Entropy Rigidity of negatively curved manifolds of finite volume

TL;DR

This paper establishes that negatively curved finite volume manifolds with maximal entropy are hyperbolic, and shows that sharing the same length spectrum implies isometry to a hyperbolic manifold, extending classical compact case results.

Contribution

It proves an entropy-rigidity theorem for finite volume negatively curved manifolds and provides a new proof for the spectral rigidity result without using entropy.

Findings

Maximal entropy characterizes hyperbolic manifolds.

Finite volume manifolds with same length spectrum are isometric to hyperbolic ones.

Extension of classical compact case theorems to finite volume setting.

Abstract

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.

Equations

$${\delta }_{\Gamma }:=\underset{R\to \infty }{\lim }\frac{1}{R}\ln \#\{\gamma \in \Gamma |d(x,\gamma x)\le R\}$$

$${\int }_0^{\infty }{e}^{{\delta }_{\Gamma }t}\mathcal{A}(t)dt=+\infty$$

$$b_{\xi }(x,y)=\underset{z\to \xi }{\lim }d(x,z)-d(z,y)$$

$$(\xi |\eta {)}_x=\frac{b_{\xi }(x,y)+b_{\eta }(x,y)}{2}$$

$$[{\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4]=\frac{D_x({\xi }_1,{\xi }_3)D_x({\xi }_2,{\xi }_4)}{D_x({\xi }_1,{\xi }_4)D_x({\xi }_2,{\xi }_3)}=\underset{\stackrel{p_1,p_2,p_3,p_4\in X}{(p_1,p_2,p_3,p_4)\to ({\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4)}}{\lim }{e}^{d(p_1,p_3)+d(p_2,p_4)-d(p_1,p_4)-d(p_2,p_3)}$$

$$d(x,y)\ge d(x,z)+d(z,x)-ϵ(\vartheta ).$$

$$\mathcal{D}=\mathcal{K}\cup {\mathcal{C}}_1\cup \cdots \cup {\mathcal{C}}_l$$

$${\mathcal{A}}_P(x,R)=\text{mbox}\mathrm{vol}\left[P\backslash {\psi }_{\xi ,R}\left(\partial H_{\xi }(x)\right)\right]=\text{mbox}\mathrm{vol}\left[{\psi }_{\xi ,R}\left({\mathcal{S}}_x\right)\right]$$

$$e^{-bt}\parallel v\parallel \le \parallel d{\psi }_{\xi ,t}(v)\parallel \le e^{-at}\parallel v\parallel$$

$$e^{-(n-1)b\Delta }\le \frac{{\mathcal{A}}_P(x,R+\Delta )}{{\mathcal{A}}_P(x,R)}\le e^{-(n-1)a\Delta }$$

$$v_P(x,y,R)=\{p\in P|d(x,py)<R\}\stackrel{C}{\asymp }{\mathcal{A}}_P^{-1}\left(x,\frac{R+b_{\xi }(x,y)}{2}\right)$$

$$v_P^{\Delta }(x,y,R)=\{p\in P|R-\frac{\Delta }{2}\le d(x,py)\le R+\frac{\Delta }{2}\}\stackrel{C^{\prime }}{\asymp }{\mathcal{A}}_P^{-1}\left(x,\frac{R+b_{\xi }(x,y)}{2}\right).$$

$$d_0(x_0,\gamma x_0)\le \lambda d(x,\gamma x)+\lambda .$$

$$[x,\gamma x]=[x_0^{+},x_1^{-}]\cup [x_1^{-},x_1^{+}]\cup \cdots \cup [x_{N-1}^{+},\gamma x_N^{-}]$$

$$d_0(x_0,\gamma x_0)\le \sum _{i=1}^N{d}_0(x_0,{\gamma }_i{x}_0)+\sum _{i=1}^{N-1}{d}_0(x_0,p_i{x}_0).$$

$$d(x,\gamma x)$$

$$\sum _{i=1}^{N}d(x,{\gamma }_{i}x)+\sum _{i=1}^{N}d(x,p_{i}x)\le (1+4D)d(x,\gamma x).$$

$$d_0(x_0,\gamma x_0)\le Cd(x,\gamma x)+C$$

$$d(x,px)\stackrel{C^{\prime }}{\asymp }{d}_0(x_0,p{x}_0).$$

$$d_0(x_0,\gamma x_0)\le \sum _{i=1}^N{d}_0(x_0,k_i{x}_0)\le NC\le Cd(x,\gamma x)+C.\square$$

$$\frac{1}{c}{d}_{\xi }(x,px)-c\le d_{{\xi }^{\prime }}(x_0,p{x}_0)\le c{d}_{\xi }(x,px)+c.$$

$$2\sinh \left(\frac{d(x,px)}{2}\right)\le d_{\xi }(x,px)\le \frac{2}{b}\sinh \left(\frac{b}{2}d(x,px)\right)$$

$$\underset{n\to +\infty }{\lim }{e}^{l({\alpha }^n)+l({\beta }^n)-l({\beta }^n{\alpha }^n)}=[{\alpha }^{-},{\beta }^{-},{\alpha }^{+},{\beta }^{+}]$$

$$d\Bigl{(}]\gamma_{n_{k}}^{-},\gamma_{n_{k}}^{+}[\,\cap B_{k}\;,\;]\alpha^{-},\beta^{+}[\,\cap B_{k}\Bigr{)}<1/k$$

$$[{\alpha }^{-},{\beta }^{-},{\alpha }^{+},{\beta }^{+}]=\underset{k\to \infty }{\lim }\frac{e^{d(a_k^{\prime },{\alpha }^{n_k}{a}_k^{\prime })+d({\beta }^{-n_k}{b}_k^{\prime },b_k^{\prime })}}{e^{d(a_k^{\prime },b_k^{\prime })+d({\beta }^{-n_k}{b}_k^{\prime },{\alpha }^{n_k}{a}_k^{\prime })}}=\underset{k\to \infty }{\lim }\frac{e^{d(a_k,{\alpha }^{n_k}{a}_k)+d({\beta }^{-n_k}{b}_k,b_k)}}{e^{d(a_k^{\prime },b_k^{\prime })+d({\beta }^{-n_k}{b}_k^{\prime },{\alpha }^{n_k}{a}_k^{\prime })}}$$

$$d(a_k^{\prime },b_k^{\prime })+d({\beta }^{-n_k}{b}_k^{\prime },{\alpha }^{n_k}{a}_k^{\prime })=l({\gamma }_{n_k}).$$

$$\frac{d{\mu }_{x^{\prime }}}{d{\mu }_x}(\xi )=e^{-{\delta }_{\Gamma }{b}_{\xi }(x^{\prime },x)}$$

$${\mu }_x({\gamma }^{-1}A)={\mu }_{\gamma x}(A)$$

$${\mu }_x(V_{\zeta }(x,R))\stackrel{c}{⪰}{e}^{-{\delta }_{\Gamma }R}$$

$${\mu }_x(V_{\zeta }(x,R))\stackrel{c}{⪰}{e}^{-{\delta }_{\Gamma }(R+r)}{v}_{P_i}(x,2r)$$

Full text

Entropy Rigidity of negatively curved manifolds of finite volume

M. Peigné, A. Sambusetti

Abstract

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^{2}\leq K_{X}\leq-1$, then $Ent_{top}(X)=n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_{0}$, the it is isometric to $X_{0}$ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.

