f(x)>0$

Since $V$ is uniformly transverse to $\mathcal{H}$, there is $\theta>0$ such that for every $x\in V$ there is $v\in T_{x}V$ making angle at least $\theta$ with distribution associated to $\mathcal{H}$. Thus, whenever $f>0$, the vector field $-\nabla f$ makes angles at most $\frac{\pi}{2}-\theta$ with tangent spaces of $V$. Let $\mathcal{F}$ be the vector field along $V$ obtained by projecting $-\nabla f$ to the tangent spaces of $V$ (whenever $f>0$). Then for every $x\in V$, \left\lVert\vbox{\hbox{\displaystyle\mathcal{F}(x)}}\right\rVert\geq\cos(\frac{\pi}{2}-\theta)\left\lVert\vbox{\hbox{\displaystyle-\nabla f(x)}}\right\rVert=2\cos(\frac{\pi}{2}-\theta)d(x,H). Since $V$ is a $C^{1}$ manifold and $\nabla f$ is smooth (whenever $\nabla f$ is well-defined, i.e. $f>0$), the vector field $\mathcal{F}$ is $C^{1}$ along $V$.

We let $F_{t}$ be the flow defined by $\mathcal{F}$. The flow $F_{t}$ is defined for all non-negative time and for every point $x\in V$ since \left\lVert\vbox{\hbox{\displaystyle\mathcal{F}(x)}}\right\rVert<2d(x,H) and $d(x,H)$ decreases under the flow. For any $x\in V$, we let $\phi(t,x)=d(F_{t}(x),H)$. Then $\phi(0,x)=d(x,H)$. Whenever $\phi(t,x)>0$, $\phi(t,x)$ is differentiable and

[TABLE]

Indeed, let $\theta_{y}$ be the angle between $\mathcal{F}(y)\in T_{y}V$ and $-\nabla_{y}d(y,H)$. Then \left\langle\nabla_{y}d(y,H),\operatorname{\mathcal{F}}(y)\right\rangle=-\cos(\theta_{y})\left\lVert\vbox{\hbox{\displaystyle\operatorname{\mathcal{F}}(y)}}\right\rVert\left\lVert\vbox{\hbox{\displaystyle\nabla_{y}d(y,H)}}\right\rVert=-2d(y,H)\cos^{2}\theta_{y}. Thus

[TABLE]

It follows that $\phi(t,x)\leq e^{-\epsilon_{0}t}d(x,H)$, where $\epsilon_{0}=2\cos^{2}(\frac{\pi}{2}-\theta)$.

Now we begin the proof. Suppose, for the sake of contradiction, that $V\cap H$ is not connected. Let $x_{1}$ and $x_{2}$ be two points on two different connected components of $V\cap H$. We let

[TABLE]

Now we consider two cases.

• Case 1*: Suppose that $r>0$. Let $\sigma$ be a path such that the furthest point to $H$ of $\sigma$ is at distance at most $re^{\epsilon_{0}}$. Then consider the path $F_{2}(\sigma)$, obtained by flowing the path $\sigma$ for time 2 by the flow $F_{t}$. This path in $V$ is continuous and connects $x_{1}$ and $x_{2}$. The furthest distance of points on this path to $H$ is at most $re^{\epsilon_{0}}e^{-2\epsilon_{0}}<r$. This contradicts with the definition of $r$.

Case 2: Suppose $r=0$. We first show that for every $x\in V$, the limit $\lim_{t\to+\infty}F_{t}(x)$ exists. Indeed, for every $x\in V$, we let $\mathcal{G}(x)$ be the projection of $\mathcal{F}(x)$ to the distribution associated to $\mathcal{H}$. Then $\mathcal{G}$ is a $C^{1}$ vector field, and \left\lVert\vbox{\hbox{\displaystyle\mathcal{G}(x)}}\right\rVert\leq\frac{1}{2}\left\lVert\vbox{\hbox{\displaystyle-\nabla f(x)}}\right\rVert=d(x,H). As $F_{t}(x)$ gets closer to $H$ in forward time, the set $\{F_{t}(x):t\geq 0\}$ is contained in the closed $d(x,H)$-neighborhood of $H$. Furthermore, let $proj:\operatorname{\mathbb{R}}^{n}\to H$ be the orthogonal projection. Then we get the estimate

[TABLE]

Hence the set $\{F_{t}(x):t\geq 0\}$ is contained in the ball radius $2\epsilon_{0}^{-1}d(x,H)$ around $x$. It follows that flowing the path $\sigma$ in forward time is trapped in a compact set. Furthermore, the inequality shows that the forward trajectory $\{F_{t}(x):t\geq 0\}$ is a Cauchy net. Therefore, $\lim_{t\to+\infty}F_{t}(x)$ exists.

