On factor rigidity and joining classification for infinite volume rank one homogeneous spaces

TL;DR

This paper classifies joinings with respect to the Burger-Roblin measure for horospherical actions on certain infinite volume hyperbolic manifolds, revealing rigidity phenomena in these dynamical systems.

Contribution

It provides a classification of locally finite joinings and establishes rigidity of certain set-valued maps for infinite volume rank one homogeneous spaces.

Findings

Classification of joinings with respect to Burger-Roblin measure

Rigidity of U-equivariant set-valued maps in geometrically finite cases

Extension of rigidity results to broader classes of infinite volume hyperbolic manifolds

Abstract

We classify locally finite joinings with respect to the Burger-Roblin measure for the action of a horospherical subgroup $U$ on $\Gamma \backslash G$, where $G = \operatorname{SO}(n,1)^\circ$ and $\Gamma$ is a convex cocompact and Zariski dense subgroup of $G$, or geometrically finite with restrictions on critical exponent and rank of cusps. We also prove in the more general case of $\Gamma$ geometrically finite and Zariski dense that certain $U$-equivariant set-valued maps are rigid.

Equations

$$X_i:={\Gamma }_i\backslash G\text{ for }i=1,2\text{ and }X:=X_1\times X_2.$$

$${\rm Y}:X_2\to X_1$$

$$\widehat{{\rm Y}}:X_2\to X_1,$$

$$(g_0^{-1}{\Gamma }_1{g}_0\cap {\Gamma }_2)\backslash G\to [(g_0,1_G)]\Delta (G)$$

$$[g]\mapsto [(g_{0}g,g)],$$

$$A=\{a_s:s\in \mathbb{R}\}\text{ where }{a}_s=\left(\begin{array}{ccc}{e}^s& & \\ & I_{n-1}& \\ & & e^{-s}\end{array}\right),$$

$$U=\{g\in G:a_{s}g{a}_{-s}\to 1_G\text{ as }s\to \infty \}.$$

$$U^{-}=\{g\in G:a_{s}g{a}_{-s}\to 1_G\text{ as }s\to -\infty \}.$$

$$U=\{u_{\text{t}}:\text{t}\in {\mathbb{R}}^{n-1}\}\text{ where }{u}_{\text{t}}=\left(\begin{array}{ccc}1& & \\ {\text{t}}^T& I& \\ \frac{1}{2}|\text{t}{|}^2& \text{t}& 1\end{array}\right)\text{ and }$$

$$U^{-}=\{v_{\text{t}}:\text{t}\in {\mathbb{R}}^{n-1}\}\text{ where }{v}_{\text{t}}=\left(\begin{array}{ccc}1& \text{t}& \frac{1}{2}|\text{t}{|}^2\\ & I& {\text{t}}^T\\ & & 1\end{array}\right).$$

$$M=\left\{\left(\begin{array}{ccc}1& & \\ & m& \\ & & 1\end{array}\right):m\in \mathrm{SO}(n-1)\right\}.$$

$$a_{-s}{u}_{\text{t}}{a}_s=u_{\text{t}{e}^s}\text{ and }{m}^{-1}{u}_{\text{t}}m=u_{\text{t}m}.$$

$$B(T)=\{g\in G:\Vert g-I\Vert \le T\}\text{ where }\Vert \cdot \Vert \text{ is the max norm on }G,$$

$$B_U(T)=\{u_{\text{t}}\in U:|\text{t}|\le T\},\text{ where }|\cdot |\text{ is the max norm on }{\mathbb{R}}^{n-1}.$$

$$d(\Gamma x,\Gamma y)=\min \{\Vert g-1_G\Vert :g\in G,\Gamma x=\Gamma yg\}.$$

$$\Lambda (\Gamma )=\overline{\Gamma v}-\Gamma v$$

$$\Lambda (\Gamma )={\Lambda }_r(\Gamma ).$$

$$\Lambda (\Gamma )={\Lambda }_r(\Gamma )\cup {\Lambda }_{\mathrm{bp}}(\Gamma ),$$

$$g^{\pm }:=g{w}_0^{\pm }.$$

$${\beta }_{\xi }(x,y)=\underset{t\to \infty }{\lim }d({\xi }_t,x)-d({\xi }_t,y),$$

$${\gamma }_{\ast }{\mu }_x={\mu }_{\gamma x}\text{ and }\frac{d{\mu }_x}{d{\mu }_y}(\xi )=e^{-\delta {\beta }_{\xi }(x,y)}$$

$$d{\mu }_{gU}^{\mathrm{PS}}(g{u}_{\text{t}})=e^{{\delta }_{\Gamma }{\beta }_{(g{u}_{\text{t}}{)}^{+}}(o,g{u}_{\text{t}}(o))}d{\nu }_o(g{u}_{\text{t}}{)}^{+}.$$

$$d{\mu }_{gU}^{\mathrm{Leb}}(g{u}_{\text{t}})=e^{(n-1){\beta }_{(g{u}_{\text{t}}{)}^{+}}(o,g{u}_{\text{t}}(o))}d{m}_o(g{u}_{\text{t}}{)}^{+}.$$

$${\mu }_g^{\mathrm{PS}}(E)=e^{{\delta }_{\Gamma }s}{\mu }_{g{a}_{-s}}^{\mathrm{PS}}(a_{s}E{a}_{-s}).$$

$${\mu }_g^{\mathrm{PS}}(B_U(e^s))=e^{{\delta }_{\Gamma }s}{\mu }_{g{a}_{-s}}^{\mathrm{PS}}(B_U(1)).$$

$$0<\underset{g\in \Omega ,g^{+}\in \Lambda (\Gamma )}{\inf }{\mu }_g^{\mathrm{PS}}(B_U(T))\le \underset{g\in \Omega ,g^{+}\in \Lambda (\Gamma )}{\sup }{\mu }_g^{\mathrm{PS}}(B_U(T))<\infty .$$

$$0<\underset{x\in \Omega }{\inf }{\mu }_x^{\mathrm{PS}}(B_U(\kappa ))\le \underset{x\in \Omega }{\sup }{\mu }_x^{\mathrm{PS}}(B_U(\kappa ))<\infty .$$

$$(g{B}_U(\kappa /2){)}^{+}\cap \Lambda (\Gamma )\ne \varnothing$$

$$(\partial ({\mathbb{H}}^n)\times \partial ({\mathbb{H}}^n)-\{(\xi ,\xi ):\xi \in \partial ({\mathbb{H}}^n)\})\times \mathbb{R}.$$

$$d{\tilde{m}}^{\mathrm{BMS}}(w)=e^{{\delta }_{\Gamma }({\beta }_{w^{+}}(o,\pi (w))+{\beta }_{w^{-}}(o,\pi (w)))}d{\nu }_o(w^{+})d{\nu }_o(w^{-})ds$$

Full text

We also prove in the more general case of $\Gamma$ geometrically finite and Zariski dense that certain $U$-equivariant set-valued maps are rigid.

J. M. Warren was supported in part by an NSERC PGS-D3 fellowship.

1. 1 Introduction

2. 2 Preliminaries and notation

3. 2.1 PS measure

4. 2.2 BMS and BR measures

5. 3 Rigidity of $U$-equivariant set-valued maps

6. 3.1 Notation

7. 3.2 $U$-equivariant implies $AM$-equivariant

8. 3.3 $AMU$-equivariant implies $U^{-}$-equivariant

9. 4 Non-concentration of the PS measure near varieties

10. 5 Joinings

11. 5.1 Proof of Theorem 1.4

12. 5.2 Notation

13. 5.3 Fibers of $\pi_{2}$ are finite

1. Introduction

In [23], Ratner classified all joinings with respect to horocycle flows on finite volume quotients of $\operatorname{PSL}_{2}(\mathbb{R})$, a problem which is closely related to that of measure classification and classification of closed orbits. These problems are well understood in the finite volume case, [24, 25, 33], but for infinite volume, a full picture is not yet clear.

In [15, 16, 17], McMullen, Mohammadi, and Oh obtained orbit closure classification in the infinite volume setting, specifically for convex cocompact acyclindral 3-manifolds. This was generalized to higher dimensions by Lee and Oh in [13], and to geometrically finite acylindical manifolds in [2] by Benoist and Oh.

Mohammadi and Oh also show equidistribution for non-closed orbits in the geometrically finite case in [18], as well as classifying joinings for geometrically finite quotients of $\operatorname{PSL}_{2}(\mathbb{R})$ or $\operatorname{PSL}_{2}(\mathbb{C})$.

There is some progress in the geometrically infinite case under certain assumptions as well. For instance, in [12], Ledrappier and Sarig classify measures for geometrically infinite regular covers of compact hyperbolic surfaces, and in [21], Pan classifies joinings for $\mathbb{Z}$ or $\mathbb{Z}^{2}$ covers of compact hyperbolic surfaces.

In addition, the following works are in the setting of either geometrically finite quotients or certain geometrically infinite quotients: [3, 5, 26]. In particular [11, 28, 32] consider the problem of measure classification, while [6] considers joinings in higher rank.

The primary purpose of this paper is to extend the classification of joinings from [18], and the proof, a la Ratner, will require proving rigidity of $U$-equivariant set-valued maps, a result which is of independent interest. Ratner proved this for factor maps in the lattice case in [22], while Flaminio and Spatzier proved a similar result for convex cocompact groups in [8]. For set-valued maps, rigidity was proven in the finite volume case by Ratner in [23], where she calls these maps measurable partitions. A partial statement is proven in the case of geometrically finite quotients of $\operatorname{SO}(n,1)^{\circ}$ for $n=2,3$ in [18]. We will prove a full rigidity result for such set-valued maps for arbitrary $n$. We prove rigidity for factor maps with no further assumptions; for set-valued maps, we need to assume the presence of a “nice” ergodic measure.

Recall that $G:=\operatorname{SO}(n,1)^{\circ}$ corresponds to the group of orientation preserving isometries of real hyperbolic space $\mathbb{H}^{n}$. Throughout the paper, we assume that

$\Gamma_{1},\Gamma_{2}$ are geometrically finite and Zariski dense discrete subgroups of $G$ with infinite co-volume

and define

[TABLE]

Our factor rigidity statement will hold for geometrically finite groups, but our proof of the joining classification will require assuming either that the $\Gamma_{i}$’s are convex cocompact, or are geometrically finite with all cusps of full rank and critical exponents larger than $n-\frac{5}{4}$.

Let $U$ denote a horospherical subgroup of $G$. In the infinite volume setting, the natural analogue of the Haar measure used in Ratner’s proofs in the finite volume case is the Burger-Roblin (BR) measure $m^{\operatorname{BR}}_{i}=m^{\operatorname{BR}}_{\Gamma_{i}}$ (defined in §2.2). The BR measure is the unique locally finite $U$-ergodic measure that is not supported on a closed $U$-orbit, [5, 26, 32].

Our rigidity theorem for factor maps is stated below. A more general version that covers set-valued maps, Theorem 3.1, is proven in §3, but requires further assumptions. Here, the action of $A$ induces the frame flow on $\Gamma_{i}\backslash\mathbb{H}^{n}$, $M$ is the compact centralizer of $A$, and $U^{-}$ is the opposite horospherical subgroup.

Theorem 1.1**.**

Let

[TABLE]

be a measurable map, and suppose that there exists a $m^{\operatorname{BR}}$-conull set on which $\Upsilon$ is $U$-equviariant. Then there exists a map

[TABLE]

a constant $\bm{\sigma}_{0}\in U$, and a $U$-invariant $m^{\operatorname{BR}}$-conull set $X_{2}^{\prime}$ such that for all $x\in X_{2}^{\prime}$,

• •

$\hat{\Upsilon}(x)=\Upsilon(x)\bm{\sigma}_{0}$,

• •

$\hat{\Upsilon}(xh)=\hat{\Upsilon}(x)h$* for all $h\in AMU$, and*

• •

$\hat{\Upsilon}(xv)=\hat{\Upsilon}(x)v$* for all $v\in U^{-}$ such that $xv\in X_{2}^{\prime}$.*

Our proofs follow the outline of Ratner’s approach from [23], but care is required in this infinite volume case.

Denote by $\Delta(U)$ the diagonal embedding into $G\times G$.

Definition 1.2**.**

Let $\mu_{i}$ be a locally finite $U$-invariant Borel measure on $\Gamma_{i}\backslash G$ for $i=1,2$. A U-joining with respect to $(\mu_{1},\mu_{2})$ is a locally finite $\Delta(U)$-invariant measure $\mu$ on $X$ such that the push-forward measure onto the $i$-th coordinate is proportional to the corresponding $\mu_{i}$, $i=1,2$. If $\mu$ is $\Delta(U)$-ergodic, we call it an ergodic $U$-joining.

The primary goal of this article is to classify $U$-joinings in this infinite volume setting for the pair $(m^{\operatorname{BR}}_{1},m^{\operatorname{BR}}_{2})$. Note that in this case, the $m^{\operatorname{BR}}_{i}$’s are infinite measures [20], so the product measure $m^{\operatorname{BR}}_{1}\times m^{\operatorname{BR}}_{2}$ is not a $U$-joining.

We now restate the definition of a finite cover self-joining as it appears in [18]:

Definition 1.3**.**

Suppose that there exists $g_{0}\in G$ so that $g_{0}^{-1}\Gamma_{1}g_{0}$ and $\Gamma_{2}$ are commensurable in $G$. In particular, we have an isomorphism

[TABLE]

defined by

[TABLE]

where $1_{G}$ denotes the identity in $G$. The pushforward of the $\operatorname{BR}$ measure $m^{\operatorname{BR}}_{g_{0}^{-1}\Gamma_{1}g_{0}\cap\Gamma_{2}}$ is a $U$-joining, which we call a finite cover self-joining. We also consider any translation of a finite cover self-joining under an element of the form $(u,1_{G})\in U\times\{1_{G}\}$ to be a finite cover self-joining.

We will obtain the following joining classification:

Theorem 1.4**.**

Suppose that $\Gamma_{1},\Gamma_{2}$ are

• •

convex cocompact, or

• •

geometrically finite with all cusps full rank and critical exponents greater than $n-\frac{5}{4}$.