AMS classification : 53C20, 37C35

Keywords: Negative curvature, entropy, length spectrum, Bowen-Margulis measure.

1 Introduction

The problem of length spectrum rigidity of Riemannian manifolds has a long history. The fact that, in negative curvature (even in constant curvature), the collection of the lengths of all closed geodesics, together with all multiplicites, does not determine the metric is well known since [34]. On the other hand, on a compact, negatively curved surface $\bar{X}$, the metric is determined up to isometry by the marked length spectrum (that is, the map ${\cal L}:{\cal C}(\bar{X})\rightarrow\mathbb{R}$ associating to each free homotopy class of loops in $\bar{X}$ the length of the shortest geodesic in the class); this was proved by Otal [25] and, independently, by Croke [8]. The same is true in dimension $n\geq 3$ for any compact, locally symmetric manifold $\bar{X}_{0}$ of negative curvature: the locally symmetric metric on $\bar{X}_{0}$ is determined, among all negatively curved metrics, by its marked length spectrum. This is consequence of Besson-Courtois-Gallot’s solution of the minimal entropy conjecture and of the fact, proved by Hamenstadt [17], that if a compact, negatively curved manifold $\bar{X}$ has the same marked length spectrum as a compact, locally symmetric space $\bar{X}_{0}$, then ${\rm vol}(\bar{X})={\rm vol}(\bar{X}_{0})$ 111 By [18], two compact, negatively curved manifolds having the same marked length spectrum have $C^{0}$-conjugated geodesic flow; moreover, if a compact manifold $\bar{X}$ has geodesic flow which is $C^{0}$-conjugated to the flow of a manifold $\bar{X}_{0}$ whose unitary tangent bundle has a $C^{1}$-Anosov splitting (e.g., a locally symmetric space), then $\bar{X}$ has the same volume as $\bar{X}_{0}$, see [17]. The fact that, for compact manifolds, volume is preserved under $C^{1}$-conjugacies is much easier and relies on Stokes’ formula, cp. [9]..

Less seems known about the length rigidity of negatively curved, finite-volume manifolds: most generalizations are not straightforward, and seem to require additional assumptions (such as bounds on the curvature and on its derivatives, or the finiteness of the Bowen-Margulis measure); we will try to point out some of these difficulties throughout the paper. For instance, the fact that having the same marked length spectrum implies the existence of a $C^{0}$-conjugacy of the geodesic flow, would certainly require some new arguments for finite-volume manifolds 222Cp. Lemma 2.4 in [18], which is central in the argument: it is based on the fact that the closed geodesics on $\bar{X}$ equidistribute towards the Bowen-Margulis measure when $\bar{X}$ is compact. This property does not hold for non uniform lattices as soon as the Bowen-Margulis measure is infinite. .

Theorem 1.1

Let $\bar{X}$ be a finite volume $n$-manifold with pinched, negative curvature $-b^{2}\leq K_{\bar{X}}\leq-1$ which is homotopically equivalent to a locally symmetric manifold $\bar{X}_{0}$,with curvature normalized between $-4$ and $-1$. If $\bar{X}$ and $\bar{X}_{0}$ have same marked length spectrum, then they are isometric.

The proof of this is probably known to experts and follows a classical scheme: one can construct a $\Gamma$-equivariant map $f:X\rightarrow X_{0}$ between the universal coverings, which induces a homeomorphism between the boundaries and preserves the cross-ratio; then, the conclusion stems, for instance, from Bourdon’s result [6] on Möbius embeddings from locally symmetric to $CAT(-1)$-spaces. However, the main difficulty, in the case of finite volume manifolds, is to show that $f$ is a quasi-isometry, the quotients $\bar{X}$ and $\bar{X}_{0}$ being non-compact; we will give a short proof of this fact in §3, by way of example, to measure the difference from the compact case.

It is tempting to approach the above problem by using a finite-volume version of Besson-Courtois-Gallot’s inequality, given by Storm [32]; however, notice that Storm’s inequality $Ent(\bar{X})^{n}Vol(\bar{X})\geq Ent(\bar{X}_{0})^{n}Vol(\bar{X}_{0})$ concerns the volume entropy333Cp. the definition of the maps $\Psi^{b}_{c}$ in [32], which clearly require that $c$ is greater than the exponential growth rate of the universal covering of the manifold under consideration. of $\bar{X}$, and not the topological entropy $Ent_{top}(\bar{X})$ of the geodesic flow on $U\bar{X}$. Recall that for compact, negatively curved manifolds, one always has $Ent(X)=Ent_{top}(\bar{X})$, but for finite-volume manifolds $Ent(\bar{X})$ is generally strictly greater than $Ent_{top}(\bar{X})$ (cp. [13], [14]); on the other hand, the topological entropy always equals the critical exponent of the group $\Gamma=\pi_{1}(X)$ acting on the universal covering:

[TABLE]

as proved in [27]. Then, the volume entropy $Ent(\bar{X})$ is not preserved, a priori, by the condition of having same marked length spectrum, or by a conjugacy of the geodesic flows. Moreover, it is not clear whether, for finite volume manifolds, the volume is preserved under a conjugacy of the flows 444This seems unclear even under the assumption of a $C^{1}$-conjugacy; cp. the proof of Proposition 1.2 in [9], where Stokes’s theorem fails, unless one knows that the conjugacy $F$ has bounded derivatives..

The upper bound on the curvature $K_{\bar{X}}\leq-1$ in Theorem 1.1 seems unreasonably strong, as it implies, when $\bar{X}_{0}$ is hyperbolic, that $\bar{X}$ has marked length spectrum which is asymptotically critical: that is, its exponential growth rate $\delta_{\Gamma}$ is greater than or equal to the corresponding exponential growth rate for $\bar{X}_{0}$ (cp. [13], Lemma 4.1). We expect that the same result holds without curvature bounds, but, even in this weaker form, we were unable to find a proof of this result in literature.

The knowledge of the full marked length spectrum can be relaxed, as we show in the following result (which implies Theorem 1.1 in the real hyperbolic case):

Theorem 1.2

Let $\bar{X}$ be a finite volume $n$-manifold with pinched, negative curvature $-b^{2}\leq K_{\bar{X}}\leq-1$. Then $Ent_{top}(\bar{X})\geq n-1$, and the equality $Ent_{top}(\bar{X})=n-1$ holds if and only if $\bar{X}$ is hyperbolic.

The entropy characterization of constant curvature (and locally symmetric) metrics has been declined in many different ways so far: in the compact case, the above theorem is due to Knieper (see [22], where this result is not explicitly stated, but can be established following the argument of the proof of Theorem 5.2.); see also [10], and [4] for a proof in the convex-cocompact case.

We want to stress here that a basic difference between Theorem 1.2 (or, more precisely, their compact versions in [22],[10],[4]) and the celebrated entropy characterization of Hamenstädt [16] of locally symmetric metrics, with same curvature normalization, is the lack of any locally symmetric manifold $\bar{X}_{0}$ of reference homotopically equivalent to $\bar{X}$. Actually, the characterization given by Theorem 1.2 is very particular to constant curvature spaces and it does not generalize, as it is, to locally symmetric spaces: indeed, it is easy to construct compact, pinched, negatively curved manifolds with $-b^{2}\leq K_{X}\leq-1$ having same entropy as, let’s say, the complex hyperbolic space, but which are not complex hyperbolic.