We also have the following estimate: for every $x\in V$ and $t\geq 0$,

[TABLE]

Indeed $d(x,F_{t}(x))\leq(\phi(0,x)-\phi(t,x))+d(proj(x),proj(F_{t}(x)))\leq d(x,H)+\epsilon_{0}^{-1}d(x,H).$

Let $\sigma$ be a continuous path connecting $x_{1}$ and $x_{2}$. We claim that $\overline{\sigma}:[0,1]\to V$, defined by $\overline{\sigma}(s)=\lim_{t\to+\infty}F_{t}(\sigma(s))$ for every $s\in[0,1]$, is continuous. The map is well-defined because of the Cauchy property remarked above and since $V$ is complete. Indeed, first we note that $\overline{\sigma}(s)\in H$ for every $s$. Let $s_{0}\in[0,1]$ and let $\epsilon>0$. Let $T_{0}=T_{0}(s_{0},\epsilon)>0$ be a time such that

[TABLE]

where $\alpha=\frac{1}{4+2\epsilon_{0}^{-1}}$. Since the flow $F_{t}$ is continuous, the path $F_{T_{0}}(\sigma)$ is continuous. Thus there exists $\delta>0$ such that for $s\in[s_{0}-\delta,s_{0}+\delta]$,

[TABLE]

It follows that $d(F_{T_{0}}(\sigma(s)),H)\leq d(F_{T_{0}}(\sigma(s_{0})),F_{T_{0}}(\sigma(s)))+d(F_{T_{0}}(\sigma(s_{0})),\overline{\sigma}(s_{0}))\leq 2\alpha\epsilon$. Hence, by the triangle inequality,

[TABLE]

Hence, $\overline{\sigma}$ is continuous at $s_{0}$. This proves $\overline{\sigma}$ is a continuous path.

On the other hand, $\overline{\sigma}(s)\in H$ since $d(\overline{\sigma}(s),H)=0$, for every $s\in[0,1]$. Hence $\overline{\sigma}$ is a path in $V\cap H$, connecting $x_{1}$ and $x_{2}$. This is a contradiction.

Therefore, $V\cap H$ is connected. ∎

For two arbitrary vector subspaces $V,W$ of a common vector space $Z$ we define

[TABLE]

In particular, the angle is [math] precisely when either $V\subset W$ or $W\subset V$. When $\dim V=\dim W$ then $\angle(V,W)$ is bilipschitz with $d^{Gr}(V,W)$. Indeed if $g\in O(\dim Z)$ is an element of $O(\dim Z)$ closest to the identity that moves $V$ to $W$, then $d(e,g)$ is at least $\angle(V,W)$ and bounded by $\dim Z$ times this angle.

Lemma B.2**.**

Let $M$ be an $m$-dimensional complete connected $C^{1}$ sub-manifold without boundary of $\mathbb{R}^{n}=\mathbb{R}^{n-k}\times\mathbb{R}^{k}$. We denote by $\mathcal{V}$ the foliation of $\mathbb{R}^{n}$ by vertical $\mathbb{R}^{k}$ spaces. Suppose that for every $p\in M$, we have $\angle(T_{p}M,T_{p}(\mathcal{V}(p))\leq\epsilon$ for a fixed $\epsilon<\frac{\pi}{2}$. Let $\pi:\mathbb{R}^{n}\to\mathbb{R}^{k}$ be the projection of $\mathbb{R}^{n}=\mathbb{R}^{n-k}\times\mathbb{R}^{k}$ to the $\mathbb{R}^{k}$ factor. If $m\geq k$, then $\pi|_{M}$ is surjective onto $\operatorname{\mathbb{R}}^{k}$, and in particular is a diffeomorphism when $m=k$.

Proof.

Transversality implies that $\pi|_{M}$ is a submersion, and hence is an open map.

Next, we show that if $m\geq k$ then \pi{\raisebox{-2.15277pt}{|}_{M}} is surjective by showing that the image of \pi{\raisebox{-2.15277pt}{|}_{M}} is closed. Suppose, for the sake of contradiction, that the image is not closed. We let $z\in\pi(M)$ and let $r:[0,1)\to\pi(M)$ be a finite line segment contained in $\pi(M)$ from $z$ to some $w\in\partial\pi(M)$. Since \pi{\raisebox{-2.15277pt}{|}_{M}} is a submersion onto its image, \pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r) is a $C^{1}$ submanifold of $M\subset\mathbb{R}^{n}$. For each $t\in[0,1)$, and each point q\in\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r(t)), note that \operatorname{ker}d_{q}(\pi{\raisebox{-2.15277pt}{|}_{M}})=T_{q}\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r(t)), and moreover d_{q}(\pi{\raisebox{-2.15277pt}{|}_{M}}) is an isomorphism on the orthogonal complement of this subspace in $T_{q}M$ which we denote by T_{q}^{\perp}\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r(t))\subset T_{q}M . Hence there is a unique vector v_{q}\in T_{q}^{\perp}\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r(t)) such that $d\pi(v_{q})=r^{\prime}(t)$. This gives a vector field on \pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r).