Then every locally finite ergodic $U$-joining on $X=\Gamma_{1}\backslash G\times\Gamma_{2}\backslash G$ with respect to $(m^{\operatorname{BR}}_{1},m^{\operatorname{BR}}_{2})$ is a finite cover self-joining.

In particular, $X$ admits a $U$-joining if and only if, up to a conjugation, $\Gamma_{1}$ and $\Gamma_{2}$ are commensurable.

Note that by the ergodic decomposition and the $U$-ergodicity of $m^{\operatorname{BR}}$ [32], classifying the ergodic $U$-joinings is sufficient to understand all $U$-joinings.

The proof strategy involves first reducing to a specific $U$-equivariant set-valued map, for which we prove an analogue of Theorem 1.1 in §3. In [18], Mohammadi and Oh prove rigidity of set-valued maps that are equivariant under a subgroup of $AM$ and under $U$ in the geometrically finite setting. Because of this extra equivariance assumption, their joining classification argument in [18, Section 7] requires arguing that a $U$-joining will be invariant under a subgroup of $AM$ before reducing the problem to rigidity of a certain set-valued map. With the more general rigidity result in §3, when proving joining classification, we avoid the need for such an argument.

This article is organized as follows. In §2, we define notation that is used throughout the paper, the Patterson-Sullivan (PS), Bowen-Margulis-Sullivan (BMS) and Burger-Roblin (BR) measures, and reference some basic properties of these measures.

In §3, we prove Theorem 1.1, by proving the more general Theorem 3.1, which includes set-valued maps. In particular, we prove that, under certain assumptions, a $U$-equivariant set-valued map must also be $AMU^{-}$-equivariant, up to a constant shift by an element of $U$, Theorems 3.4 and 3.6. This will be key in the final steps of the proof of Theorem 1.4. Note that the results in this section are proved in the more general setting of $\Gamma_{i}$’s being geometrically finite and Zariski dense, not necessarily convex cocompact, although in general, for set-valued maps, we need to assume the existence of an ergodic joining-like measure on $X$, see Theorem 3.1.

In §4, we prove general results about the behaviour of PS measure on varieties that will be important in the proof of Theorem 1.4. In particular, we prove that Lebesgue integrals on small neighbourhoods of varieties are controlled by the PS measure, Lemma 4.6. Understanding this behaviour is a crucial step needed to generalize the results from [18] to higher dimensions.

In §5, we show that the fibers of the projection $\pi_{2}$ onto the second coordinate must be finite, Theorem 5.1. This allows us to use the results of §3 to prove Theorem 5.2, which is a more precise formulation of Theorem 1.4.

2. Preliminaries and notation

For convenience, we remind the reader of the following notation that appeared in the introduction:

• •

$G=\operatorname{SO}(n,1)^{\circ}$ is the connected component of the identity in $\operatorname{SO}(n,1)$. It is the group of orientation preserving isometries of $\mathbb{H}^{n}$, and $1_{G}$ denotes the identity element in $G$.

• •

$\Gamma_{1},\Gamma_{2}$ are geometrically finite and Zariski dense discrete subgroups of $G$ with infinite co-volume.

• •

$X_{i}:=\Gamma_{i}\backslash G$ and $X:=X_{1}\times X_{2}$.

• •

For $H\subset G$, $\Delta(H)$ denotes the diagonal embedding of $H$ into $G\times G$.

Define

[TABLE]

where $I_{n-1}$ is the $(n-1)\times(n-1)$ identity matrix.

We denote by $U$ the contracting horospherical subgroup, that is,

[TABLE]

Similarly, we denote by $U^{-}$ the expanding horospherical subgroup,

[TABLE]

Both groups are isomorphic to $\mathbb{R}^{n-1}$, and we use the following parametrizations:

[TABLE]

[TABLE]

We also define

[TABLE]

We will often abuse notation by writing $m\in M$ to refer to the matrix $\begin{pmatrix}1&&\\ &m&\\ &&1\end{pmatrix}$. Observe that these parametrizations satisfy

[TABLE]

We view $G$ as being embedded in $\operatorname{SL}_{n+1}(\mathbb{R})$. For $T>0$, we denote balls in $G$ by

[TABLE]

and in $U$ by

[TABLE]

We write $\textbf{t}\in B_{U}(T)$ as shorthand for $u_{\textbf{t}}\in B_{U}(T)$.

On $\Gamma\backslash G$,

[TABLE]

2.1. PS measure

We use many definitions and notations as in [18, Section 2], but provide a paraphrased version here for the convenience of the reader. See [18, Section 2] for more details about these constructions.

Let $\partial(\mathbb{H}^{n})\cong\mathbb{S}^{n-1}$ denote the geometric boundary of $\mathbb{H}^{n}$. For any discrete subgroup $\Gamma$ of $G$, we can define the limit set of $\Gamma$, $\Lambda(\Gamma)$, as the accumulation points of any orbit in $\partial(\mathbb{H}^{n})$, that is,

[TABLE]

for any $v\in\mathbb{H}^{n}$, where the closure is taken in $\mathbb{H}^{n}\cup\partial(\mathbb{H}^{n})$. This is independent of $v$ because $\Gamma$ acts by isometries on $\mathbb{H}^{n}$.

We denote by $\Lambda_{\operatorname{r}}(\Gamma)$ the set of radial limit points of $\Gamma$. $\xi\in\Lambda(\Gamma)$ is a radial limit point if some (hence every) geodesic ray towards $\xi$ has accumulation points in some compact subset of $\Gamma\backslash G$. A parabolic limit point $\xi\in\Lambda(\Gamma)$ is one that is fixed by a parabolic element of $\Gamma$, that is, some element of $\Gamma$ that fixes exactly one element of $\partial(\mathbb{H}^{n})$. A parabolic limit point $\xi\in\Lambda(\Gamma)$ is called bounded parabolic if the stabilizer of $\xi$ in $\Gamma$ acts cocompactly on $\Lambda(\Gamma)-\{\xi\}$. We denote the set of bounded parabolic limit points by $\Lambda_{\operatorname{bp}}(\Gamma)$. In the case of $\Gamma$ convex cocompact,

[TABLE]

If $\Gamma$ is geometrically finite, then

[TABLE]

and $\Lambda_{\operatorname{bp}}(\Gamma)$ is countable, [3].

Fix a reference point $o\in\mathbb{H}^{n}$ and a reference vector $w_{o}\in T_{o}^{1}(\mathbb{H}^{n})$, the unit tangent space of $\mathbb{H}^{n}$ at $o$. Consider the maximal compact subgroup $K:=\operatorname{Stab}_{G}(o)$. Then $\mathbb{H}^{n}$ can be viewed as $G/K$. Similarly, $M$ is $\operatorname{Stab}_{G}(w_{0})$, and $T^{1}(\mathbb{H}^{n})$ can be identified with $G/M$.

With these identifications and the parametrizations in §2, $A$ implements the geodesic flow on $T^{1}(\mathbb{H}^{n})$. That is, if $\{g^{t}:t\in\mathbb{R}\}$ is the geodesic flow on $T^{1}(\mathbb{H}^{n})$, then $g^{t}(w_{0})=[a_{s}M]$, where $[\cdot]$ denotes the coset in $G/M$.

For $w\in T^{1}(\mathbb{H}^{n})$, $w^{\pm}\in\partial(\mathbb{H}^{n})$ denotes the forward or backward endpoints of the geodesic $w$ determines, i.e. $w^{\pm}=\lim\limits_{t\to\infty}g^{t}(w)$. For $g\in G$, we define

[TABLE]

For $x=\Gamma g\in\Gamma\backslash G$, we write $x^{\pm}\in\Lambda(\Gamma)$ if $g^{\pm}\in\Lambda(\Gamma)$ for some representative of the coset. This is well-defined by definition of $\Lambda(\Gamma)$.

Let $x,y\in\mathbb{H}^{n}$ and $\xi\in\partial(\mathbb{H}^{n})$. The Busemann function based at $\xi$ is the function

[TABLE]

where $d$ is the hyperbolic metric and $\xi_{t}$ is a geodesic ray in $\mathbb{H}^{n}$ towards $\xi$.

For $\Gamma<G$ discrete, a $\Gamma$-invariant conformal density of dimension $\delta>0$ is a family $\{\mu_{x}:x\in\mathbb{H}^{n}\}$ of pairwise mutually absolutely continuous finite measures on $\partial(\mathbb{H}^{n})$ satisfying

[TABLE]

for all $x,y\in\mathbb{H}^{n}$ and $\xi\in\partial(\mathbb{H}^{n})$, where $\gamma_{*}\nu_{x}(E)=\nu_{x}(\gamma^{-1}(E))$ for all Borel measurable subsets $E\subseteq\partial(\mathbb{H}^{n})$.

Let $\delta_{\Gamma}$ denote the critical exponent of $\Gamma$. Up to scalar multiplication, there exists a unique $\Gamma$-invariant conformal density of dimension $\delta_{\Gamma}$, denoted by $\{\nu_{x}:x\in\mathbb{H}^{n}\}$, and called the Patterson-Sullivan density.

For each $g\in G$, this density allows us to define the Patterson-Sullivan (PS) measure on a horocycle $gU$ by

[TABLE]

If $x^{-}\in\Lambda_{\operatorname{r}}(\Gamma)$, then the map $u\mapsto xu$ is injective on $U$, and we can define the PS measure on $xU\subseteq\Gamma\backslash G$ by push forward. However, in general, there is some subtlety in defining the PS measure on $xU$; see [18, Section 2.3] for more discussion of this. We note that $\mu^{\operatorname{PS}}_{gU}$ can be viewed as a measure on $U\cong\mathbb{R}^{n-1}$ via $d\mu^{\operatorname{PS}}_{g}(\textbf{t})=d\mu^{\operatorname{PS}}_{gU}(gu_{\textbf{t}})$.

The Lebesgue density is $\{m_{x}:x\in\mathbb{H}^{n}\}$, where $m_{x}$ is the unique probability measure on $\partial(\mathbb{H}^{n})$ that is invariant under $\operatorname{Stab}_{G}(x)$. The Lebesgue density is a $G$-invariant conformal density of dimension $n-1$. We similarly define the Lebesgue measure on $gU$:

[TABLE]

This is independent of the orbit and is in fact a scalar multiple of the Lebesgue measure on $U\cong\mathbb{R}^{n-1}$, denoted by $d\textbf{t}$.

Note that for every Borel measurable subset $E\subseteq U$, every $g\in G$, and every $s\in\mathbb{R}$, the properties of conformal densities imply that

[TABLE]

In particular,

[TABLE]

We record the following properties of PS measure:

Lemma 2.1**.**

[8, Cor. 1.4]** For every $g\in G$, every proper subvariety of $U$ is a null set for $\mu^{\operatorname{PS}}_{g}$.

Lemma 2.2**.**

The map $g\mapsto\mu^{\operatorname{PS}}_{gU}$ is continuous, where the topology on the space of all regular Borel measures on $U$ is given by $\mu_{n}\to\mu\iff\mu_{n}(f)\to\mu(f)$ for all $f\in C_{c}(U)$.

Proof.

The proof follows by the definition of the PS measure, since it is defined using stereographic projection and the Busemann function, which are continuous. ∎

Corollary 2.3**.**

[18, Cor. 2.2]** For any compact set $\Omega\subset G$ and any $T>0$,

[TABLE]

Lemma 2.4**.**

For every compact subset $\Omega\subseteq X_{2}$, there exists $\kappa=\kappa(\Omega)>0$ such that

[TABLE]

Proof.

Because $G\ni g\mapsto d_{\partial\mathbb{H}^{n}}(g^{+},\Lambda(\Gamma))$ is continuous, there exists $\kappa>0$ such that

[TABLE]

for every $x=\Gamma g\in\Omega$, and thus $\mu^{\operatorname{PS}}_{x}(B_{U}(\kappa))>0$ for all $x\in\Omega$.

Since $x\mapsto\mu^{\operatorname{PS}}_{x}$ is also continuous, it follows that $\{\mu^{\operatorname{PS}}_{x}(B_{U}(\kappa)):g\in\Omega\}$ is bounded with a positive lower bound. ∎

2.2. BMS and BR measures

The map $w\mapsto(w^{+},w^{-},\beta_{w^{-}}(o,\pi(w)))$, where $\pi(w)\in\mathbb{H}^{n}$ is the base point of $w$, is a homeomorphism between $T^{1}(\mathbb{H}^{n})$ and

[TABLE]

This identification allows us to define the BMS and BR measures on $T^{1}(\mathbb{H}^{n})\cong G/M$, denoted by $\tilde{m}^{\operatorname{BMS}}$ and $\tilde{m}^{\operatorname{BR}}$:

[TABLE]

[TABLE]

By lifting to $M$-invariant measures on $G$, these will induce locally finite Borel measures on $\Gamma\backslash G$, denoted by $m^{\operatorname{BMS}}$ and $m^{\operatorname{BR}}$; see [18, Section 2.4] for more details. $m^{\operatorname{BMS}}$ is a finite measure [31] (which we will assume to be normalized to a probability measure) and $m^{\operatorname{BR}}$ is infinite if and only if $\Gamma$ is not a lattice, [20].

We have that

[TABLE]

and

[TABLE]

Convex cocompactness is equivalent to $\operatorname{supp}m^{\operatorname{BMS}}$ being compact.

The relationship between $m^{\operatorname{BMS}}$ and $m^{\operatorname{BR}}$ will be important in what follows. Note that by comparing equations (2.2) and (2.3), we see that the most significant difference is in the appearance of $d\nu_{o}(w^{+})$ vs. $dm_{o}(w^{+})$, that is, the major difference is in how they see the $U$ direction. In particular, $m^{\operatorname{BR}}$ is $U$-ergodic when $\Gamma$ is Zariski dense [32], while $m^{\operatorname{BMS}}$ is not even $U$-invariant. Moreover,

[TABLE]

3. Rigidity of $U$-equivariant set-valued maps

In this section, we assume only that the $\Gamma_{i}$’s are geometrically finite and Zariski dense subgroups of $G$. We will prove the following theorem, which includes Theorem 1.1 in the first case. Note that the codomain of $\Upsilon$ in the following statement is a complete metric space with the Hausdorff metric.