The same difference holds with the existing, finite volume versions of Besson-Courtois-Gallot’s characterization of locally symmetric spaces, in particular with Boland-Connell-Souto’s papers [3] and Storm’s [32]: these two works, together, imply that if a finite volume manifold $\bar{X}$ with curvature $K_{\bar{X}}\leq-1$ has volume entropy $Ent(\bar{X})=n-1$, then it is hyperbolic, provided that one knows beforehand that $\bar{X}$ is homotopically equivalent to a hyperbolic manifold $\bar{X}_{0}$. Besides the difference between volume and topological entropy stressed above, this strong supplementary topological assumption on $\bar{X}$ is not made in Theorem 1.2.

Let us also point out that Knieper’s approach in [22] does not allow to deduce the above characterization in the finite volume case. Although G. Knieper’s horospherical measure $\mu_{H}$ can be perfectly defined in this context (following §3 of [22]), it can easily be infinite, as well as the Bowen-Margulis measure $\mu_{BM}$: given a finite volume surface $\bar{X}$ with convergent fundamental group $\Gamma$ and with a cusp whose metric, in horospherical coordinates, writes as ${\cal A}^{2}(t)dx^{2}+dt^{2}$, it is not difficult to show that $\mu_{H}$ is infinite as soon as

[TABLE]

(cp. examples in §3, [14]). Therefore, all formulas in [22] relating $Ent_{top}(\bar{X})$ to the trace of the second fundamental form of unstable horospheres need to be justified in some other way 555For instance, Corollary 4.2 in [22] only holds for $\mu_{H}$-integrable functions, and cannot be applied as it is to constant functions or to $tr\,U^{+}(v)$ to deduce Theorem 5.1, when $\|\mu_{H}\|=\infty$..

On the other hand, we will give in §4 a proof of Theorem 1.2 using the barycenter method, initiated by Besson-Courtois-Gallot in [1]-​​[2], together with some careful estimates of the Patterson-Sullivan measure, which will not need neither the finiteness of $\mu_{BM}$ (or $\mu_{H}$) nor the conservativity of the geodesic flow with respect to $\mu_{BM}$.

Also, notice that if we drop the assumption $K_{\bar{X}}\geq-b^{2}$ in Theorem 1.2, the manifold $\bar{X}$ might as well be of infinite type (i.e. with infinitely generated fundamental group, or even without any cusp, see examples in [24]), hence very far from being a hyperbolic manifold of finite-volume.

Notations.

Given functions $f,g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, we will systematically write $f\stackrel{{\scriptstyle C}}{{\prec}}g$ (or $g\stackrel{{\scriptstyle C}}{{\succ}}f$) if there exists $C>0$ and $R_{0}>0$ such that $f(R)\leq Cg(R)$ for $R>R_{0}$. We write $f\stackrel{{\scriptstyle C}}{{\asymp}}g$ when $g\stackrel{{\scriptstyle C}}{{\prec}}f\stackrel{{\scriptstyle C}}{{\prec}}g$ for $R\gg 0$ (or simply $f\asymp g$ and $f\prec g$ when the constants $C$ and $R_{0}$ are unessential)

2 Geometry at infinity in negative curvature

Throughout all the paper, $X$ will be a $n$-dimensional, complete, simply connected manifold with strictly negative curvature $-b^{2}\leq K_{X}\leq-1$.

Let $X(\infty)$ the ideal boundary of $X$: for $x,y\in X$ and $\xi\in X(\infty)$, we will denote by $[x,y]$ (resp. $[x,\xi[$) the geodesic segment from $x$ to $y$ (resp. the ray from $x$ to $\xi$), and by $x\xi(t)$ the parametrization of geodesic ray $[x,\xi[$ by arc length. Let

[TABLE]

be the Busemann function centered at $\xi$; the level set $\partial H_{\xi}(x)=\{y\,|\,b_{\xi}(x,y)|=0\}$ (resp. the suplevel set $H_{\xi}(x)=\{y\,|\,b_{\xi}(x,y)\geq 0\}$ is the horosphere (resp. the horoball) with center $\xi$ and passing through $x$. We will denote by $d_{\xi}$ the horospherical distance between two points on a same horosphere centered at $\xi$, and we define the radial semi-flow $(\psi_{\xi,t})_{t\geq 0}$ in the direction of $\xi$ as follows: for any $x\in X$, the point $\psi_{\xi,t}(x)$ lies on the geodesic ray $[x,\xi[$ at distance $t$ from $x$.

Finally, recall that for any fixed $x\in X$, the Gromov product between two points $\xi,\eta\in X(\infty),\xi\neq\eta$, is defined as

[TABLE]

where $y$ is any point on the geodesic $]\xi,\eta[$ joining $\xi$ to $\eta$; as $K_{X}\leq-1$, the expression $D_{x}(\xi,\eta)=e^{-(\xi|\eta)_{x}}$ defines a distance on $X(\infty)$, which we will call the visual distance from $x$, cp. [5]. Accordingly, the cross-ratio on $X(\infty)^{4}$ is defined as

[TABLE]

for all $\xi_{1},\xi_{2},\xi_{3},\xi_{4}\in X(\infty)$, and it is easily seen that it is independent from the choice of the base point $x$, cp.[26], [5].

We will repeatedly make use of the following, classical result in strictly negative curvature: there exists $\epsilon(\vartheta)=\log(\frac{2}{1-\cos\vartheta})$ such that any geodesic triangle $xyz$ in $X$ making angle $\vartheta=\angle_{z}(x,y)$ at $z$ satisfies:

[TABLE]

2.1 On the geometry of finite volume manifolds

Consider a lattice $\Gamma$ of $X$. The quotient manifold $\bar{X}=\Gamma\backslash X$ has finite volume, it is thus a geometrically finite manifold which admits some particular decomposition which we now recall. The following classical results are due to B. Bowditch [7], and we state them in the particular case of finite volume manifolds :

(a) the limit set of $L(\Gamma)$ of $\Gamma$ is the full boundary at infinity $X(\infty)$ and is the disjoint union of the radial limit set $L_{rad}(\Gamma)$ with finitely many orbits of bounded parabolic fixed points $L_{bp}(\Gamma)=\Gamma\xi_{1}\cup\ldots\cup\Gamma\xi_{l}$; this means that each $\xi_{i}\in L_{bp}(\Gamma)$ is the fixed point of some maximal parabolic subgroup $P_{i}$ of $\Gamma$, acting co-compactly on $X(\infty)\small\setminus\{\xi_{i}\}$;

(b) (Margulis’ lemma) there exist closed horoballs $H_{\xi_{1}},\ldots,H_{\xi_{l}}$ centered respectively at $\xi_{1},\ldots,\xi_{l}$, such that $\gamma H_{\xi_{i}}\cap H_{\xi_{j}}=\emptyset$ for all $1\leq i,j\leq l$ and all $\gamma\in\Gamma\small\setminus P_{i}$;

(c) The finite volume manifold $\bar{X}$ can be decomposed into a disjoint union of a compact set $\bar{\cal K}$ and finitely many “cusps” $\bar{\cal C}_{1},...,\bar{\cal C}_{l}$: each $\bar{\cal C}_{i}$ is isometric to the quotient of $H_{\xi_{i}}$ by a corresponding maximal bounded parabolic group $P_{i}$. We refer to $\bar{\cal K}$ and to $\bar{\cal C}=\cup_{i}\bar{\cal C}_{i}$ as to the thick part and the cuspidal part of $\bar{X}$.