We observe that for any $v\in T_{q}M$ we may write it as $v=v_{\perp}+v_{\operatorname{ker}}$ for v_{\perp}\in T_{q}^{\perp}\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r(t)) and $v_{\operatorname{ker}}\in\operatorname{ker}d_{q}\pi=(T_{q}\mathcal{V}(q))^{\perp}$. Let $w\in T_{q}\mathcal{V}(q)$. If $v_{\perp}=0$, then $\left\langle v,w\right\rangle=\left\langle v_{\perp},w\right\rangle=0$. So suppose $v_{\perp}\neq 0$. Since $\left\langle v,w\right\rangle=\left\langle v_{\perp},w\right\rangle$ and \left\lVert\vbox{\hbox{\displaystyle v}}\right\rVert\geq\left\lVert\vbox{\hbox{\displaystyle v_{\perp}}}\right\rVert, we have \left\langle\frac{v}{\left\lVert\vbox{\hbox{\displaystyle v}}\right\rVert},w\right\rangle\leq\left\langle\frac{v_{\perp}}{\left\lVert\vbox{\hbox{\displaystyle v_{\perp}}}\right\rVert},w\right\rangle. Thus $\angle(v,w)\geq\angle(w,v_{\perp})$ for any $w\in T_{q}\mathcal{V}(q)$ and $v\in T_{q}M$. Hence we have

[TABLE]

Therefore \left\lVert\vbox{\hbox{\displaystyle r^{\prime}(t)}}\right\rVert\leq\left\lVert\vbox{\hbox{\displaystyle v_{q}}}\right\rVert\leq\frac{1}{\cos(\epsilon)}\left\lVert\vbox{\hbox{\displaystyle r^{\prime}(t)}}\right\rVert where $\epsilon<\frac{\pi}{2}$ was fixed. Let us fix any point q_{0}\in\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(z) and let $\sigma:[0,1)\to M$ be the integral curve of the vector field $q\mapsto v_{q}$ starting at $q_{0}$. Then $\sigma$ projects to $r$. Moreover, $\sigma$ has finite length since the norm of $d\pi$ is bounded below on $v_{q}$ for each q\in\pi{\raisebox{-2.15277pt}{|}_{M}}^{-1}(r). By completeness of $M$, there is a limit point $\sigma(1)\in M$ which must map to the endpoint $w$ of $r$. This contradicts the assumption that $w$ did not belong to $\pi(M)$.

Finally, we show that when $m=k$, $\pi|_{M}$ is a diffeomorphism onto its image. By Lemma B.1, the intersection of $M$ and each horizontal $\mathbb{R}^{n-k}\times\{v\}$ is path connected for every $v\in\mathbb{R}^{k}$. But by transversality, each intersection is a zero dimensional manifold. Thus the intersection of $M$ and each horizontal $\mathbb{R}^{n-k}\times\{v\}$ consists of at most one point, and hence exactly one point, for every $v\in\mathbb{R}^{k}$. Therefore \pi{\raisebox{-2.15277pt}{|}_{M}} is a diffeomorphism onto its image. ∎

Finally, we need the following lemma about Hausdorff convergence of a sequence of transverse intersections of submanifolds.

Lemma B.3**.**

Suppose $\left\{M_{n}:n\geq 0\right\}$ is a sequence of connected embedded $C^{1}$ submanifolds of a manifold $X_{0}$ such that $M_{n}\to M_{0}$ in the pointed Hausdorff topology and $TM_{n}\to TM_{0}$ pointwise. Suppose $N$ is another connected embedded $C^{1}$ submanifold of $X_{0}$ such that $M_{n}\cap N\neq\varnothing$ and the intersection is transverse for all $n\geq 0$. Then $M_{n}\cap N\to M_{0}\cap N$ in the pointed Hausdorff topology.

Proof.

Consider any point of $x\in M_{0}\cap N$. Note that it is enough to show that for a pre-compact open subset $U_{x}\subset X_{0}$ containing $x$, $M_{n}\cap N\cap U_{x}\to M_{0}\cap N\cap U_{x}$ in the Hausdorff topology. Let $m=\dim(M_{0})$, $m+k=\dim(X_{0})$, $r=\dim(N)$. Then $\dim(M_{0}\cap N)=r-k\geq 0$.