Theorem 3.1**.**

Let $\ell\in\mathbb{N}$, and suppose that

[TABLE]

is a measurable map and is $U$-equivariant on a $m^{\operatorname{BR}}$-conull set. Suppose further that either

1. (1)

$\ell=1$, or 2. (2)

there exists a $\Delta(U)$-ergodic measure $\mu$ on $X=X_{1}\times X_{2}$ such that

• •

if $Z\subseteq X_{2}$ is $m^{\operatorname{BR}}$-conull, then $\bigcup\limits_{x_{2}\in Z}(\Upsilon(x_{2})\times\{x_{2}\})$ is $\mu$-conull, and

• •

if $W\subseteq X$ is $\mu$-conull, then $\pi_{2}(W)$ is $m^{\operatorname{BR}}$-conull.

Then there exists a map

[TABLE]

a constant $\bm{\sigma}_{0}\in U$, and a $U$-invariant $m^{\operatorname{BR}}$-conull set $X_{2}^{\prime}$ such that for all $x\in X_{2}^{\prime}$,

• •

$\hat{\Upsilon}(x)=\Upsilon(x)\bm{\sigma}_{0}$,

• •

$\hat{\Upsilon}(xh)=\hat{\Upsilon}(x)h$* for all $h\in AMU$, and*

• •

$\hat{\Upsilon}(xv)=\hat{\Upsilon}(x)v$* for all $v\in U^{-}$ such that $xv\in X_{2}^{\prime}$.*

Throughout this section, suppose that $\ell$ and $\Upsilon$ are as in Theorem 3.1, and that $\tilde{X}_{2}\subseteq X_{2}$ is such that

[TABLE]

for all $u\in U$ and $x\in\tilde{X}_{2}$.

By a standard argument for constructing measurable cross-sections, we may also assume that there exist measurable maps $\upsilon_{i}:\tilde{X_{2}}\to X_{1}$ such that

[TABLE]

for all $x\in\tilde{X_{2}}.$ The following construction shows that we can further assume that the maps $\upsilon_{i}$ are defined $m^{\operatorname{BMS}}$-a.e.: let $\{B_{n}:n\in\mathbb{N}\}$ be a countable collection of balls that cover $X_{2}$ and satisfy $m^{\operatorname{BMS}}(\partial B_{n})=0$. Proceed inductively: for $x\in(B_{n}\cap AMU^{-}\cap\tilde{X}_{2})-B_{n-1}$ and $u$ so that $xu\in B_{n}-B_{n-1}$, define

[TABLE]

This shows that the $\upsilon_{i}$’s can be measurably defined on $\tilde{X}_{2}U-\bigcup\limits_{n=1}^{\infty}\partial B_{n}$, a $m^{\operatorname{BMS}}$-conull set.

We will assume throughout this section that either

1. (1)

$\ell=1$, or 2. (2)

there exists a $\Delta(U)$-ergodic measure $\mu$ on $X=X_{1}\times X_{2}$ such that

• •

if $Z\subseteq X_{2}$ is $m^{\operatorname{BR}}$-conull, then $\bigcup\limits_{x_{2}\in Z}(\Upsilon(x_{2})\times\{x_{2}\})$ is $\mu$-conull, and

• •

if $W\subseteq X$ is $\mu$-conull, then $\pi_{2}(W)$ is $m^{\operatorname{BR}}$-conull.

Remark 3.2**.**

These further assumptions are needed only in the proof of Lemma 3.15.**

Remark 3.3**.**

In our application to joining classification in §5, the conditions in case (2) will be satisfied with $\mu$ an ergodic $U$-joining for $(m^{\operatorname{BR}}_{1},m^{\operatorname{BR}}_{2})$ and

[TABLE]

That this $\Upsilon$ will take values in cardinality $\ell$ subsets of $X_{1}$ for some $\ell\in\mathbb{N}$ is not immediately clear, and this is proven in §5.3.**

§3.2 is dedicated to the proof of the following theorem:

Theorem 3.4**.**

There exists a $U$-invariant $m^{\operatorname{BR}}$-conull set $Y\subseteq X_{2}^{\prime}$, a constant $\bm{\sigma}_{0}\in U$, and $\tilde{\Upsilon}:X_{2}\to\{\text{cardinality }\ell\text{ subsets of }X_{1}\}$ satisfying:

1. (1)

$\tilde{\Upsilon}=\Upsilon$* on $Y$,* 2. (2)

for all $x\in Y,$ and all $h\in AMU$, $\tilde{\Upsilon}(xh)u_{\bm{\sigma}_{0}}=\tilde{\Upsilon}(x)u_{\bm{\sigma}_{0}}h.$

The heart of the argument lies in the following proposition.

Proposition 3.5**.**

For all sufficiently small $\eta>0$, there exists $\overline{\varepsilon}>0$ with $\overline{\varepsilon}\to 0$ as $\eta\to 0$ and a $U$-invariant $m^{\operatorname{BR}}$-conull set $\hat{X}_{h}$ satisfying: for every $h\in B_{AM}(\eta)$, every $x\in\hat{X}_{h}$, and every $1\leq i\leq\ell$, there exists a unique $1\leq k(i)\leq\ell$ and ${\bm{\tau}}_{h}(x,\upsilon_{i}(x))\in B_{U}(\overline{\varepsilon})$ such that

[TABLE]

Moreover, ${\bm{\tau}}_{h}(x,\upsilon_{i}(x))={\bm{\tau}}_{h}(xu_{\textbf{t}},\upsilon_{i}(x)u_{\textbf{t}})$ for all $\textbf{t}\in U$.

An analogous statement is proven for convex cocompact groups and assuming $\Upsilon$ is a factor map in [8], following Ratner’s approach from [22]. In [23], Ratner proves this in the case of finite volume set-valued maps, which she refers to as measurable partitions; we follow the general lines of her approach in this section. Lemma 3.7, which appears in §3.2 and is a modification of [18, Lemma 6.2], is vital to our approach for generalizing this to the infinite volume case.

In §3.3, we complete our rigidity statement by proving the following theorem.

Theorem 3.6**.**

Let $\hat{\Upsilon}(x)=\tilde{\Upsilon}(x)u_{\bm{\sigma}_{0}}$, where $\tilde{\Upsilon}$ is as in Theorem 3.4. There exists a $m^{\operatorname{BR}}$-conull set $X_{2}^{\prime\prime}\subseteq X_{2}^{\prime}$ such that for all $x\in X_{2}^{\prime\prime}$ and for every $v_{\textbf{r}}\in U^{-}$ with $xv_{\textbf{r}}\in X_{2}^{\prime\prime}$, we have

[TABLE]

3.1. Notation

We provide here a summary of important notation in this section for ease of reference for the reader. This is only a list of notation; full explanations are given in the following section. In particular, the reader may first skip this list and only refer to it when needed.

The constant $\ell$, the set-valued map $\Upsilon$, the set $\tilde{X_{2}}$, and the functions $\upsilon_{i}$.

Fix $\ell\in\mathbb{N}$. Then $\tilde{X}_{2}\subseteq X_{2}$ is a $U$-invariant and $m^{\operatorname{BR}}$-conull set, and

[TABLE]

is a measurable map such that

[TABLE]

for all $u\in U$ and $x\in\tilde{X}_{2}$. The $\upsilon_{i}$’s are measurable maps $\upsilon_{i}:\tilde{X_{2}}\to X_{1}$ such that

[TABLE]

for all $x\in\tilde{X_{2}}.$

The measure $\mu$

If $\ell\neq 1$, we assume that there exists a $\Delta(U)$-ergodic measure $\mu$ on $X=X_{1}\times X_{2}$ such that

• •

if $Z\subseteq X_{2}$ is $m^{\operatorname{BR}}$-conull, then $\bigcup\limits_{x_{2}\in Z}(\Upsilon(x_{2})\times\{x_{2}\})$ is $\mu$-conull, and

• •

if $W\subseteq X$ is $\mu$-conull, then $\pi_{2}(W)$ is $m^{\operatorname{BR}}$-conull.

The set $\mathcal{P}_{d,m}$

For $d,m>0$, define $\mathcal{P}_{d,m}$ to be the set of functions $\Theta:U\to\mathbb{R}$ of the form

[TABLE]

where the $P_{i}:U\to\mathbb{R}$ are polynomials of degree at most $d$.

The set $X_{2}^{\prime}$ and the constant $\rho_{0}$

The set $X_{2}^{\prime}\subseteq\tilde{X}_{2}$ is a $U$-invariant and $m^{\operatorname{BR}}$-conull set on which there exists a constant $\rho_{0}>0$ such that for all $x\in X_{2}^{\prime}$,

[TABLE]

(See Lemma 3.8.)

The functions $\Theta_{x,h,i}$,$\overline{\Theta}_{x,h,i}$, and $q_{i,j}$

Define

[TABLE]

[TABLE]

and

[TABLE]

The constant $\rho$

Let $\rho>0$ be such that if $d(xu_{\textbf{t}},yu_{\textbf{t}})<\rho$, then there exists some finite collection of polynomials $p_{i}$ of degree at most $n$ such that

[TABLE]

The constants $\varepsilon^{\prime},\xi$ and the set $K$

Let $0<\varepsilon^{\prime}<1/2$ and let $0<\xi<\varepsilon^{\prime}/10$. $K\subseteq X_{2}^{\prime}$ is a compact set with $m^{\operatorname{BMS}}(K)>1-\xi$ on which every $\upsilon_{i}$ is uniformly continuous.

The constant $C$

Let $C$ be the constant from Lemma 3.7 for the compact set $KB_{U}(1)$, with $d$ and $m$ chosen so that all polynomials arising from the ${\Theta}_{x,h,i}$’s and the $q_{i,j}$’s are elements of $\mathcal{P}_{d,m}$.

The constants $C^{\prime},\alpha$

Constants such that for all balls $V\subseteq\mathbb{R}^{n-1}$ and all $\varepsilon>0$,

[TABLE]

for all $i,j$, where $\lambda$ is the Lebesgue measure on $U\cong\mathbb{R}^{n-1}$. See equation (3.1).

The constants $R$, $\varepsilon$ and $\tilde{\rho}$

Let $R$ be the injectivity radius of $K$. Define

[TABLE]

and

[TABLE]

The constants $\eta,\beta$

Let $\beta>0$ be such that for all $x,y\in K$,

[TABLE]

for every $1\leq i\leq\ell$ and $0<\eta<\varepsilon^{\prime}$ is such that for all $x\in X_{2},$

[TABLE]

The set $P_{\xi}$

Define

[TABLE]

It is a $U$-invariant $m^{\operatorname{BR}}$-conull set.

The set $L_{r,K}$

Define

[TABLE]

It is $U$-invariant and $m^{\operatorname{BR}}$-conull.

The set $\hat{X}_{h}$

For $h\in B_{AM}(\eta)$,

[TABLE]

It is $U$-invariant and $m^{\operatorname{BR}}$-conull.

The subgroups $A^{\prime},M^{\prime}$

Let $A^{\prime}\subseteq A$ and $M^{\prime}\subseteq M$ be countable dense subgroups.

3.2. $U$-equivariant implies $AM$-equivariant

In this section, we prove Theorem 3.4.

For $d,m>0$, define $\mathcal{P}_{d,m}$ to be the set of functions $\Theta:U\to\mathbb{R}$ of the form

[TABLE]

where the $P_{i}:U\to\mathbb{R}$ are polynomials of degree at most $d$.

The following lemma is critical in adapting to the infinite volume setting. Roughly speaking, it says that PS measure “sees” the growth of polynomials: they cannot be “small” everywhere within the support of the PS measure, because the PS measure is ‘friendly’ in the sense of [10].

Lemma 3.7**.**

Fix $d,m>0$. For any compact set $\Omega$, let $\kappa=\kappa(\Omega)$ be as in Lemma 2.4. Then there exists some $C=C(\Omega,d,m)>0$ satisfying the following: for every $x\in L_{r,\Omega}$ and $T>0$ such that $xa_{-\log(T/\kappa)}\in\Omega$ and for every $\Theta\in\mathcal{P}_{d,m}$, we have

[TABLE]

where $L_{r,\Omega}=\{x:\text{there exists }s_{n}\to\infty\text{ with }xa_{-s_{n}}\in\Omega\}$.

Proof.

Observe that for any $\kappa,T>0$,

[TABLE]

Now, assume for contradiction that the claim is false. Then, by scaling the $\Theta$’s if necessary, we may assume that we have:

• •

sequences $x_{i}\in L_{r,\Omega}$, $s_{i}\to\infty$ such that $y_{i}=x_{i}a_{-\log{(s_{i}/\kappa)}}\in\Omega$

• •

$\tilde{\Theta}_{i}\in\mathcal{P}_{d,m}$ with $\sup\limits_{B_{U}(\kappa)}\tilde{\Theta}_{i}(\textbf{t})=1$

satisfying $\frac{1}{\mu^{\operatorname{PS}}_{y_{i}}(B_{U}(\kappa))}\int_{B_{U}(\kappa)}\tilde{\Theta}_{i}(\textbf{t})d\mu^{\operatorname{PS}}_{y_{i}}(\textbf{t})\to 0$ as $i\to\infty$.

Since the $\tilde{\Theta}_{i}$’s are given by uniformly bounded polynomials of bounded degree, they form an equicontinuous family. Thus, by dropping to a subsequence if necessary, we may assume that there exists $y\in\Omega$ and $\tilde{\Theta}\in\mathcal{P}_{d,m}$ with $\sup\limits_{\textbf{t}\in B_{U}(\kappa)}\tilde{\Theta}(\textbf{t})=1$ such that $y_{i}\to y$ and $\tilde{\Theta}_{i}\to\tilde{\Theta}$.

Since $g\mapsto\mu^{\operatorname{PS}}_{g}$ is continuous by Lemma 2.2 and $\sup\limits_{g\in\Omega}\mu^{\operatorname{PS}}_{g}(B_{U}(\kappa))<\infty$ by Lemma 2.4, we then have that

[TABLE]

Thus,

[TABLE]

Since $\mu^{\operatorname{PS}}_{y}(B_{U}(\kappa))>0$ by definition of $\kappa$, this implies

[TABLE]

a contradiction to Lemma 2.1. ∎

Recall the setup: $\ell\in\mathbb{N}$ and

[TABLE]

is such that there exists a $U$-invariant $m^{\operatorname{BR}}$-conull set $\tilde{X}_{2}\subseteq X_{2}$ with

[TABLE]

for all $x\in\tilde{X}_{2}$ and $u\in U$. There are measurable maps $\upsilon_{i}:\tilde{X}_{2}\to X_{1}$ such that

[TABLE]

for all $x\in\tilde{X}_{2}$.