For any fixed $x\in X$, let ${\cal D}={\cal D}(\Gamma,x)$ the Dirichlet domain of $\Gamma$ centered at $x$; this is a convex fundamental subset of $X$, and we may assume that ${\cal D}$ contains the geodesic rays $[x,\xi_{i}[$. Each parabolic group $P_{i}$ acts co-compactly on the horosphere $\partial H_{\xi_{i}}$ which bounds the horoball $H_{\xi_{i}}$; setting ${\cal S}_{i}={\cal D}\cap\partial H_{\xi_{i}}$ and ${\cal C}_{i}={\cal D}\cap H_{\xi_{i}}\simeq{\cal S}_{i}\times\mathbb{R}_{+}$, the fundamental domain ${\cal D}$ can be decomposed into a disjoint union

[TABLE]

where ${\cal K}$ is a convex, relatively compact set containing $x$ in its interior and projecting to the thick part $\bar{\cal K}$ of $\bar{X}$ , while ${\cal C}_{i}$ and ${\cal S}_{i}$ are, respectively, connected fundamental domains for the action of $P_{i}$ on $H_{\xi_{i}}$ and $\partial H_{\xi_{i}}$, projecting respectively to $\bar{\cal C}_{i}$ and $\bar{\cal S}_{i}$.

2.2 Growth of parabolic subgroups

The subgroups $P_{1},s,P_{l}$ will play a crucial role in the sequel; the growth of their orbital functions is best expressed by introducing the horospherical area function. Let us recall the necessary definitions:

Definition 2.1** (Horospherical Area)**

Let $P$ be a bounded parabolic group of isometries of $X$ fixing $\xi\in X(\infty)$: that is, $P$ acts cocompactly on $X(\infty)\smallsetminus\{\xi\}$ (as well as on every horosphere centered at $\xi$). Given $x\in X$, let ${\cal S}_{x}$ be a fundamental, relatively compact domain for the action of $P$ on $\partial H_{\xi}(x)$: the horospherical area function of $P$ is the function

[TABLE]

where vol denotes the Riemannian measure of horospheres.

Remark 2.2

When $-b^{2}\leq K_{X}\leq-a^{2}<0$, well-known estimates of the differential of the radial flow (cp. [19]) yield, for any $t\in\mathbb{R}$ and $v\in T^{1}X$

[TABLE]

Therefore we deduce that, for any $\Delta>0$,

[TABLE]

The following Proposition shows how the horospherical area ${\cal A}_{P}$ is related to the orbital function of $P$, cp. [13]:

Proposition 2.3

*Let $P$ be a bounded parabolic group of $X$ fixing $\xi$, with $diam({\cal S}_{x})\leq d$.

There exist $R_{0}$ and $\Delta_{0}$ only depending on $n,a,b,d$ and constants $C=C(n,a,b,d)$ and $C^{\prime}=C^{\prime}(n,a,b,d,\Delta)$ such that, for any $R\geq b_{\xi}(x,y)+R_{0}$ and any $\Delta>\Delta_{0}$, the numbers $v_{P}(x,y,R)$ and $v^{\Delta}_{P}(x,y,R)$ of orbit points of $Py$ falling, respectively, in the balls $B(x,R)$ and in the annuli $A^{\Delta}(x,R)$ satisfy:*

[TABLE]

[TABLE]

3 Length spectrum and rigidity

This section is devoted to the proof of Theorem 1.1.

Let $\Gamma$ be the fundamental group of the manifolds $\bar{X}$ and $\bar{X}_{0}$, acting by isometries on their Riemannian universal coverings $X$ and $X_{0}$ respectively. We will construct a $\Gamma$-equivariant homeomorphism $f_{\infty}:\partial X(\infty)\rightarrow X_{0}(\infty)$ and apply the following:

Theorem 3.1** (cp. [6])**

Let $X$ be a CAT$(-1)$-space and $X_{0}$ a symmetric space of rank one, with curvature $-4\leq K_{X_{0}}\leq 1$. Assume that $f_{\infty}:X(\infty)\rightarrow X_{0}(\infty)$ is a $\Gamma$-equivariant homeomorphism which preserves the cross-ratio: then there exists a $\Gamma$-equivariant isometry $f:X\to X_{0}$ whose extension on $X(\infty)$ coincides with $f_{\infty}$.

For this, we fix $x\in X$ and $x_{0}\in X_{0}$ and consider the natural $\Gamma$-equivariant bijection $\phi:\Gamma x\rightarrow\Gamma x_{0}$. The main difficulty here is to show the following:

Proposition 3.2

The map $\phi$ is a quasi-isometry between the orbits, with respect to the distances induced by the Riemannian distances of $X$ and $X_{0}$ respectively.

We assume Proposition 3.2 for a moment. Since $\bar{X}$ and $\bar{X}_{0}$ have finite volume, the limit set of $\Gamma$ coincides with the full boundaries $X(\infty)$ and $X_{0}(\infty)=\mathbb{S}^{n-1}$ and the map $\phi$ extends to a bi-Hölder and $\Gamma$-equivariant homeomorphism $f_{\infty}$ between these boundaries, endowed with their natural visual metric from $x$ and $x_{0}$.

Now, the fact that $\bar{X}$ and $\bar{X}_{0}$ have the same marked length spectrum implies that $f_{\infty}$ preserves the cross ratio; this follows for instance from [26]. For the sake of completeness, we will give a proof of this fact at the end of this section (Proposition 3.5), based on an argument from [21] (where the same is proved for symmetric spaces).

We conclude by 3.1 that there exists an isometry between the quotients $\bar{X}$ and $\bar{X}_{0}$.$\hfill\Box$

**Proof of Proposition 3.2.

**Let us first show that there exists $\lambda>1$ such that, for all $\gamma\in\Gamma$, we have

[TABLE]

Consider the decomposition of $\bar{X}$ described in subsection 2.1: we denote by $\mathcal{H}$ the set of pairwise disjoint horoballs which project on the cuspidal part of $\bar{X}$, so that $\widetilde{\mathcal{K}}:=X\small\setminus\cup_{H\in\mathcal{H}}H=\Gamma{\mathcal{K}}$ is the subset of $X$ projecting to the thick part $\bar{\mathcal{K}}$ of $\bar{X}$.

We assume that $d(H,H^{\prime})\geq 1$ for any $H\neq H^{\prime}$ in $\mathcal{H}$, and set ${\rm diam}({\cal K})=D$.

For any $\gamma\in\Gamma$, the geodesic segment $[x,\gamma x]$ intersects at most $N\leq d(x,\gamma x)$ distinct horoballs $H\in\mathcal{H}$ and can be decomposed as

[TABLE]

where $x_{0}^{+}=x$, $x_{N}^{-}=x$, and where $[x_{i}^{-},x_{i}^{+}]$ is equal to $[x,\gamma x]\cap H_{i}$ for some horoball $H_{i}\in\mathcal{H}$ and each $[x_{i}^{+},x_{i+1}^{-}]$ is included in $\widetilde{\mathcal{K}}$. Then, there exist elements $g_{i}\in\Gamma$ and $p_{i}\in P_{1}\cup\cdots\cup P_{l}$, for $1\leq i\leq N-1$, with $g_{N}:=\gamma$, such that $x_{i}^{-}\in g_{i}{\cal K}$, $x_{i}^{+}\in g_{i}p_{i}{\cal K}$; moreover, set $\gamma_{i}:=p_{i-1}^{-1}g_{i-1}^{-1}g_{i}$ for $1\leq i\leq N$ with the convention $p_{0}=g_{0}=1$.