Choose a coordinate chart $\phi:U_{x}\to(-1,1)^{m+k}\subset\mathbb{R}^{m+k}$ where $U_{x}\subset X_{0}$ is a pre-compact open set, $\phi(M_{0}\cap U_{x})=(-1,1)^{m}\times\left\{0\right\}^{k}$ and $\phi(N\cap U_{x})=\left\{0\right\}^{m+k-r}\times(-1,1)^{r}$. We have to show that $\phi(M_{n}\cap U_{x})\cap(\left\{0\right\}^{m+k-r}\times(-1,1)^{r})$ converges to $\phi(M_{0}\cap U_{x})\cap(\left\{0\right\}^{m+k-r}\times(-1,1)^{r})=\left\{0\right\}^{m+k-r}\times(-1,1)^{r-k}\times\left\{0\right\}^{k}$ in the Hausdorff topology. Note that since $\phi(M_{n}\cap U_{x})$ converges to $\phi(M_{0}\cap U_{x})$ in the pointed Hausdorff topology and $U_{x}$ is pre-compact. So, for any $0<\epsilon<1$ there is an $n$ large enough such that $\phi(M_{n}\cap U_{x})$ is contained in $(-1,1)^{m}\times(-\epsilon,\epsilon)^{k}$.

Since $\phi$ is $C^{1}$ and the distributions $TM_{n}$ converge to $TM_{0}$ pointwise, $M_{n}\cap U_{x}$ is transverse to $\left\{0\right\}^{m+k-r}\times\mathbb{R}^{r}$ for $n$ large enough. Indeed, if $\left\{y_{n}\right\}$ is a sequence such that $y_{n}\in\phi(M_{n}\cap U_{x})$ and $y_{n}\to y_{0}$, then $T_{y_{n}}\phi(M_{n}\cap U_{x})$ converges to $T_{y}\phi(M_{0}\cap U_{x})=\mathbb{R}^{m}\times\left\{0\right\}^{k}$. Thus for $n$ sufficiently large, $M_{n}\cap U_{x}$ is transverse to $\left\{0\right\}^{m+k-r}\times\mathbb{R}^{r}$.

Now suppose that $n$ is large enough. As the image of $\phi(M_{n}\cap U_{x})$ has tangent space nowhere tangent to ${0}\times\mathbb{R}^{r}$, each connected component of $\phi(M_{n}\cap U_{x})$ is a graph of a function whose domain is a subset of $(-1,1)^{m}\times\left\{0\right\}^{k}$. We now claim that for sufficiently large $n$, each connected component of $\phi(M_{n}\cap U_{x})$ is a graph of a function whose domain is $(-1,1)^{m}\times\left\{0\right\}^{k}$. Since $M_{n}$ is embedded in $X_{F}$, $\phi(M_{n}\cap U_{x})$ is an embedded submanifold of $\phi(U_{x})$. If the claim is false, then there is a connected component $C_{n}$ of $\phi(M_{n}\cap U_{x})$ and $y_{n}\in(-1,1)^{m}$ such that $(y_{n},z_{n})\in\partial C_{n}$. As $C_{n}$ is properly embedded in $\phi(U_{x})$ and $\epsilon<1$, $(y_{n},z_{n})$ is an interior point of $C_{n}$ and $T_{(y_{n},z_{n})}C_{n}$ is transverse to $\left\{0\right\}\times\operatorname{\mathbb{R}}^{r}$. This contradicts that $\phi(M_{n}\cap U_{x})\subset(-1,1)^{m}\times\left\{0\right\}^{k}$. Consequently, we have $\partial(\phi(M_{n}\cap U_{x}))\subset\partial((-1,1)^{m})\times[-\epsilon,\epsilon]^{k}$ and $\partial\pi_{m}(\phi(M_{n})\cap U_{x})\subset\partial((-1,1)^{m})$ (where $\pi_{m}$ is projection onto the first $m$ coordinates in $\operatorname{\mathbb{R}}^{m+k}$). Therefore, each connected component of $\phi(M_{n}\cap U_{x})$ is a graph of a function whose domain is entire $(-1,1)^{m}\times\left\{0\right\}^{k}$.

We let $d_{n}=d^{\operatorname{Haus}}(\phi(M_{n}\cap N\cap U_{x}),\phi(M_{0}\cap N\cap U_{x}))$. We then have $d_{n}\to 0$ when $n\to\infty$. For each connected component of $\phi(M_{n}\cap U_{x})$, there is a function $\theta_{n}:(-1,1)^{m}\to(-1,1)^{k}$ such that the connected component is precisely the graph $\{(x,\theta_{n}(x)):x\in(-1,1)^{m}\}$. It follows that the intersection of this connected component with $N$ is the set $S=\{(x,\theta_{n}(x)):x\in\left\{0\right\}^{m+k-r}\times(-1,1)^{r-k}\}$. It is clear that

[TABLE]

which tends to [math] when $n\to\infty$. This shows that $\phi(M_{n}\cap N\cap U_{x})$ converges to $\phi(M_{0}\cap N\cap U_{x})$ in the Hausdorff topology. ∎