Lemma 3.8**.**

There exists a $U$-invariant $m^{\operatorname{BR}}$-conull set $X_{2}^{\prime}\subseteq\tilde{X}_{2}$ and a constant $\rho_{0}>0$ such that for all $x\in X_{2}^{\prime}$ and all $1\leq i,j\leq\ell$,

[TABLE]

That is, there is a positive minimum distance in the $U$ direction within $\Upsilon(x)$.

Proof.

Define $f:\tilde{X}_{2}\to\mathbb{R}\cup\{+\infty\}$ by

[TABLE]

Suppose that $\upsilon_{i}(x)=\upsilon_{j}(x)u_{\textbf{t}}$ for some $i,j$ and $u_{\textbf{t}}\in U,\textbf{t}\neq 0$. By $U$-equivariance of $\Upsilon$, for any $u\in U$, there exist $i^{\prime},j^{\prime}$ such that

[TABLE]

Thus,

[TABLE]

so $f(xu)\leq f(x)$. Swapping the roles of $x$ and $xu$ shows that $f(x)=f(xu)$.

Hence, by the ergodicity of $m^{\operatorname{BR}}$, there exists a $m^{\operatorname{BR}}$-conull set $X_{2}^{\prime}\subseteq\tilde{X}_{2}$ on which $f$ is constant. If $f\equiv+\infty$ on $X_{2}^{\prime}$, then define $\rho_{0}=1$; otherwise, let $\rho_{0}$ be the value of $f$ on $X_{2}^{\prime}$. It is positive by definition of $f$. ∎

Restricting to $x\in X_{2}^{\prime}$, where $X_{2}^{\prime}$ is as in Lemma 3.8, is necessary to ensure the uniqueness of ${\bm{\tau}}_{h}$ in Proposition 3.5.

For $x\in X_{2}^{\prime},h\in AM,$ and $1\leq i\leq\ell$, define

[TABLE]

and

[TABLE]

We will primarily work with $\Theta_{x,h,i}$, but $\overline{\Theta}_{x,h,i}$ comes into play when we use the $U$-equivariance of $\Upsilon$. Also define

[TABLE]

The main idea of the proof is to show that $q_{i,j}(\textbf{t})$ stays bounded as $|\textbf{t}|\to\infty$, showing that the points $\upsilon_{i}(x),\upsilon_{j}(xh)h^{-1}$ stay in the same $U$ orbit.

The following lemma is well known by the polynomial divergence of $U$ orbits. Recall from (2.1) that

[TABLE]

where $\|\cdot\|$ denotes the max norm.

Lemma 3.9**.**

There exists $\rho>0$ such that if $d(xu_{\textbf{t}},yu_{\textbf{t}})<\rho$, then there exists some finite collection of polynomials $p_{i}$ of degree at most $n$ such that

[TABLE]

Note that the $\Theta_{x,h,i}$’s and $q_{i,j}$’s can be controlled by polynomials in this sense, but not necessarily the $\overline{\Theta}_{x,h,i}$’s, since the $u_{\textbf{t}}$’s are inside the $\upsilon_{i}$’s here.

Let $0<\varepsilon^{\prime}<1/2$ and $0<\xi<\varepsilon^{\prime}/10$. By Lusin’s theorem, there exists a compact set $K\subseteq X_{2}^{\prime}$ with

[TABLE]

on which every $\upsilon_{i}$ is uniformly continuous. Let $d,m>0$ be such that all of the polynomials arising from the $\Theta_{x,h,i}$’s and $q_{i,j}$’s are elements of $\mathcal{P}_{d,m}$, and let

[TABLE]

Recall that polynomials are $(C^{\prime},\alpha)$ good on $\mathbb{R}^{n-1}$ [9, 10]: there exist constants $C^{\prime},\alpha$ that depend only on the degree of the polynomial $f$ and the dimension of the space such that for all balls $V$ and all $\varepsilon>0$,

[TABLE]

where $\lambda$ denotes the Lebesgue measure.

Choose $C^{\prime}$ and $\alpha$ such that (3.1) holds when $f=q_{i,j}$ for all $i,j$. Let $R$ be the injectivity radius of $K$, let $\rho_{0}$ be as in Lemma 3.8, and define

[TABLE]

and

[TABLE]

We remark that these have been defined to achieve three things:

• •

the $\Theta_{x,h,i}$’s and $q_{i,j}$’s will be controlled by polynomials and we will stay within the injectivity radius throughout our arguments;

• •

the definition of $\varepsilon$, together with Corollary 3.11, will give a contradiction in the proof of Lemma 3.12; and

• •

$\overline{\varepsilon}:=(C^{\prime}\ell)^{1/\alpha}\tilde{\rho}$, which will arise in the proof of Proposition 3.5, is less than $\frac{1}{10}\rho_{0}$, giving uniqueness of ${\bm{\tau}}_{h}$ in that proof.

Now, let $\beta>0$ be such that for all $x,y\in K$,

[TABLE]

for every $1\leq i\leq\ell$ and let $0<\eta<\varepsilon^{\prime}$ be such that for all $x\in X_{2},$

[TABLE]

Define

[TABLE]

By (2.4), it is a $m^{\operatorname{BR}}$-conull set because it is $U$-invariant and has $m^{\operatorname{BMS}}(P_{\xi})=1$, [27, Theorem 17]. The following lemma will allow us to control the $\Theta_{x,h,i}$’s by understanding $P_{\xi}$.

For $r>0$, let

[TABLE]

where we recall that $B(r)$ denotes the ball of radius $r$ in $G$ using the max norm.

Lemma 3.10**.**

If $x\in P_{\xi}\cap P_{\xi}h^{-1}$ for $h\in B_{AM}(\eta)$, then there exists $T_{1}=T_{1}(x,h)$ such that for all $T\geq T_{1}$,

[TABLE]

Proof.

Since $xh\in P_{\xi}$, there exists $T_{0}>0$ such that for all $T\geq T_{0}$,

[TABLE]

Write $h=a_{s}m$. Recall that $\mu^{\operatorname{PS}}_{ya_{s}}(E)=e^{\delta_{\Gamma}s}\mu^{\operatorname{PS}}_{y}(a_{s}Ea_{-s})$, $mu_{\textbf{t}}m^{-1}=u_{\textbf{t}m}$, and $a_{s}u_{\textbf{t}}a_{-s}=u_{\textbf{t}e^{-s}}$. Note also that $\mu^{\operatorname{PS}}_{ym}(E)=\mu^{\operatorname{PS}}_{y}(mEm^{-1})$. Using these, we have

[TABLE]

Where the last line is because we are using the max norm on $U$, not the Euclidean norm. Similarly,

[TABLE]

Putting this together, we conclude that for all $T\geq\max\{(\sqrt{n-1})e^{-s}T_{0},T_{0}\}$,

[TABLE]

Since $x\in P_{\xi}$, we can choose $T_{1}\geq\max\{(\sqrt{n-1})e^{-s}T_{0},T_{0}\}$ so that for all $T\geq T_{1}$,

[TABLE]

Intersecting the sets on the left hand sides of equations (3.6) and (3.7) yields the claim. ∎

Define

[TABLE]

By Poincaré recurrence and ergodicity of $A$, $m^{\operatorname{BMS}}(L_{r,K})=1$. It is also $U$-invariant, hence $m^{\operatorname{BR}}$-conull by (2.4). Staying within this set will be necessary for our applications of Lemma 3.7 throughout this section.

For $h\in B_{AM}(\eta)$, define

[TABLE]

The set $\hat{X}_{h}$ is also $U$-invariant and $m^{\operatorname{BR}}$-conull. We will show that Proposition 3.5 holds with this $\hat{X}_{h}$. Recall that by the definitions of $P_{\xi}$ and $X_{2}^{\prime}$, if $x\in\hat{X}_{h}$, this means that both $x$ and $xh$ have many returns to $K$ under $U$ and $\Upsilon$ is $U$-equivariant at both $x$ and $xh$.

Corollary 3.11**.**

If $x\in\hat{X}_{h}$ with $h\in B_{AM}(\eta)$ and $T_{1}$ is as in Lemma 3.10, then for all $T\geq T_{1}$ and for all $1\leq i\leq\ell$,

[TABLE]

Proof.

Let $D(T):=\{\textbf{t}\in B_{U}(T):xu_{\textbf{t}},xu_{\textbf{t}}h\in K\}$. On $D(T)$, ${\Theta}_{x,h,i}(\textbf{t})<\varepsilon^{2}$ because, pointwise, there exists $j(\textbf{t})$ such that

[TABLE]

(because $x,xh\in X_{2}^{\prime}$), and it is clear for the $\overline{\Theta}$’s by definition of $K$ and choice of $\eta$ in (3.4). On $B(T)-D(T)$, it is bounded by 1. Thus,

[TABLE]

∎

Lemma 3.12**.**

For $h\in B_{AM}(\eta)$ and $x\in\hat{X}_{h}$ (defined in (3.8)), there exists $T_{0}>0$ such that for all $i$ and all $T\geq T_{0}$,

[TABLE]

Proof.

Suppose not and let $T_{1}$ be as in Lemma 3.10. Let $\kappa=\kappa(KB_{U}(1))$ from Lemma 3.7. Since $x\in L_{r,K}$, there exists $T\geq T_{1}$ sufficiently large so that $xa_{-\log(T/\kappa)}\in K$ and $\sup\limits_{\textbf{t}\in B_{U}(T)}\Theta_{x,h,i}(\textbf{t})\geq\tilde{\rho}$.

Let $\Omega_{x,h,i}(\textbf{t})=\min\{\rho^{2},\Theta_{x,h,i}(\textbf{t})\}$. Recall by definition of $\rho$ that this means $\Omega_{x,h,i}(\textbf{t})$ is given by polynomials in the sense of Lemma 3.7. Thus, we have that

[TABLE]

and $\Theta_{x,h,i}\geq\Omega_{x,h,i}$, so the same is true for that function. This contradicts Corollary 3.11 by the definition of $\tilde{\rho}=10\varepsilon/C$. ∎

Recall the definition

[TABLE]

Corollary 3.13**.**

For $h\in B_{AM}(\eta)$, $x\in\hat{X}_{h}$ (defined in (3.8)) and $1\leq i\leq\ell$, there exists $1\leq k(i)\leq\ell$ and $T_{1}>0$ such that for all $T\geq T_{1}$,

[TABLE]

satisfies

• •

$\lambda(J(T,i,k(i)))\geq\frac{1}{\ell}\lambda(B_{U}(T))$, where $\lambda$ is the Lebesgue measure on $U$, and

• •

$\sup\limits_{\textbf{t}\in J(T,i,k(i))}q_{i,k(i)}(\textbf{t})<\tilde{\rho}$.

Proof.

Suppose not. Then there exists $h\in B_{AM}(\eta),x\in\hat{X}_{h}$, and $1\leq i\leq\ell$ such that for all $1\leq k(i)\leq\ell$ and for all $T_{1}>0$, there exists $T\geq T_{1}$ such that either

[TABLE]

or

[TABLE]

By the pigeonhole principle, there exists $k(i)$ and $T_{n}\to\infty$ such that for all $n$,

[TABLE]

Thus, it must be that

[TABLE]

However, this contradicts Lemma 3.12. ∎

Lemma 3.14**.**

If $x,y\in X_{i}$ are such that $d(xu_{\textbf{t}},yu_{\textbf{t}})$ stays bounded for all $\textbf{t}\in U$, then there exists $u\in U$ such that

[TABLE]

Proof.

By direct computation, we will show that if $x=yg$, then $g\in C_{G}(U)=U$. Write

[TABLE]

where $\textbf{b},\textbf{d},\textbf{f},\textbf{i}\in\mathbb{R}^{n-1}$ are row vectors, and $E$ is a $(n-1)\times(n-1)$ matrix. By assumption, $\|u_{-\textbf{t}}gu_{\textbf{t}}-1_{G}\|$ stays bounded for all $\textbf{t}\in U$. We will investigate the entries of $u_{-\textbf{t}}gu_{\textbf{t}}$.

The (1,1) entry is $a+\textbf{b}\cdot\textbf{t}+\frac{1}{2}c|\textbf{t}|^{2}$. Since this stays bounded for all t, we conclude that

[TABLE]

The (2,1) entry is $-a\textbf{t}^{T}+\textbf{d}^{T}+E\textbf{t}^{T}-\frac{1}{2}c|\textbf{t}|^{2}\textbf{t}^{T}+\frac{1}{2}|\textbf{t}|^{2}\textbf{f}^{T}$. Again, since this stays bounded, we conclude that

[TABLE]

where $I$ denotes the $(n-1)\times(n-1)$ identity matrix.

The (3,2) entry is $\frac{1}{2}|\textbf{t}|^{2}\textbf{b}-\textbf{t}E+\textbf{i}+\frac{1}{2}c|\textbf{t}|^{2}-(\textbf{t}\cdot\textbf{f})\textbf{t}+j\textbf{t}$. From this, we conclude that

[TABLE]

But combining all of our conclusions up to this point tells us that $g$ is block lower triangular with $E=aI=jI$, so

[TABLE]

Hence

[TABLE]

With our above assumptions, the (3,2) entry is simply i. The (3,1) entry simplifies to $h+(\textbf{i}-\textbf{d})\cdot\textbf{t}$, so

[TABLE]

from which we finally conclude that

[TABLE]

completing the proof. ∎

We are now ready to prove Proposition 3.5.

Proof of Proposition 3.5.

If $J(T,i,k(i))$ is as in Corollary 3.13, then by definition of $C^{\prime},\alpha$ from (3.1) for $f=q_{i,k(i)}$, we have for all $T>0$,

[TABLE]

which yields

[TABLE]

Thus, $q_{i,k(i)}(\textbf{t})$ stays bounded for all t, and so $\upsilon_{i}(x)$ and $\upsilon_{k(i)}(xh)h^{-1}$ are in the same $U$-orbit by Lemma 3.14. In particular, since the bound above holds at $\textbf{t}=0$, this tells us that there exists a ${\bm{\tau}}_{h}(x,\upsilon_{i}(x))\in B_{U}(\overline{\varepsilon})$ such that

[TABLE]

where $\overline{\varepsilon}=(C^{\prime}\ell)^{1/\alpha}\tilde{\rho}$.