Notice that all the geodesics $[x,\gamma_{i}x]$ are included in a $D^{\prime}=D^{\prime}(D)$-neighbourhood of $\widetilde{\cal K}$: in fact, the length of the broken geodesic $[x_{i}^{+},g_{i}p_{i}x]\cup[g_{i}p_{i}x,g_{i+1}x]\cup[g_{i+1}x,x_{i+1}^{-}]$ exceeds the length of $[x_{i}^{+},x_{i+1}^{-}]$ at most of $2D$, so (the curvature being bounded above by $-1$) it stays $D^{\prime}(D)$ close to $[x_{i}^{+},x_{i+1}^{-}]$; by construction this last segment does not enter any horoball of ${\cal H}$, so $[g_{i}p_{i}x,g_{i+1}x]$ and $[x,\gamma_{i+1}x]=p_{i}^{-1}g_{i}^{-1}[g_{i}p_{i}x,g_{i+1}x]$ stay in the $D^{\prime}$-neighbourhood of $\widetilde{\cal K}$.

Now we have $\gamma=\gamma_{1}p_{1}\gamma_{2}\cdots\gamma_{N-1}p_{N-1}\gamma_{N}$, so

[TABLE]

On the other hand

[TABLE]

which in turn yields, as $N\leq d(x,\gamma x)$,

[TABLE]

To obtain inequality (5), it is thus sufficient to check that it holds for each $\gamma_{i}$ and $p_{i}$ which appears in the sums (6) and (7). This is proved in the two following lemmas:

Lemma 3.3

For any $D^{\prime}>0$, there exists $C>0$ such that

[TABLE]

for any $\gamma\in\Gamma$ such that $[x,\gamma x]$ lies in the $D^{\prime}$-neighbourhood of $\widetilde{\cal K}$.

Lemma 3.4

There exists $C^{\prime}>0$ such that, for any parabolic isometry $p\in P_{1}\cup\cdots\cup P_{l}$

[TABLE]

Switching the roles of $(X,d)$ and $(X_{0},d_{0})$, we obtain the opposite inequality $d_{0}(x_{0},\gamma x_{0})\leq\lambda d(x,\gamma x)+\lambda$, which concludes the proof of Proposition 3.2.$\hfill\Box$

Proof of Lemma 3.3.

Let $\gamma\in\Gamma$ such that $[x,\gamma x]$ lies in the $D^{\prime}$-neighbourhood of $\widetilde{\cal K}$, and let $x_{0}=x$, $x_{N}=\gamma x$ and $x_{1},\cdots,x_{N-1}$ be the points on the geodesic segment $[x,\gamma x]$ such that $d(x,x_{i})=iD$ for $0\leq i\leq N-1$, with $N-1=[d(x,\gamma x)]$. There exist isometries $h_{0}=1,h_{1},\cdots,h_{N-1},h_{N}=\gamma$ in $\Gamma$ such that $d(x_{i},h_{i}x)\leq D+D^{\prime}$; setting $k_{i}=h_{i-1}^{-1}h_{i}$, we then have $\gamma=k_{1}k_{2}\cdots k_{N}.$ Now, for any $1\leq i\leq N$, we have $d(x,k_{i}x)\leq 1+D+D^{\prime}$; so every $k_{i}$ belongs to the finite set $B:=\{k\in\Gamma\,|\,d(x,kx)\leq 1+D+D^{\prime}\}$. Setting $C:=\max\{d_{0}(x_{0},kx_{0})\,|\,k\in B\}$, we obtain

[TABLE]

Proof of Lemma 3.4.

Let us first notice that if $p\in\Gamma$ acts on $X$ as a parabolic (resp. a hyperbolic) isometry, then it acts in the same way on $X_{0}$: actually, the infimum of the length of curves in $\bar{X}$ in the free homotopy class of a parabolic element $p$ is [math] and this condition is preserved since $\bar{X}$ and $\bar{X}_{0}$ have the same length spectrum.

Then, let $\xi_{1},...,\xi_{l}\in X(\infty)$ be the fixed points of the maximal parabolic subgroups $P_{1},..,P_{l}$ of $\Gamma$ such that the geodesic rays $[x,\xi_{i}[$ are included in the Dirichlet domain ${\cal D}$, as described in the subsection §2.1, and call $\xi^{\prime}_{i}$ the corresponding parabolic fixed points of $X_{0}(\infty)$; in order to simplify the notations, we set $P=P_{i}$, $\xi=\xi_{i}$ and $\xi^{\prime}=\xi_{i}^{\prime}$.

Fix a finite generating set $S$ for $P$ and let $|\cdot|_{S}$ be the corresponding word metric. As $P$ acts cocompactly by isometries on $(\partial H_{\xi}(x),d_{\xi})$ and on $(\partial H_{\xi^{\prime}}(x_{0}),d_{\xi^{\prime}})$ we know that these metric spaces are both quasi-isometric to $(P,|\cdot|_{S})$. In particular, there exists a constant $c>0$ such that, for any $p\in P$

[TABLE]

Now, by the bounds on curvature $-b^{2}\leq K_{X}\leq-1$ we get (cp. [19])

[TABLE]

hence $d(x,px)/d(x_{0},px_{0})\stackrel{{\scriptstyle C^{\prime}}}{{\asymp}}1$ for a constant $C^{\prime}>0$ only depending on $b$ and $c$.$\Box$

Proposition 3.5

Let $\alpha$ and $\beta$ be two hyperbolic isometries in $\Gamma$ with, respectively, repelling and attractive fixed points $\alpha^{-},\alpha^{+},\beta^{-},\beta^{+})$. Then

[TABLE]

where $l(\gamma)$ denotes the length of the closed geodesic corresponding to $\gamma$ for any $\gamma\in\Gamma$.

The set of couples $(\alpha^{-},\alpha^{+})$ of all hyperbolic fixed points of $\Gamma$ being dense in $X(\infty)^{2}$, this shows that $[f_{\infty}(\xi_{1}),f_{\infty}(\xi_{2}),f_{\infty}(\xi_{3}),f_{\infty}(\xi_{4})]=[\xi_{1},\xi_{2},\xi_{3},\xi_{4}]$ $\forall\xi_{1},\xi_{2},\xi_{3},\xi_{4}\in X(\infty)$.

**Proof of Proposition 3.5.

**Fix $x\in X$. For $n\geq 0$, set $\gamma_{n}=\beta^{n}\alpha^{n}$, and let $\gamma_{n}^{-}$, $\gamma_{n}^{+}$ be its repelling and attractive fixed points. Consider two sequences of points $a_{k}\in\;]\alpha^{-},\alpha^{+}[$ and $b_{k}\in\,]\beta^{-},\beta^{+}[$ such that $\displaystyle\lim_{k\to+\infty}a_{k}=\alpha^{-}$ and $\displaystyle\lim_{k\to+\infty}b_{k}=\beta^{+}$; moreover, we can choose a sequence $n_{k}\rightarrow\infty$ such that $\displaystyle\lim_{k\to+\infty}\alpha^{n_{k}}a_{k}=\alpha^{+}$ and $\displaystyle\lim_{k\to+\infty}\beta^{-n_{k}}b_{k}=\beta^{-}$.