Note that the restriction ${\bm{\tau}}_{h}(x,\upsilon_{i}(x))\in B_{U}(\overline{\varepsilon})$ ensures that this quantity is unique, because the constants have been chosen such that

[TABLE]

where $\rho_{0}$ is from Lemma 3.8, and is the minimum distance in the $U$-direction in $\Upsilon(x)$. Thus, if there is another element $\upsilon_{k(i)^{\prime}}(xh)\in\Upsilon(xh)$ such that

[TABLE]

for some t, we must have that $|\textbf{t}|\geq\rho_{0}$, hence $u_{\textbf{t}}\not\in B_{U}(\overline{\varepsilon})$.

$U$-invariance of ${\bm{\tau}}_{h}$ follows from the $U$-equivariance of $\Upsilon$ on $X_{2}^{\prime}$: let $u_{\textbf{s}}$ be such that $u_{\textbf{t}}h=hu_{\textbf{s}}$. Then

[TABLE]

and $\upsilon_{k(i)}(xh)u_{\textbf{s}}\in\Upsilon(xu_{\textbf{t}}h)$ by $U$-equivariance. ∎

Let $A^{\prime}\subseteq A$ and $M^{\prime}\subseteq M$ be countable dense subgroups. Recall the assumptions made at the beginning of the section: either

1. (1)

$\ell=1$, or 2. (2)

there exists a $\Delta(U)$-ergodic measure $\mu$ on $X=X_{1}\times X_{2}$ such that

• •

if $Z\subseteq X_{2}$ is $m^{\operatorname{BR}}$-conull, then $\bigcup\limits_{x_{2}\in Z}(\Upsilon(x_{2})\times\{x_{2}\})$ is $\mu$-conull, and

• •

if $W\subseteq X$ is $\mu$-conull, then $\pi_{2}(W)$ is $m^{\operatorname{BR}}$-conull.

It is in the following lemma that these assumptions are needed.

Lemma 3.15**.**

For every $h\in B_{A^{\prime}M^{\prime}}(\eta)$, there exists a $U$-invariant $m^{\operatorname{BR}}$-conull set $W_{h}\subseteq X_{2}^{\prime}$ and a constant ${\bm{\tau}}_{h}\in B_{U}(\overline{\varepsilon})$ such that for all $x\in W_{h}$ and all $1\leq i\leq\ell$, ${\bm{\tau}}_{h}(x,\upsilon_{i}(x))={\bm{\tau}}_{h}$.

Proof.

We first prove this in case (1), that is, we asssume that $\ell=1$. Then the second variable in ${\bm{\tau}}_{h}$ is redundant; instead, consider ${\bm{\tau}}_{h}:\hat{X}_{h}\to U$ as simply ${\bm{\tau}}_{h}(x)$, where $\hat{X}_{h}$ is as in equation (3.8). By Proposition 3.5,

[TABLE]

for all $x\in\hat{X}_{h}$, $u\in U$. Thus, by $U$-ergodicity of $m^{\operatorname{BR}}$, there exists a $m^{\operatorname{BR}}$-conull set $W_{h}\subseteq X_{2}^{\prime}$ and a constant ${\bm{\tau}}_{h}\in B_{U}(\overline{\varepsilon})$ such that for all $x\in W_{h}$,

[TABLE]

This completes the proof of the first case.

Now, suppose we are in case (2), so such an ergodic measure $\mu$ exists. Define

[TABLE]

$\hat{W}_{h}$ is exactly the domain of ${\bm{\tau}}_{h}$, and is $\mu$-conull because $\hat{X}_{h}$ is $m^{\operatorname{BR}}$-conull and our assumptions on $\mu$. Thus, ${\bm{\tau}}_{h}$ is defined $\mu$-a.e. on $X_{1}\times X_{2}$.

As noted in Proposition 3.5, ${\bm{\tau}}_{h}$ is $\Delta(U)$-invariant. Thus, by ergodicity of $\mu$, there exists a $U$-invariant $\mu$-conull set $\tilde{W}_{h}\subseteq\hat{W}_{h}$ and a constant ${\bm{\tau}}_{h}\in B_{U}(\overline{\varepsilon})$ such that

[TABLE]

Now, define

[TABLE]

By the second assumption about $\mu$, it is $m^{\operatorname{BR}}$-conull. It satisfies the desired conditions by construction. ∎

The following lemma will allow us to drop the restriction that $|h|<\eta$.

Lemma 3.16**.**

For any $h\in B_{A^{\prime}M^{\prime}}(\eta)$, there exists a $U$-invariant $m^{\operatorname{BR}}$-conull set $Y_{h}$ with the property that for every $n\in\mathbb{N}$, there exists ${\bm{\tau}}_{h^{n}}\in U$ such that for all $x\in Y_{h}$,

[TABLE]

Moreover, ${\bm{\tau}}_{h^{n}}={\bm{\tau}}_{h}+e^{-s}{\bm{\tau}}_{h^{n-1}}m^{-1}$ and if $x\in Y_{h}$, so is $xh^{n}$ for all $n\in\mathbb{N}$. Thus, ${\bm{\tau}}_{h}$ is defined in a way that satisfies (3.9) for all $h\in A^{\prime}M^{\prime}$.

Proof.

Define $Y_{h}:=\bigcap\limits_{n\in\mathbb{N}}W_{h}h^{-n}$ where $W_{h}$ is as in Lemma 3.15. Observe that ${\bm{\tau}}_{h^{n}}$ satisfies

[TABLE]

Since $x{h^{n}}\in W_{h}$ as well, we can proceed by induction:

[TABLE]

where $h=a_{s}m$. This shows that

[TABLE]

extends the definition to $h^{n}$ if it did not already exist (i.e. if $h^{n}\not\in B_{AM}(\eta)$), or alternatively that if it is already defined, then ${\bm{\tau}}_{h^{n}}$ satisfies this identity. ∎

Lemma 3.17**.**

Define $Y=\bigcap\limits_{h\in A^{\prime}M^{\prime}}Y_{h}$. For all $x\in Y$,

[TABLE]

is uniformly continuous.

Proof.

For $\varepsilon^{\prime}>0$, define $\overline{\varepsilon}=(C^{\prime}\ell)^{1/\alpha}\tilde{\rho}$ as in the proof of Proposition 3.5. In particular, since $\tilde{\rho}=10\varepsilon/C$, the definition of $\varepsilon$ in equation (3.2) implies that

[TABLE]

Now, if $w,z\in xA^{\prime}M^{\prime}$ with $d(w,z)<\eta$, where $\eta$ is defined in (3.4), then there exists $h\in B_{A^{\prime}M^{\prime}}(\eta)$ such that $z=wh$. Then, by Proposition 3.5,

[TABLE]

where ${\bm{\tau}}_{h}\in B_{U}(\overline{\varepsilon})$. Thus, $d(\Upsilon(z),\Upsilon(w))$ is bounded in terms of $\eta$ and $\overline{\varepsilon}$ whenever $d(w,z)<\eta$, and both $\eta$ and $\overline{\varepsilon}$ are independent of $z$ and $w$. ∎

Corollary 3.18**.**

There exists $\tilde{\Upsilon}:X_{2}\to\{\text{cardinality }\ell\text{ subsets of }X_{1}\}$ such that:

1. (1)

$\Upsilon(x)=\tilde{\Upsilon}(x)$* for all $x\in Y$ (which is $m^{\operatorname{BR}}$-conull and both $U$ and $A^{\prime}M^{\prime}$-invariant);* 2. (2)

$\tilde{\Upsilon}|_{xAM}$* is continuous for every $x\in Y$;* 3. (3)

and for all $x\in Y$ and $h\in AM$, there exists ${\bm{\tau}}_{h}$ such that

[TABLE]

Moreover, this extension of the function $h\mapsto{\bm{\tau}}_{h}$ is continuous on $AM$.

Proof.

Since $X_{1}$ is complete (because $G=\operatorname{SO}(n,1)^{\circ}$ is complete and $\Gamma_{1}$ is closed), $\{\text{cardinality }\ell\text{ subsets of }X_{1}\}$ is a complete metric space with the Hausdorff metric. Thus, $\Upsilon|_{xA^{\prime}M^{\prime}}$ extends continuously to $xAM$ by the uniform continuity in Lemma 3.17. Call this continuous extension $\tilde{\Upsilon}$. Clearly, (1) and (2) are satisfied.

Let $x\in Y$, $h\in AM$, and let $h_{n}\in A^{\prime}M^{\prime}$ be such that $h_{n}\to h$. Then we have that

[TABLE]

By the continuity of $\tilde{\Upsilon}$ on $xAM$, the right hand side converges to

[TABLE]

which defines ${\bm{\tau}}_{h}$ in a way that satisfies (3). ∎

Corollary 3.19**.**

For every $h\in AM-M$, there exists $\bm{\sigma}_{h}\in U$ satisfying $\bm{\sigma}_{h^{n}}=\bm{\sigma}_{h}$ for all $n\in\mathbb{N}$ and such that

[TABLE]

for all $x\in Y$. Moreover, $\bm{\sigma}$ is continuous on $AM-M$.

Proof.

Let $h\in AM-M$ and write $h=a_{s}m$. By assumption on $h$, $s\neq 0$, so $I-e^{-s}m^{-1}$ is invertible. Define

[TABLE]

It follows from the recurrence formula for ${\bm{\tau}}_{h}$ in Lemma 3.16 that $\bm{\sigma}_{h^{n}}=\bm{\sigma}_{h}$, and it satisfies the desired equality for $x\in Y$ by definition. It is continuous on $AM-M$ because $h\mapsto{\bm{\tau}}_{h}$ is by Corollary 3.18, as is the inversion $a_{s}m\mapsto(I-e^{-s}m^{-1})^{-1}$. ∎

Lemma 3.20**.**

There exists $\bm{\sigma}_{0}\in U$ such that for all $h\in AM$ and all $x\in Y$,

[TABLE]

Proof.

We will first how prove the lemma under the assumption that there exists $x\in K\cap Y$, a sequence $n_{k}\to\infty$, and $a\in A^{\prime}-\{1_{G}\}$ such that:

• •

$xa^{n_{k}}\in K\cap Y$ for all $k$, and

• •

$xa^{n_{k}}\to x$.

First, consider $h\in AM-M$. Since $x,xa^{n_{k}}\in K\cap Y$, we have that

[TABLE]

On the other hand,

[TABLE]

where the convergence follows because $a\neq 1_{G}$ means that $a^{-n_{k}}ua^{n_{k}}\to 1_{G}$ for all $u\in U$. Thus, $\bm{\sigma}_{h}=\bm{\sigma}_{0}$ for all $h\in AM-M$.

The statement then follows for all $h\in AM$ using the continuity in Corollary 3.18.

We will now show how to establish the existence of such $x$ and $n_{k}\to\infty$. Let $K^{\prime}\subseteq K$ be a compact set consisting of density points of $K$ and satisfying

[TABLE]

That is, for all $x\in K^{\prime}$, there exists $r_{x}>0$ such that for all $r\leq r_{x}$,

[TABLE]

where $B(r)$ denotes the ball of radius $r$ in $G$; see §2.

Let $\{x_{n}:n\in\mathbb{N}\}$ be a countable dense subset of $K^{\prime}$. For $m,k\in\mathbb{N}$, define

[TABLE]

Let $a\in A^{\prime}-\{1_{G}\}$. By Birkhoff’s theorem applied to the family $\{f_{m,k}\}$, there exists $Z\subseteq X$ with $m^{\operatorname{BMS}}(Z)=1$ such that for all $x\in Z$ and all $m,k$, there exists a sequence $n_{j}\to\infty$ such that for every $j$,

[TABLE]

Now, let $x\in K^{\prime}\cap Y\cap Z$. By the density of $\{x_{n}\}$, there exists a subsequence $x_{m_{j}}\to x$. Then, since $x\in Z$, we can find $n_{k}\to\infty$ such that

[TABLE]

This establishes the existence of such $x,n_{k}$, completing the proof. ∎

3.3. $AMU$-equivariant implies $U^{-}$-equivariant

In this section, we establish Theorem 3.6.

We will first show how the following proposition, which is identical to Theorem 3.6 except for the use of $m^{\operatorname{BMS}}$ instead of $m^{\operatorname{BR}}$, follows from the proof of [18, Theorem 6.1], with only slight modifications required due to the arbitrary dimension in our case.

Proposition 3.21**.**

Let $\hat{\Upsilon}(x)=\tilde{\Upsilon}(x)u_{\bm{\sigma}_{0}},$ where $\tilde{\Upsilon}$ is as in Theorem 3.4. There exists a $m^{\operatorname{BMS}}$-conull set $X_{2}^{\prime\prime}\subseteq X_{2}^{\prime}$ such that for all $x\in X_{2}^{\prime\prime}$ and for every $v_{\textbf{r}}\in U^{-}$, we have

[TABLE]

We refer the reader to the proof of [18, Theorem 6.1] for the details, and provide here only the changes that need to be made to accommodate the arbitrary dimension.

The heart of the proof of [18, Theorem 6.1] is [18, Prop. 6.4], and it is here where all changes due to dimension appear. The first difference is the time-change map $\beta_{\textbf{r}}(\textbf{t})$. In this case, the necessary time change is given by

[TABLE]

It is chosen so that

[TABLE]

where $g_{\textbf{r}}\in AMU^{-}$, using the notation as in equation (6.11) in the proof of [18, Prop. 6.4]. Direct computation shows that for $|\textbf{t}|\leq e^{s}$ and $|\textbf{r}|<\varepsilon$, we still have that

[TABLE]

so the proof carries through, up to Step 4.

In Step 4, the matrix computation to prove equation (6.19) in the proof of [18, Prop. 6.4] is more cumbersome, but not fundamentally different. We provide an outline of the approach below.