Now, for each $k$, let $B_{k}$ be a compact ball centered at $x$ containing $a_{k}$ and $b_{k}$. Notice that $\gamma_{n}^{-}$ and $\gamma_{n}^{+}$ tend respectively to $\alpha^{-}$ and $\beta^{+}$, so the sequence of geodesics $]\gamma_{n}^{-},\gamma_{n}^{+}[$ tend to $]\alpha^{-},\beta^{+}[$: namely, for $k$ fixed, the distance between $]\gamma_{n}^{-},\gamma_{n}^{+}[$ and $]\alpha^{-},\beta^{+}[$, restricted to the compact set $B_{k}$, tends to [math] when $n\rightarrow\infty$. We can then choose $n_{k}$ large enough so that

[TABLE]

Call $a^{\prime}_{k}$, $b^{\prime}_{k}$ the projections of $a_{k},b_{k}$ on $]\gamma_{n_{k}}^{-},\gamma_{n_{k}}^{+}[$; so, the sequences $(a^{\prime}_{k})_{k}$ and $(b^{\prime}_{k})_{k}$ also converge respectively to $\alpha^{-}$ and $\beta^{+}$, and the sequences $(\alpha^{n_{k}}a_{k}^{\prime})_{k}$, $(\beta^{-n_{k}}b_{k}^{\prime})_{k}$ respectively to $\alpha^{+}$ and $\beta^{-}$.

We then have:

[TABLE]

by definition of the cross-ratio. Notice that the numerator gives exacty $e^{l(\alpha^{n_{k}})+l(\beta^{n_{k}})}$, as the points $a_{k}$ and $b_{k}$ lie on the axes of $\alpha,\beta$ respectively. On the other hand, for $k\gg 0$

[TABLE]

Indeed, let $V_{\beta}(b_{k})$ be the hyperplane orthogonal to the axis of $\beta$, passing through $b_{k}$. When $k$ is large enough, the point $\alpha^{n_{k}}a_{k}^{\prime}$ is close to $\alpha^{+}$, in particular it belongs to the half space bounded by $\beta^{-n_{k}}(V_{\beta}(b_{k}))$ which contains $\beta^{+}$; consequently, the point $\gamma_{n_{k}}a_{k}^{\prime}=\beta^{n_{k}}\alpha^{n_{k}}a_{k}^{\prime}$ belongs to the half-space $V_{\beta}(b_{k})$ which contains $\beta^{+}$, so $b_{k}^{\prime}$ lies on the geodesic $(\gamma_{n_{k}}^{-},\gamma_{n_{k}}^{+})$ between $a_{k}^{\prime}$ and $\gamma_{n_{k}}a_{k}^{\prime}$. As $d(b_{k}^{\prime},\gamma_{n_{k}}a_{k}^{\prime})=d(\beta^{-n_{k}}b_{k}^{\prime}\,,\,\alpha^{n_{k}}a_{k}^{\prime})$, the equality (11) readily follows. Letting $k\to+\infty$ in (10) then achieves the proof.$\Box$

4 Entropy rigidity

This section is devoted to the proof of Theorem 1.2.

The proof is through the method of barycenter, initiated by Besson-Courtois-Gallot [1], [2], and follows the lines of [10] (Theorem 1.6, holding for compact quotients). The main difficulty in the finite volume, non compact case is to show that the map produced by the barycenter method is proper: we will recall in §4.2 the main steps of the construction, referring the reader to [10] for the estimates which are now well established, while we will focus on the new estimates necessary to prove properness. For this, we will need accurates estimates of the Patterson-Sullivan measure of some subsets of $X(\infty)$, which we will describe in the first subsection.

4.1 On the Patterson measure of non uniform lattices

The Patterson-Sullivan measures of $\Gamma$ are a family of finite measures $\mu=(\mu_{x})$, indexed by points of $X$ and with support included in the limit set $L\Gamma\subset X(\infty)$, satisfying the following properties (cp. for instance [33], [29] for details about their construction):

they are absolutely continuous w.r. to each other: for any $x,x^{\prime}\in X$

[TABLE] 2. 2.

they are $\Gamma$-equivariant: for every $\gamma\in\Gamma$ and every Borel set $A\subset X(\infty)$

[TABLE]

When $\Gamma$ is a lattice, we will use the decomposition of $X$ explained in §2.1 to describe the local behavior of Patterson-Sullivan measures of $\Gamma$ on the limit set $\Lambda_{\Gamma}=X(\infty)$.

For $x\in X$ and $\zeta\in X(\infty)$, we consider the point $x\zeta(R)$ at distance $R$ from $x$ on the geodesic ray $[x,\zeta[$, and define the “spherical cap” $V_{\zeta}(x,R)\subset X(\infty)$ as the set of points of $X(\infty)$ whose projection on the geodesic ray $[x,\zeta[$ falls between $x\zeta(R)$ and $\zeta$. The proposition below gives an uniform estimate, which will be crucial in the sequel, for the measure $\mu_{x}(V_{\zeta}(x,R))$ of “small” spherical caps, i.e. when $R\gg 0$.

So, let ${\cal D}={\cal K}\cup{\cal C}_{1}\cup\cdots\cup{\cal C}_{l}$ be a decomposition of the Dirichlet domain of $\Gamma$ centered at some fixed point $x$, corresponding to the maximal, bounded parabolic subgroups $P_{1},...,P_{l}$ of $\Gamma$ with fixed points $\xi_{1},...,\xi_{l}$ as described in 2.1. If $x\zeta(R)$ projects to the thick part $\bar{\cal K}$ of $\bar{X}$, then formulas (12) and (13) easily give the uniform lower estimate:

[TABLE]

(where $c$ is a positive constant, depending on the minimal mass of a spherical cap at distance less than $D=diam({\cal K})$ from $x$). On the other hand, when $x\zeta(R)$ projects to the cuspidal part, we have:

Proposition 4.1

There exists a constant $c>0$ satisfying the following property. Let $\zeta\!\in\!X(\infty)$ and assume that the point $x\zeta(R)$ belongs to $\gamma{\cal C}_{i}$, $R>0$. Then:

[TABLE]

where $r=b_{\xi_{i}}(x,\gamma^{-1}x\zeta(R))$.

This estimate stems from a series of technical lemmas, and might be deduced from work developed in [28] and [31] (notice however that, in [28], $\mu_{x}$ has no atomic part, and in [31] the parabolic subgroups are assumed to satisfy an additional, strong regularity assumption). Since the estimate is of independent interest, we will report for completeness the proof of Proposition 4.1, in full generality, in the Appendix.

4.2 Entropy rigidity : proof of Theorem 1.2

Let $\bar{X}=\Gamma\backslash X$, fix a point $x_{0}\in X$ and call for short $b_{\xi}(x)=b_{\xi}(x,x_{0})$.

The function $b_{\xi}$ is strictly convex if $K_{X}\leq-1<0$, since for every point $y$ we have:

[TABLE]

where $g$ denotes the metric tensor of $X$; moreover, it is known that if the equality holds in (16) at every point $y$ and for every direction $\xi$, then the sectional curvature is constant, and $X$ is isometric to the hyperbolic space $\mathbb{H}^{n}$. The idea of the proof is to show that the condition $\delta_{\Gamma}=n-1$ forces the equality in (16).

Recall that, for every measure $\mu$ on $X(\infty)$ whose support is not reduced to one point, we can consider its barycenter, denoted $bar[\mu]$, that is the unique point of minimum of the function $y\mapsto{\cal B}_{\mu}(y)=\int_{X(\infty)}e^{b_{\xi}(y)}d\mu(\xi)$ (notice that this is $C^{2}$ and strictly convex function, as $b_{\xi}(y)$ is). If $supp(\mu)$ is not a single point, it is easy to see that $\lim_{y\rightarrow\xi}{\cal B}_{\mu}(y)=+\infty$ for all $\xi\in X(\infty)$ cp. [10].

Consider now the map $F:X\rightarrow X$ defined by

[TABLE]

where $(\mu_{x})_{x}$ is the family of Patterson-Sullivan measures associated with the lattice $\Gamma$.

In [10] it is proved that the map $F$ satisfies the following properties:

(a)* $F$ is equivariant with respect to the action of $\Gamma$, i.e. $F(\gamma x)=\gamma F(x)$;*

(b)* $F$ is $C^{2}$, with Jacobian:*

[TABLE]

where $k_{x}(u,v)$ is the bilinear form on $T_{x}X$ defined as

[TABLE]

Notice that the eigenvalues of $k_{x}$ are all greater or equal than 1, by (16).

Property (a) stems from the equivariance (i) of the family of Patterson-Sullivan measures with respect to the action of $\Gamma$, and from the cocycle formula for the Busemann function: $b_{\xi}(x_{0},x)+b_{\xi}(x,y)=b_{\xi}(x_{0},y)$. Property (b) comes from the fact that the Busemann function is $C^{2}$ on Hadamard manifolds, and is proved by direct computation, which does not use cocompactness.

By equivariance, the map $F$ defines a quotient map $\bar{F}:\bar{X}\rightarrow\bar{X}$, which is homotopic to the identity through the homotopy

[TABLE]

where $\lambda_{x}$ is the visual measure from $x$ (with total mass equal to the volume of $S^{n-1}$).

Actually, the map $F_{t}=bar\left[e^{-b_{\xi}(x)}\left(t\mu_{x}+(1-t)\lambda_{x}\right)\right]$ defines a map between the quotient manifolds, as it is still $\Gamma$-equivariant; moreover, we have $bar\left[e^{-b_{\xi}(x)}\lambda_{x}\right]=x$ since, for all $v\in T_{x}X$:

[TABLE]

We will now prove that:

Proposition 4.2

The homotopy map $\bar{F}_{t}$ is proper.

Assuming for a moment Proposition 4.2, the proof of Theorem 1.2 follows by the degree formula: since $\bar{F}$ is properly homotopic to the identity, it has degree one, so

[TABLE]

as $det(k_{x})\geq 1$ everywhere. So, if $\delta_{\Gamma}=\delta_{\Gamma}(\mathbb{H}^{n})=n-1$, we deduce that $det(k_{x})=1$ everywhere and $k=g$, hence the equality in the equation (16) holds for every $y=F(x)$ and $\xi$. Since $F$ is surjective, this shows that $X$ has constant curvature $-1$.$\Box$

Proof of Proposition 4.2.

Denote by $\bar{z}$ the projection of a point $z\in X$ to $\bar{X}$, and set $\delta=\delta_{\Gamma}$; recall that $\delta=n-1$, but we will use this property only at the end of the proof.

Let $\mu_{x}^{t}=e^{-b_{\xi}(x)}\left(t\mu_{x}+(1-t)\lambda_{x}\right)$: we need to show that if $t_{k}\rightarrow t_{0}$ and if $\bar{x}_{k}\rightarrow\infty$ in $\bar{X}$, then $\bar{y}_{k}=\bar{F}_{t_{k}}(\bar{x}_{k})=\overline{bar[\mu_{x_{k}}^{t_{k}}]}$ goes to infinity too.

Now, assume by contradiction that the points $\bar{y}_{k}$ stay in a compact subset of $\bar{X}$: so (up to a subsequence) $\bar{x}_{k},\bar{y}_{k}$ lift to points $x_{k},y_{k}$ such that $y_{k}\rightarrow y_{0}\in X$ and $d(y_{0},x_{k})=d(\bar{y}_{0},\bar{x}_{k})=R_{k}\rightarrow\infty$.

By the cocycle relation $b_{\xi}(y,x)=b_{\xi}(y,y_{0})+b_{\xi}(y_{0},x)$ and by the density formula for the Patterson-Sullivan measures $\frac{d\mu_{x}}{d\mu_{y_{0}}}(\xi)=e^{-\delta b_{\xi}(x,y_{0})}$, we have

[TABLE]

We will now estimate the two terms in (19) and show that $(d{\cal B}_{\mu_{x_{k}}^{t_{k}}})_{y_{k}}$ does not vanish for $R_{k}\gg 0$, a contradiction. So, let $\zeta_{k}$ be the endpoints of the geodesic rays $y_{0}x_{k}$ and let $v_{k}=(\nabla b_{\zeta_{k}})_{y_{k}}$. Also, consider the spherical caps $V_{\zeta_{k}}(y_{0},R_{k}/2)$ and $V_{\zeta_{k}}(y_{0},R_{k})$.

Let us first consider the contributions of the two integrals of the right hand side in (19) over $X\smallsetminus V_{\zeta_{k}}(y_{0},R_{k}/2)$. If $\xi\in X(\infty)\smallsetminus V_{\zeta_{k}}(y_{0},R_{k}/2)$, the projection of $\xi$ over $y_{0}\zeta_{k}$ falls closer to $y_{0}$ than to $x_{k}$, hence $b_{\xi}(y_{0},x_{k})\leq 0$; moreover, $|b_{\xi}(y_{k},y_{0})|\leq d(y_{k},y_{0})\rightarrow 0$, so the first integral on $X\smallsetminus V_{\zeta_{k}}(y_{0},R_{k}/2)$ for $x=x_{k}$, $y=y_{k}$ and $v=v_{k}$ gives:

[TABLE]

for $k\gg 0$. Analogously, the second integral on $X\smallsetminus V_{\zeta_{k}}(y_{0},R_{k}/2)$ yields

[TABLE]

for $k\gg 0$, since $|b_{\xi}(y_{k},x_{k})-b_{\xi}(y_{0},x_{k})|\leq d(y_{k},y_{0})$. So, these contributions are bounded.

We now compute the contributions of the integrals over $V_{\zeta_{k}}(y_{0},R_{k}/2)\smallsetminus V_{\zeta_{k}}(y_{0},R_{k})$. For all $\xi\in V_{\zeta_{k}}(y_{0},R_{k}/2)$ we have that $(\nabla b_{\xi})_{y_{0}}(\nabla b_{\zeta_{k}})_{y_{0}}$ is close to $1$, for $R_{k}\gg 0$; moreover, as

[TABLE]

we deduce that $(db_{\xi})_{y_{k}}(v_{k})>\frac{1}{2}$ on $V_{\zeta_{k}}(y_{0},R_{k}/2)$ for $k\gg 0$, hence these contributions are positive.