For $g_{s}:=g_{s,i}$ in that proof, write

[TABLE]

for some row vectors $\textbf{v},\textbf{w}\in\mathbb{R}^{n-1}$, $C\in M,\lambda>0$. Multiplying it out gives

[TABLE]

Writing $g_{s}=\begin{pmatrix}a&\textbf{q}&b\\ \textbf{x}^{T}&B&\textbf{z}^{T}\\ c&\textbf{y}&d\end{pmatrix}$, we investigate the entries of $I-u_{-\textbf{t}}g_{s}u_{\textbf{t}}$. We still have that

[TABLE]

The magnitude of the (3,1) entry is

[TABLE]

By considering the $t^{4}$ term and equation (3.10), we conclude that

[TABLE]

Using this and the $t^{2}$ terms of the (2,1) entry

[TABLE]

we similarly conclude that

[TABLE]

Using the (3,2) entry

[TABLE]

we get

[TABLE]

Continuing in this manner, we end up with the following conclusions:

1. (1)

$|b|=O(e^{-4s})$ by the (3,1) entry 2. (2)

$|\frac{1}{2}|\textbf{z}|-|\textbf{q}||=O(e^{-2s})$ from the (2,1) entry 3. (3)

$|\frac{1}{2}|\textbf{q}|-|\textbf{z}||=O(e^{-2s})$ by the (3,2) entry 4. (4)

$||\textbf{q}|-|\textbf{z}||=O(e^{-3s})$ from the (3,1) entry 5. (5)

$|\textbf{q}|=O(e^{-2s})$ and $|\textbf{z}|=O(e^{-2s})$ using the three lines above 6. (6)

$||B|-|a||=O(e^{-s})$ from the (2,1) entry 7. (7)

$||B|-|d||=O(e^{-s})$ from the (3,2) entry 8. (8)

$||a|-|d||=O(e^{-s})$ from the two lines above

We will further show that

[TABLE]

From these facts, it will follow that $d(1_{G},a_{s}g_{s}a_{-s})=O(e^{-s})$, completing the proof of Proposition 3.21.

Because $d(1_{G},u_{-\textbf{t}}g_{s}u_{\textbf{t}})=O(1)$, we know that $\lambda=1+O(1)$, so $|\lambda\textbf{v}|=O(1)$ implies that

[TABLE]

Similarly,

[TABLE]

Then, from the fact that $C=I-\lambda\textbf{v}^{T}\textbf{w}+O(1)$, we conclude

[TABLE]

Thus,

[TABLE]

Now, using that $a=\lambda$, $||a|-|d||=O(e^{-s})$, and $d=\lambda^{-1}+O(e^{-s})$ by equation(3.11), it follows that

[TABLE]

This in turn implies (because $\lambda=1+O(1)$) that

[TABLE]

so $\lambda^{2}=1+O(e^{-s})$. Using the Taylor series for $\sqrt{1+x}$, we conclude that

[TABLE]

Combining the above with $|B-aI|=|B-\lambda I|=O(e^{-s})$ implies

[TABLE]

as desired.

We will now explain how to slightly change the proof of [18, Theorem 6.1] to yield a $U$-invariant $m^{\operatorname{BMS}}$-conull set, hence a $m^{\operatorname{BR}}$-conull set in light of equation (2.4). This will establish Theorem 3.6.

Let $K_{\eta}$ be the compact subset chosen in equation (6.4) in [18] and $\Omega_{\eta}$ as in equation (6.6). More specifically,

[TABLE]

is a compact set with

[TABLE]

such that there exists $T_{\eta}>1$ so that for every $x\in\Omega_{\eta}$ and $T\geq T_{\eta}$,

[TABLE]

We will need to thicken $\Omega_{\eta}$ slightly in the $U$-direction. By [18, Lemma 4.4], there exists $r_{0}>0$ and $R>0$ such that for all $x\in\Omega_{\eta}$ and all $T\geq R$,

[TABLE]

Remark 3.22**.**

*Despite the stated dependence in [18, Lemma 4.4], $R$ is in fact independent of $x$. The apparent dependence in that statement arises from [18, Theorem 4.1], which is actually weaker than the result cited from [30]. The original proof in [30] shows that there is no such dependence on the base point. ***

We will show that if $y\in\Omega_{\eta}B_{U}(r_{0})$, then for all $T\geq T_{\eta}+r_{0}$,

[TABLE]

Equation (3.13) implies that

[TABLE]

Suppose now that $x\in\Omega_{\eta}$, $u\in B_{U}(r_{0})$ and $T\geq T_{\eta}+r_{0}$. Then

[TABLE]

so together with the above we conclude that

[TABLE]

and therefore

[TABLE]

which establishes (3.14).

Thus, by adjusting constants slightly, we can use $\Omega_{\eta}B_{U}(r_{0})$ in place of $\Omega_{\eta}$ in [18, Prop. 6.4]. Now, as in the proof of [18, Theorem 6.1], by Birkhoff’s theorem, there exists an $A$-invariant $m^{\operatorname{BMS}}$-conull set $Z$ such that for all $x\in Z$,

[TABLE]

Let $x\in Z$ and $u_{\textbf{t}}\in U$. Suppose that $|e^{-s}\textbf{t}|<r_{0}$ and that $xa_{s}\in\Omega_{\eta}$. Then

[TABLE]

From this, we conclude that if $x\in ZU$, it will have infinitely many returns under $A$ to the set $\Omega_{\eta}B_{U}(r_{0})$, the set which replaced $\Omega_{\eta}$ in [18, Prop. 6.4]. Thus, if we define

[TABLE]

then the proof of [18, Theorem 6.1] carries through. By equation (2.4), $X_{2}^{\prime\prime}$ is $m^{\operatorname{BR}}$-conull, so we have established Theorem 3.6.

4. Non-concentration of the PS measure near varieties

In this section, we prove several lemmas showing that PS measure does not concentrate near varieties. This will be needed in the next section, and is a key step for extending the results from [18] to higher dimensions. The main result is Lemma 4.6.

In the geometrically finite case, we will need to control the PS measure of the unit ball in $U$ based at a point that may be far out in a cusp. We will use a variation Sullivan’s shadow lemma for this purpose.

Suppose that $\Gamma\backslash\mathbb{H}^{n}$ is geometrically finite. For $\xi\in\Lambda_{\operatorname{bp}}(\Gamma)$, let $g\in G$ be such that $g^{-}=\xi$. For $R>0$, define

[TABLE]

where $K$ is the maximal compact subgroup $\operatorname{Stab}_{G}(o)$ as in §2.1. The rank of the horoball $\mathcal{H}(\xi,r)$ is the rank of $\operatorname{Stab}_{\Gamma}(\xi)$, which is a finitely generated abelian group. It is always strictly less than $2\delta_{\Gamma}$.

As in [3], it follows from the thick-thin decomposition of the convex core that there exists a compact set $\mathcal{K}_{0}$, a constant $R_{0}\geq 1$, and a finite set $\{\xi_{1},\ldots,\xi_{m}\}\subseteq\Lambda_{\operatorname{bp}}(\Gamma)$ such that

[TABLE]

The following version of Sullivan’s shadow lemma is due to Maucourant and Schapira, [14]:

Lemma 4.1**.**

[14, Lemma 5.1, Remark 5.2]** There exists a constant $R\geq 1$ such that for all $x\in\operatorname{supp}m^{\operatorname{BMS}}$,

[TABLE]

where $r$ is the rank of the cusp containing $xa_{-\log T}$, and is zero if $xa_{-\log T}\in\mathcal{K}_{0}$.

Let

[TABLE]

We will need to extend Lemma 4.1 to the following:

Corollary 4.2**.**

Suppose that all cusps have rank $n-1$. There exists a constant $R>0$ such that for all $x\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, and all $T>1$ we have

[TABLE]

where $r$ is the rank of the cusp containing $xa_{-\log T}$, and is zero if $xa_{-\log T}\in\mathcal{K}_{0}$.

Proof.

Let $x\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$. By definition of $N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, there exists

[TABLE]

Thus,

[TABLE]

and so by Lemma 4.1 and since $T>1$, there exists $R_{0}>0$ such that

[TABLE]

where $r^{\prime}$ is the rank of the cusp containing $x^{\prime}a_{-\log T}.$

Note that

[TABLE]

We will first consider the upper bound by cases. Suppose that $r^{\prime}=0$. Then the upper bound yields

[TABLE]

where the last inequality follows because if $r=0$, $e^{(r-{\delta_{\Gamma}})d(xa_{-\log T},\mathcal{K}_{0})}=1$, and otherwise, $e^{(r-{\delta_{\Gamma}})d(xa_{-\log T},\mathcal{K}_{0})}>1$, and also $e^{(n-1-\delta_{\Gamma})}\geq 1.$

Now, suppose instead that $r^{\prime}=n-1$. Then by using (4.3), we obtain

[TABLE]

where the last equality is because either $r=n-1=r^{\prime}$, or $r=0$, in which case $d(xa_{-\log T},\mathcal{K}_{0})=0$.

We will now consider the lower bound by cases. Again, first suppose that $r^{\prime}=0$. Then $d(x^{\prime}a_{-\log T},\mathcal{K}_{0})=0$, so we have

[TABLE]

where the last line follows from (4.3) if $r=n-1$, and if $r=0$, then

[TABLE]

and $e^{-(n-1-\delta_{\Gamma})}<1$.

Thus, letting $R=R_{0}2^{\delta_{\Gamma}}e^{(n-1-\delta_{\Gamma})}$ establishes the claim. ∎

Corollary 4.3**.**

Suppose that all cusps have rank $n-1$, and let $R$ be as in Corollary 4.2. Then for every $y\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$ and every $\varepsilon>0$, we have

1. (1)

$\mu^{\operatorname{PS}}_{y}(B_{U}(\varepsilon))\leq R^{2}\varepsilon^{2\delta_{\Gamma}-n+1}\mu^{\operatorname{PS}}_{y}(B_{U}(1))$* if $\varepsilon<1$,* 2. (2)

$\mu^{\operatorname{PS}}_{y}(B_{U}(\varepsilon))\leq R^{2}\varepsilon^{2\delta_{\Gamma}}\mu^{\operatorname{PS}}_{y}(B_{U}(1))$* if $\varepsilon\geq 1$.*

Proof.

First, note that by assumption on the rank of the cusps,

[TABLE]

By Corollary 4.2, we have that

[TABLE]

where $r$ is the rank of the cusp containing $y$, and is zero if $y\in\mathcal{K}_{0}$. Similarly, if $r_{\varepsilon}$ denotes the rank of the cusp containing $ya_{-\log\varepsilon}$, then

[TABLE]

We also have

[TABLE]

Let $0<\varepsilon<1$ and assume first that $r_{\varepsilon}=n-1$, so that $r_{\varepsilon}-{\delta_{\Gamma}}\geq 0$. Then by (4.5) and (4.6), we have

[TABLE]

where the last line follows because if $r=0$, then $d(y,\mathcal{K}_{0})=0$, and otherwise, $r_{\varepsilon}=r=n-1.$

Now, suppose that $r_{\varepsilon}=0$. Then we have by (4.5),

[TABLE]

where the last line follows because $\delta_{\Gamma}\geq 2\delta_{\Gamma}-(n-1)$ and $\varepsilon<1$. This establishes the first case.

Now, assume that $\varepsilon\geq 1$, so $\log\varepsilon\geq 0$. We again consider cases. First, suppose that $r_{\varepsilon}=n-1$, Then by (4.5) and (4.6), we have

[TABLE]

where the second to last line follows because $e^{(n-1-r)d(y,\mathcal{K}_{0})}=0$ when $r\in\{0,n-1\}$, and the final line because $\delta_{\Gamma}>(n-1)/2$.

Now, suppose that $r_{\varepsilon}=0$. Then again by (4.5) and (4.6), we have

[TABLE]

which completes the second case. ∎

For $d\in\mathbb{N}$ and $c>0$, define

[TABLE]

Note that it is a compact subset of $C(B_{U}(1))$. For $f\in\mathcal{F}_{d,m}$ and $r>0$, define

[TABLE]

Lemma 4.4**.**

Assume that all cusps have rank $n-1$ and that $\delta_{\Gamma}>n-\frac{5}{4}$. Let $d,m>0$. Then there exist constants $\overline{c}>0$ and $\alpha>0$ such that for every $y\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, every $0<\varepsilon<1/4$, and every $f\in\mathcal{F}_{2,m}$,

[TABLE]

Proof.

By Corollary 4.3, $\alpha_{1}:=2\delta_{\Gamma}-(n-1)$ is such that for all $y\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$,

[TABLE]

Let $\{z_{1},\ldots,z_{k}\}$ be a maximal $\varepsilon$-separated set in $N_{\varepsilon}(f)$.

Claim: There exists some constant $d^{\prime}>0$ such that $k\leq d^{\prime}\varepsilon^{\frac{3-2n}{2}}.$

Proof of claim.

By the mean value theorem, there exists $c^{\prime}$ such that for all $\textbf{s},\textbf{t}\in B_{U}(1)$,

[TABLE]

Thus, since $z_{i}\in N_{\varepsilon}(f)$, for all $1\leq i\leq k$ and for all $\textbf{t}\in z_{i}B_{U}(\varepsilon)$, we have

[TABLE]

Hence, we have

[TABLE]

Thus, there exists some constant $d>0$ such that $dk(\varepsilon/4)^{n-1}\leq\lambda(N_{(1+c^{\prime})\varepsilon}(f))$, where $\lambda$ denotes the Lebesgue measure.

∎

Because $\{z_{1},\ldots,z_{m}\}$ is a maximal $\varepsilon$-separated subset of $N_{\varepsilon}(f)$, we have that

[TABLE]

By Corollary 4.3, there exists $d_{1}>0$ such that for all $w\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$,

[TABLE]

Then we also have that

[TABLE]

This is because if $z_{i}\not\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, then $z_{i}B_{U}(2\varepsilon)\cap\operatorname{supp}m^{\operatorname{BMS}}=\emptyset$, and so the left hand side is zero.

Also by Corollary 4.3, there exists $d_{2}>0$ such that for all $w\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}}),$

[TABLE]

From this, we obtain

[TABLE]

where $\overline{c}=R^{4}d^{\prime}d_{1}d_{2}$. Let $\alpha=\frac{3-2n+2\alpha_{1}}{2}$. Since $\alpha_{1}=2\delta_{\Gamma}-(n-1)$, the assumption $\delta_{\Gamma}>n-\frac{5}{4}$ ensures $\alpha>0$. ∎

Lemma 4.5**.**

Let $K\subseteq N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$ be compact and let $d,m>0$. Then for every $\eta>0$, there exists $\varepsilon>0$ such that for all $y\in K$ and for all $f\in\mathcal{F}_{d,m}$,

[TABLE]

Proof.

Suppose not. Then there exists $\eta>0$ and sequences $f_{i}\in\mathcal{F}_{d,m}$, $y_{i}\in K$, and $\varepsilon_{i}\to 0$ such that

[TABLE]

By compactness of $K$ and of $\mathcal{F}_{d,m}$, we may assume that there exists $f\in\mathcal{F}_{d,m}$ and $y_{\infty}\in K$ such that $f\to f_{i}$ uniformly and $y_{i}\to y_{\infty}$.

Let $V=\{t\in B_{U}(1):f(\textbf{t})=0\}$. Since $f_{i}\to f$ uniformly, for each $i$ there exists $\varepsilon_{i}^{\prime}>0$ such that

[TABLE]

and $\varepsilon_{i}^{\prime}\to 0$. Thus, we have that for all $i$,

[TABLE]

By the continuity of $g\mapsto\mu^{\operatorname{PS}}_{g}$ in Lemma 2.2, it follows that

[TABLE]

However, $\mu^{\operatorname{PS}}_{y_{\infty}}(B_{U}(1))>0$ because $y_{\infty}\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, and $\mu^{\operatorname{PS}}_{y_{\infty}}(V)=0$ by Lemma 2.1, so this is a contradiction. ∎

We will use Lemma 4.5 in the context of $\Gamma$ convex cocompact and $K=N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, as follows.

Lemma 4.6**.**

Assume that $\Gamma$ is either

• •

convex cocompact, or

• •

geometrically finite with all cusps of rank $n-1$ and $\delta_{\Gamma}>n-\frac{5}{4}$.

Let $g\in\mathcal{F}_{d,m}$ for some $d,m$. For every $f\in C_{c}(\Gamma\backslash G)$ and for every $\eta>0$, there exists $\varepsilon>0$ and $T_{0}=T_{0}(f,\varepsilon)>0$ such that for all $T\geq T_{0}$, if $ya_{-s}\in K:=N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})\cap\{x:x^{-}\in\Lambda_{\operatorname{r}}(\Gamma)\}$, where $s=\log T$, we have

[TABLE]

($\ll_{f}\eta$ means $\leq k\eta$ for some constant $k$ that depends only on $f$.)

To prove Lemma 4.6, we need a fact from [18], which requires the following definition. Let

[TABLE]

and let $P_{r}$ denote the ball of radius $r$ in $P$. For $\varepsilon_{0},\varepsilon_{1}>0$, we say that $zP_{\varepsilon_{1}}B_{U}(\varepsilon_{0})$ is an admissible box if it is the injective image of $P_{\varepsilon_{1}}B_{U}(\varepsilon_{0})$ in $\Gamma\backslash G$ under the map $g\mapsto zg$, and $\mu^{\operatorname{PS}}_{zp}(zpB_{U}(\varepsilon_{0}))\neq 0$ for all $p\in P_{\varepsilon_{1}}$. In the statement of the next lemma, we will assume our functions are supported within an admissible box. By a partition of unity argument, there is no loss of generality by making this assumption.

Lemma 4.7**.**

[18, Claim A in Theorem 4.6]** Let $\xi\in C_{c}(X_{1})$ be supported within the admissible box $zP_{\varepsilon_{1}}B_{U}(\varepsilon_{0})$ with $0<\varepsilon_{1}\leq\varepsilon_{0}$. Suppose that $x^{-}\in\Lambda_{\operatorname{r}}(\Gamma)$, and let $\Omega$ be a compact set such that there exists $t_{n}\to\infty$ with $xa_{-t_{n}}\in\Omega$ for all $n$. Let $s_{0}$ be such that

[TABLE]

For $\rho>0$ and every $y\in x_{0}U$, suppose that $f_{y}\in C(yB_{U}(\rho))$ is such that

[TABLE]

Then there exists $c>0$ depending only on $\operatorname{supp}\xi$ (in particular, $\operatorname{diam}(\operatorname{supp}\xi)$ is a possible choice) such that for all $y\in x_{0}U$,

[TABLE]

Proof of Lemma 4.6.

Let $f\in C_{c}(X_{1})$. By a partition of unity argument, we may assume that $\operatorname{supp}(f)$ is contained in some admissible box $zP_{\varepsilon_{1}}U_{\varepsilon_{0}}$.

Fix $\eta>0$. By Lemma 4.4 (in the geometrically finite case) or 4.5 (in the convex cocompact case), there exists $\varepsilon>0$ such that for all $w\in N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}})$, we have

[TABLE]

Let $c>0$ be as from Lemma 4.7 above. Let $T_{0}=T_{0}(f,\varepsilon)>0$ be such that

[TABLE]

where $s_{0}=\log T_{0}$.

Let $y\in K$, let $T\geq T_{0}$, and define $s=\log T$. Let $I_{T}$ be a maximal set of points in $ya_{-s}N_{\varepsilon}({g})$ such that the balls $\{zB_{U}(\varepsilon/16):z\in I_{T}\}$ are disjoint. Thus,

[TABLE]

covers $ya_{-s}N_{\varepsilon}({g})$. Let $\{f_{z}:z\in I_{T}\}$ be a partition of unity subordinate to this cover. Then we have:

[TABLE]

where $\kappa$ is the multiplicity of the cover given by the Besicovitch covering theorem; $\kappa$ depends only on the dimension $n$. ∎

5. Joinings

In this section, we prove Theorem 1.4. In particular, throughout this section, we assume either that the $\Gamma_{i}$’s are convex cocompact, or that they are geometrically finite with all cusps of rank $n-1$ and critical exponents $\delta_{\Gamma_{i}}>n-\frac{5}{4}.$

Let $\mu$ be an ergodic $U$-joining for $(m^{\operatorname{BR}}_{1},m^{\operatorname{BR}}_{2})$. In §5.3, we prove that it must be the case that $m^{\operatorname{BR}}$-a.e. fiber of $\pi_{2}$, the projection map from $X\to X_{2}$, is finite, as otherwise the joining measure $\mu$ would be invariant under a nontrivial connected subgroup of $U\times\{1_{G}\}$, which is impossible by [18, Lemma 7.16]. More precisely, in §5.3, we prove:

Theorem 5.1** (c.f. [18], Theorem 7.17).**

There exists a positive integer $\ell>0$ and a $m^{\operatorname{BR}}$-conull subset $\tilde{X}_{2}\subseteq X_{2}$ so that $\pi_{2}^{-1}(x_{2})$ has cardinality $\ell$ for every $x_{2}\in\tilde{X}_{2}$. Moreover, the fiber measures $\mu_{x_{2}}^{\pi_{2}}$ are uniform measures for each $x_{2}\in\tilde{X}_{2}$.

This will allow us to reduce to considering $U$-equivariant set-valued maps, which we proved rigidity for in §3. Specifically, we will define for $x\in\tilde{X}_{2}$ from the previous theorem

[TABLE]

In §5.1, we prove the following more precise formulation of Theorem 1.4, assuming Theorem 5.1:

Theorem 5.2**.**

Let $\hat{\Upsilon}(x):=\tilde{\Upsilon}(x)u_{\bm{\sigma}_{0}}$, where $\tilde{\Upsilon}$ and $\bm{\sigma}_{0}$ are as in Theorem 3.4. Then there exists $q_{0}\in G$ such that $\Gamma_{2}\cap q_{0}^{-1}\Gamma_{1}q_{0}$ has finite index in $\Gamma_{2}$ and satisfying: if $\gamma_{i}\in\Gamma_{2}$, $1\leq i\leq\ell$, are such that $\Gamma_{1}q_{0}\Gamma_{2}=\bigcup\limits_{1\leq i\leq\ell}\Gamma_{1}q_{0}\gamma_{i}$, then

[TABLE]

on a $m^{\operatorname{BR}}$-conull subset of $X_{2}$. Moreover, the joining $\mu$ is a $\Delta(U)$-invariant measure supported on $\{(\hat{\Upsilon}(x_{2}),x_{2}):x_{2}\in X_{2}\}$, and hence is a finite cover self-joining as in Definition 1.3.

5.1. Proof of Theorem 1.4

We show in this section how to use Theorem 5.1 to prove Theorem 5.2, which is a more precise statement of Theorem 1.4.

By Theorem 5.1, there exists a $m^{\operatorname{BR}}$-conull set $\tilde{X}_{2}$ and a natural number $\ell>0$ such that

[TABLE]

has cardinality $\ell$ for all $x\in\tilde{X}_{2}$. Moreover, this is a $U$-equivariant condition, so we may assume that $\tilde{X}_{2}$ is $U$-invariant and that

[TABLE]

for all $x\in\tilde{X}_{2}$ and $u\in U$. By a standard argument for constructing cross sections, there exist measurable maps $\upsilon_{1},\ldots,\upsilon_{\ell}:\tilde{X}_{2}\to X_{1}$ such that

[TABLE]

The proof of Theorem 5.2 now follows as in [18, Prop. 7.23], where references to Theorems 6.1 and equation (7.21) are replaced with references to Theorems 3.4 and 3.6.

5.2. Notation

We provide here for the readers convenience a list of important notation that will be used in §5.3. Full explanations appear in that section, which should be read first, using this section for reference when needed.

The measure $\mu$

$\mu$ is an ergodic $U$-joining on $X=X_{1}\times X_{2}$ for the pair $(m^{\operatorname{BR}}_{1},m^{\operatorname{BR}}_{2})$.

$\psi$ and $\Psi$

$\psi\in C_{c}(X_{1})$ is non-negative with $m^{\operatorname{BR}}(\psi)>0$. Let

[TABLE]

The set $\Omega_{1}$

A compact set $\Omega_{1}\subseteq\{x:x^{-}\in\Lambda_{\operatorname{r}}(\Gamma)\}$ with $m^{\operatorname{BR}}(\Omega_{1})>0$ such that

[TABLE]

holds uniformly for all $x\in\Omega_{1}$.

The set $Q$

A compact set $Q\subseteq X$ with $\mu(Q)\in(0,\infty)$, $\pi_{1}(Q)\subseteq\Omega_{1}$, and such that for all $f\in C_{c}(X)$ and all $x\in Q$,

[TABLE]

$\varepsilon$, $\eta_{0}$ and the sets $Q_{+},Q_{++}$

Fix $0<\varepsilon<1$ satisfying

[TABLE]

and $\eta_{0}>0$ such that

[TABLE]

where

[TABLE]

Define

[TABLE]

$\phi$ and $\Phi$

$\phi\in C_{c}(X_{1})$ is such that

[TABLE]

where $\mathbf{1}_{E}$ denotes the characteristic function of a set $E$ in $X$. Let $\Phi=\phi\circ\pi_{1}$.

The set $Q_{\varepsilon}$ and the family $\mathcal{F}$

Let $\mathcal{F}=\{\mathbf{1}_{Q},\mathbf{1}_{Q_{+}},\mathbf{1}_{Q_{++}},\Phi\}$. $Q_{\varepsilon}\subseteq Q$ is a compact set with

[TABLE]

such that for each $f\in C_{c}(X)\cup\mathcal{F}$ and $\theta>0$, there exists $T_{0}=T_{0}(f,\theta)$ such that if $T\geq T_{0}$, then

[TABLE]

for all $x\in Q_{\varepsilon}$.

The functions $\varphi_{m}$

Suppose that we have a sequence $g_{m}\in G-U$ with $g_{m}\to 1_{G}$ and a point $x=(x_{1},x_{2})\in Q_{\varepsilon}$ such that $(x_{1}g_{m},x_{2})\in Q_{\varepsilon}$ for all $m$. For each $m\geq 0$,

[TABLE]

The values $T_{m}$

Define $T_{m}:=\sup\{T>0:\varphi_{m}(B_{U}(T))\subseteq B(1)\}$.

The functions $\tilde{\varphi}_{m}$ and $\tilde{\varphi}$

On $B_{U}(1)$, define

[TABLE]

By equicontinuity of the entries of the $\tilde{\varphi_{m}}$’s, we may assume that there exists some $\tilde{\varphi}$ defined on $B_{U}(1)$ such that $\tilde{\varphi}_{m}\to\tilde{\varphi}$ uniformly on $B_{U}(1)$.

The sets $N_{\varepsilon^{\prime}}(\tilde{\varphi}-1_{G})$, and $I_{\varepsilon^{\prime}}(T)$

For $\varepsilon^{\prime}>0$, define

[TABLE]

For $T>0$, let $s=\log T$ and define

[TABLE]

5.3. Fibers of $\pi_{2}$ are finite

Recall that $\mu$ is an ergodic $U$-joining on $X=X_{1}\times X_{2}$ for $(m^{\operatorname{BR}}_{1},m^{\operatorname{BR}}_{2})$. Let

• •

$\psi\in C_{c}(X_{1})$ be non-negative with $m^{\operatorname{BR}}(\psi)>0$,

• •

and let $\Psi=\psi\circ\pi_{1}\in C(X)$.

Recall that

[TABLE]

Let $\Omega_{1}\subseteq\{x:x^{-}\in\Lambda_{\operatorname{r}}(\Gamma)\}$ be a compact set with $m^{\operatorname{BR}}(\Omega_{1})>0$ such that [18, Lemma 4.6] holds for $\psi$ uniformly across all $x\in\Omega_{1}$. That is, the convergence

[TABLE]

holds uniformly for all $x\in\Omega_{1}$. (By Egorov’s theorem, such a compact set exists within any set with positive $m^{\operatorname{BR}}$ measure, see [18, Remark 4.8].)

By the Hopf ratio ergodic theorem, there exists a compact set $Q\subseteq X$ such that

• •

$\mu(Q)\in(0,\infty)$,

• •

$\pi_{1}(Q)\subseteq\Omega_{1}$,

• •

and for all $f\in C_{c}(X)$ and all $x\in Q$,

[TABLE]

Fix $0<\varepsilon<1$ satisfying

[TABLE]

(this condition is needed to ensure a non-empty intersection in the claim in the proof of Theorem 5.3) and $\eta_{0}>0$ such that

[TABLE]

where

[TABLE]

Also define

[TABLE]

Let $\phi\in C_{c}(X_{1})$ be such that

[TABLE]

where $\mathbf{1}_{E}$ denotes the characteristic function of a set $E$ in $X$. Let

[TABLE]

and define

[TABLE]

By the Hopf ratio ergodic theorem again together with Egorov’s theorem, there exists a compact set

[TABLE]

with

[TABLE]

such that for each $f\in C_{c}(X)\cup\mathcal{F}$, the convergence in equation (5.3) holds uniformly for all $x\in Q_{\varepsilon}$. That is, for all $f\in C_{c}(X)\cup\mathcal{F}$ and $\theta>0$, there exists $T_{0}=T_{0}(f,\theta)$ such that if $T\geq T_{0}$, then

[TABLE]

for all $x\in Q_{\varepsilon}$.

The proof of Theorem 5.1 will follow as in the proof of [18, Theorem 7.17] once we establish the following generalization of [18, Theorem 7.12]:

Theorem 5.3** (c.f. [18], Theorem 7.12).**

Suppose that there exists $x=(x_{1},x_{2})\in Q_{\varepsilon}$ and a sequence $g_{m}\in G-U$ with $g_{m}\to 1_{G}$ such that $(x_{1}g_{m},x_{2})\in Q_{\varepsilon}$ for all $m$. Then $\mu$ is invariant under a nontrivial connected subgroup of $U\times\{1_{G}\}$.

The proof of Theorem 5.3 requires several lemmas, which in turn require more setup.

Suppose that we have a sequence $g_{m}\in G-U$ with $g_{m}\to 1_{G}$ and a point $x=(x_{1},x_{2})\in Q_{\varepsilon}$ such that $(x_{1}g_{m},x_{2})\in Q_{\varepsilon}$ for all $m$. For each $m\geq 0$, define

[TABLE]

In particular, $\varphi_{m}(\textbf{t})$ satisfies

[TABLE]

Define

[TABLE]

Since $g_{m}\not\in U=C_{G}(U)$ (where $C_{G}(U)$ denotes the centralizer in $G$ of $U$), $\varphi_{m}(\textbf{t})$ is not constant, and so $T_{m}<\infty$. Moreover,

[TABLE]

because $g_{m}\to 1_{G}$.

On $B_{U}(1)$, define

[TABLE]

By definition of $T_{m}$, each of the entries in the $\tilde{\varphi}_{m}$’s gives rise to a sequence of uniformly bounded polynomials with degree at most 2 on a compact domain, hence an equicontinuous family. Thus, we may assume that there exists some $\tilde{\varphi}$ defined on $B_{U}(1)$ such that

[TABLE]

uniformly on $B_{U}(1)$. Observe that $\tilde{\varphi}$ maps into $C_{G}(U)=U$ by construction of the $\varphi_{m}$’s, so $\tilde{\varphi}\in\mathcal{F}_{2,k}$ for some $k>0$, where $\mathcal{F}_{2,k}$ is defined in §4.

Define

[TABLE]

For $T>0$, let $s=\log T$ and define

[TABLE]

Lemma 5.4**.**

For every $0<\eta^{\prime}<1/2$, there exists $\varepsilon^{\prime}>0$ and $T_{0}>0$ such that for all $T\geq T_{0}$, for all $F\in\{\Psi,\Phi\}$, and for all $y\in Q_{\varepsilon}$, we have that

[TABLE]

for some constant $c_{1}\geq 1$, where $s=\log T$.

Proof.

Let $\kappa=\kappa(\Omega_{1})$ be as in Lemma 2.4. Since $\pi_{1}(Q_{\varepsilon})\subseteq\Omega_{1}$, it follows from equation (5.2) that there exists $T_{1}\geq\kappa$ such that for all $T\geq T_{1}$ and all $y\in Q_{\varepsilon}$,

[TABLE]

By definition of $\kappa$, for all $x\in\Omega_{1},$

[TABLE]

From this, it follows that there exists $T_{2}\geq T_{1}$ such that for all $T\geq T_{2}$,

[TABLE]

where $N_{1/2}(\operatorname{supp}m^{\operatorname{BMS}}):=(\operatorname{supp}m^{\operatorname{BMS}})B_{U}(1/2).$

Now, let $f=\psi$ if $F=\Psi$, and $f=\phi$ if $f=\Phi$. Fix $0<\eta^{\prime}<1$ and let

[TABLE]

where $D$ is the implied constant from Lemma 4.6 applied to $f$ and $N_{\varepsilon^{\prime}}(\tilde{\varphi}-1_{G})$. That is, there exists $\varepsilon^{\prime}>0$ and $T_{0}\geq T_{2}$ such that for all $T\geq T_{0}$ and all $w\in\Omega_{1}$,

[TABLE]

where $s=\log T$. In particular, this implies that for all $T\geq T_{0}$ and $w\in\Omega_{1}$,

[TABLE]

where we have used that $\mu^{\operatorname{PS}}_{w}(B_{U}(T))=e^{\delta_{\Gamma_{1}}s}\mu^{\operatorname{PS}}_{wa_{-s}}(B_{U}(1))$.

By subtracting equation (5.5) for $\psi$ from (5.4), we conclude that for all $T\geq T_{0}$ and for all $w\in\Omega_{1}$,

[TABLE]

Then from equations (5.5) and (5.6), we have that for all $T\geq T_{0}$ and $y\in\Omega_{1}$,

[TABLE]

for $c_{1}=\max\{m^{\operatorname{BR}}(\phi)/m^{\operatorname{BR}}(\psi),1\}$, as desired. ∎

By definition of $Q_{\varepsilon}$, we have that for all $F\in\mathcal{F}$ and for all $\theta>0$, there exists $T_{0}=T_{0}>0$ such that if $T\geq T_{0}$, then for all $y\in Q_{\varepsilon}$,

[TABLE]

We can now improve this to integration over sets of the form $I_{\varepsilon^{\prime}}(T)$ as follows.

Corollary 5.5**.**

For all $\theta>0$, there exists $\varepsilon^{\prime}>0$ and $T_{0}>0$ such for all $T\geq T_{0}$, all $y\in Q_{\varepsilon}$, and every $F\in\mathcal{F}=\{\mathbf{1}_{Q},\mathbf{1}_{Q_{+}},\mathbf{1}_{Q_{++}},\Phi\}$, we have that

[TABLE]

Proof.

Let $\varepsilon^{\prime}$ and $T_{0}$ be as in Lemma 5.4. Let $T\geq T_{0}$ and let $s=\log T$. Define $N_{s}=a_{-s}N_{\varepsilon^{\prime}}(\tilde{\varphi}-1_{G})a_{s}\cap B_{U}(T)$. By equation (5.7), there exists $\Theta(T)$ which tends to zero uniformly over $Q_{\varepsilon}$ such that

[TABLE]

Thus, by subtracting $\int_{N_{s}}F(y\Delta(u_{\textbf{t}}))d\textbf{t}$ and dividing by $\int_{I_{\varepsilon^{\prime}}(T)}\Psi(y\Delta(u_{\textbf{t}}))d\textbf{t}$, we obtain

[TABLE]

The conclusion then follows from Lemma 5.4, where for the last term we note that for all $f\in\mathcal{F},0\leq F\leq\Phi$. ∎

We can now prove Theorem 5.3.

Proof of Theorem 5.3.

Recall the notation from §5.2. Define

[TABLE]

Note that $T_{m}\to\infty$ as $m\to\infty$.

It follows from Corollary 5.5 (by writing out with error terms and dividing) that there exists $\varepsilon^{\prime}>0$ and $T_{0}>0$ such that for every $T\geq T_{0}$, every $y\in Q_{\varepsilon}$, and every $F_{1},F_{2}\in\mathcal{F}=\{\mathbf{1}_{Q},\mathbf{1}_{Q_{+}},\mathbf{1}_{Q_{++}}\}$,

[TABLE]

Moreover, this $T_{0}$ can be chosen so that $\lambda(\{\textbf{t}\in I_{\varepsilon^{\prime}}(T_{0}):x\Delta(u_{\textbf{t}})\in Q_{++}\})>0$, where $\lambda$ denotes the Lebesgue measure on $U$.

Claim: Let $h_{m}=(g_{m},1_{G})$. For all $m$ with $T_{m}^{\prime}\geq T_{0}$ and all $T_{0}\leq T\leq T_{m}^{\prime}$,

[TABLE]

Proof of claim.

Recall that $\varphi_{m}(\textbf{t})=u_{\textbf{t}}^{-1}g_{m}u_{\textbf{t}}$ satisfies

[TABLE]

By definition of $T_{m}^{\prime}$, if $|\textbf{t}|\leq T_{m}^{\prime}$, then $\|(\varphi_{m}(\textbf{t}),1_{G})-(1_{G},1_{G})\|\leq\eta_{0}/4$, so

[TABLE]

By applying equation (5.8) to $F_{1}=\mathbf{1}_{Q_{++}}$ and $F_{2}=\mathbf{1}_{Q}$ with $y=x$, we have that

[TABLE]

where $\lambda$ is the Lebesgue measure.

And by applying it with $F_{1}=\mathbf{1}_{Q_{+}}$ and $F_{2}=\mathbf{1}_{Q}$ with $y=xh_{m}$, we have that

[TABLE]

Since $\{\textbf{t}\in I_{\varepsilon^{\prime}}(T):x\Delta(u_{\textbf{t}})\in Q\}$ and $\{\textbf{t}\in I_{\varepsilon^{\prime}}(T):xh_{m}\Delta(u_{\textbf{t}})\in Q\}$ are both subsets of $\{\textbf{t}\in I_{\varepsilon^{\prime}}(T):x\Delta(u_{\textbf{t}})\in Q_{++}\}$ from the definition of $T_{m}^{\prime}$, the choice of $\varepsilon$ implies that both subsets have greater than half the Lebesgue measure of the larger set (which is positive by choice of $T_{0}$), and thus their intersection cannot be empty. ∎

By the claim, for all sufficiently large $m$, there exists $\textbf{t}_{m}\in I_{\varepsilon^{\prime}}(T_{m}^{\prime})$ such that

[TABLE]

By the compactness of $Q$ and by dropping to a subsequence if necessary, we may assume that there exists $x_{\infty}\in Q$ such that $x\Delta(u_{\textbf{t}_{m}})\to x_{\infty}$.

Let

[TABLE]

so that

[TABLE]

Then again by the compactness of ${B_{U}(1)}$ and the uniform convergence of $\tilde{\varphi}_{m}\to\tilde{\varphi}$, we may assume that there exists $\textbf{t}_{\infty}\in{B_{U}(1)}$ such that

[TABLE]

By definition of $I_{\varepsilon^{\prime}}(T_{m}^{\prime})$, $\tilde{\textbf{t}}_{m}\not\in N_{\varepsilon^{\prime}}(\tilde{\varphi}-1_{G})$ for all $m$, so

[TABLE]

Moreover, the image of $\tilde{\varphi}$ is contained within $C_{G}(U)=U$, so it follows from [18, Lemma 7.7] applied to $(\tilde{\varphi}(\textbf{t}_{\infty}),1_{G})$ that $\mu$ is quasi-invariant under a nontrivial connected subgroup of $U\times\{1_{G}\}$. Strict invariance follows from [18, Lemma 7.3]. ∎

We now prove Theorem 5.1, following the approach of [18, Theorem 7.17].

Proof of Theorem 5.1..

We begin by showing that $m^{\operatorname{BR}}$-a.e. fiber measure $\mu_{x_{2}}^{\pi_{2}}$ is atomic. Define

[TABLE]

and assume for contradiction that $m^{\operatorname{BR}}(B)>0$. Then, as in [18, Remark 4.8], we may find a compact set

[TABLE]

with $m^{\operatorname{BR}}(\Omega_{1})>0$ and satisfying (5.2).

Write

[TABLE]

where $(\mu_{x_{2}}^{\pi_{2}})^{a}$ is the purely atomic part and $(\mu_{x_{2}}^{\pi_{2}})^{c}$ is the continuous part. Define

[TABLE]

Then there exists

[TABLE]

compact with $\mu(Q)\in(0,\infty)$, $\pi_{1}(Q)\subseteq\Omega_{1}$, and satisfying equation (5.3) as in the beginning of this section, and similarly can define $Q_{\varepsilon}\subseteq Q$.

Since $Q_{\varepsilon}\subseteq\mathcal{B}$, there exists $x=(x_{1},x_{2})\in Q_{\varepsilon}$ and a sequence $(x_{1,m},x_{2})\in Q_{\varepsilon}$ with $x_{1,m}\neq x_{1}$ and

[TABLE]

We will show that this implies that $\mu$ is invariant under a non-trivial connected subgroup of $U\times\{1_{G}\}$, a contradiction to [18, Lemma 7.16].

Write

[TABLE]

where $g_{m}\to 1_{G}$, $g_{m}\neq 1_{G}$. There are two possible cases.

First, suppose that $g_{m}\in U$ for all sufficiently large $m$. Then by [18, Lemma 7.7], $\mu$ will be quasi-invariant under the subgroup generated by $\{(g_{m},1_{G})\}$, which implies invariance under a non-trivial connected subgroup of $U\times\{1_{G}\}$ becauase $g_{m}\to 1_{G}$ and $U$ is unipotent. This is a contradiction.

Thus, it must be that there exists a subsequence $g_{m_{k}}\not\in U$ for all $m_{k}$. Then by Theorem 5.3, $\mu$ is invariant under a nontrivial connected subgroup of $U\times\{1_{G}\}$, again a contradiction.

In all cases, we obtain a contradiction, and so it must have been that

[TABLE]

that is, $m^{\operatorname{BR}}$-a.e. fiber measure is atomic. Now, define

[TABLE]

We have shown that $m^{\operatorname{BR}}$-a.e. fiber measure is atomic, and $Z$ is $\Delta(U)$ invariant, so it follows from ergodicity of the joining $\mu$ that $\mu(Z^{c})=0$. This implies that there exists some $\ell\in\mathbb{N}$ so that $m^{\operatorname{BR}}$-a.e. $x_{2}$ has

[TABLE]

and the fiber measure $\mu_{x_{2}}^{\pi_{2}}$ is the uniform distribution on $\ell$ points, as desired. ∎