Finally, let us compute the contributions of these integrals on the caps $V_{\zeta_{k}}(y_{0},R_{k})$. For $\xi\in V_{\zeta_{k}}(y_{0},R_{k})$, consider the ray $y_{0}\xi$ from $y_{0}$ to $\xi$, and the projection $P(t)$ on the geodesic $y_{0}\zeta_{k}$ of the point $\xi(t):=y_{0}\xi(t)$. We have, by (1)

[TABLE]

with $\epsilon=\epsilon(\pi/2)$. Therefore we deduce that, for $k\gg 0$, we have

[TABLE]

It is clear that the right-hand side of (21) goes to infinity when $R_{k}\gg 0$; we will now prove that the right-hand side of (20) also diverges for $R_{k}\rightarrow\infty$. This will conclude the proof, as it will show that $d{\cal B}_{\mu_{x_{k}}^{t^{k}}}(v_{k})$ does not vanish for $k\gg 0$ (being a convex combination of two positively diverging terms).

So, let ${\cal D}={\cal K}\cup{\cal C}_{1}\cup\cdots\cup{\cal C}_{l}$ be a decomposition of the Dirichlet domain of $\Gamma$ centered at $y_{0}$ as in (2), corresponding to maximal, bounded parabolic subgroups $P_{1},...,P_{l}$ with fixed points $\xi_{1},...,\xi_{l}$. We know that $\bar{x}_{k}$ belongs to some cusp of $\bar{X}$, so $x_{k}\in\gamma{\cal C}_{i}$ for some $\gamma$; let then $r_{k}=b_{\xi_{i}}(y_{0},\gamma^{-1}x_{k})\leq R_{k}$.

If $\gamma\xi_{i}$ falls in $V_{\zeta_{k}}(y_{0},R_{k})$ and $\delta\gg 0$, as $K_{X}\geq-b^{2}$ we can use Propositions 4.1 and 2.3 to deduce that

[TABLE]

Since $K_{X}\leq-1$, we know that $\mathcal{A}_{P_{i}}(x_{0},y_{0},r_{k})\preceq e^{-(n-1)r_{k}}$, so we obtain

[TABLE]

On the other hand, when $\gamma\xi_{i}\not\in V_{\zeta_{k}}(y_{0},R_{k})$, we have, by Propositions 4.1 and 2.3:

[TABLE]

Both lower bounds tend to $+\infty$ as $k\to+\infty$, since $R_{k}\to+\infty$ and $\delta\leq n-1$; thus, the integral in (20) diverges. This concludes the proof that the map $\bar{F}_{t}$ is proper.$\Box$

5 Appendix

We report here, for completeness, a proof of the estimate given in Proposition 4.1.

To prove Proposition 4.1, we will need a series of elementary lemmas, where some equalities hold up to some constant: so we will use the symbol $f\stackrel{{\scriptstyle C}}{{\approx}}g$ to mean that two quantities $f$ and $g$ differ of at most $C$. To avoid cumbersome notations, we will give the same name to these constants in all the lemmas, meaning that they all hold for the choice of a suitable constant $C$ large enough. All these constants will depend on the upper bound of the sectional curvature $K_{X}\leq-1$ and, possibly, on other parameters of $\bar{X}=\Gamma\backslash X$ which we will specify case by case.

Recall that a parabolic group $P$ of isometries fixing $\xi\in X(\infty)$ is called bounded if it acts cocompactly on $X(\infty)-\{\xi\}$ (as well as on every horosphere $\partial H$ centered at $\xi$). If ${\cal D}(P,x)$ is the Dirichlet domain of $P$ centered at $x$, the sets ${\cal S}_{x}={\cal D}(P,x)\cap\partial H_{\xi}(x)$ and the trace at infinity ${\cal S}_{x}(\infty)=\overline{{\cal D}(P,x)}\cap X(\infty)$ of ${\cal D}(P,x)$ are compact, fundamental domains for the action of $P$ on $\partial H_{\xi}(x)$ and on $X(\infty)$, respectively.

The following Lemmas can be found, for instance, in [31] (Lemmes 2.6, 2.7 and 2.9):

Lemma 5.1

*There exists a constant $C>0$ with the following property.

Let $x\in X$ and $\zeta\in X(\infty)$ be fixed. Then, for any $\xi\in V_{\zeta}(x,R)$ we have:*

[TABLE]

Lemma 5.2

*There exists a constant $C>0$ with the following property.

Let $x\in X$ and $\zeta\in X(\infty)$ be fixed. Then:*

(i) for any $x^{\prime}$ such that $d(x,x^{\prime})<C$ we have

[TABLE]

(ii) for any $\xi\in V_{\zeta}(x,R+2C)$ we have

[TABLE]

provided that $R>C$.

Lemma 5.3

Let $P$ a bounded parabolic subgroup of $X$ fixing $\xi$, and let $S_{x}(\infty)$ as above. There exists a constant $C>0$ (depending on the diameter of ${\cal S}_{x}$) with the following properties: for any $p\in P$

(i) if $d(x,px)>2R$, with $R>C$, then $\forall\eta\in S_{x}(\infty)$ we have $p\eta\in V_{\xi}(x,R-C)$ and

[TABLE]

(ii) if $d(x,px)\leq 2R$, then $\forall\eta\in S_{x}(\infty)$ we have $p\eta\in X(\infty)\setminus V_{\xi}(x,R+C)$ and

[TABLE]

where $x_{R}=x\xi(R)$ is the point at distance $R$ from $x$ on the geodesic ray $[x,\xi[$.

Proof of Proposition 4.1.

Let ${\cal D}={\cal K}\cup{\cal C}_{1}\cup\cdots\cup{\cal C}_{l}$ be a decomposition of the Dirichlet domain of $\Gamma$ centered at $x$, corresponding to the maximal, bounded parabolic subgroups $P_{1},...,P_{l}$ of $\Gamma$ with parabolic fixed points $\xi_{1},...,\xi_{l}$, and with ${\cal C}_{i}={\cal D}\cap H_{\xi_{i}}$, as described in 2.1. Moreover, let ${\cal S}_{i}(\infty)=\overline{{\cal D}}\cap X(\infty)$ be the fundamental domains for the action of $P_{i}$ on $X(\infty)\setminus\{\xi_{i}\}$, and let $z_{t}:=x\zeta(t)$ and $x_{i,t}:=x\xi_{i}(t)$.

We assume that $z_{R}$ belongs to $\gamma H_{\xi_{i}}$; so, call for short $\xi=\xi_{i}$, $P=P_{i}$, $x_{R}=x_{i,R}$, ${\cal S}(\infty)={\cal S}_{i}(\infty)$ and set $r=b_{\xi}(x,\gamma^{-1}z_{R})$ hereafter.

Now, first notice that $|b_{\eta}(x,z_{R})-R|$ is bounded, uniformly in $\eta\in V_{\zeta}(x,R)$, since for $t\gg 0$ we have

[TABLE]

for $\epsilon=\epsilon(\frac{\pi}{2})$ as in (1). Thus, the density formula (12) yields

[TABLE]

for some constant $c>0$ only depending on the upper bound of the curvature.

It is thus sufficient to show that

[TABLE]

For this, we will analyse two different cases:

Case 1: $\zeta\in\Gamma\xi$.

$a)$ Assume first $\gamma=1$, so $\zeta=\xi$ and $z_{R}=x_{R}\in H_{\xi}$.

We have, by Lemma 5.3:

[TABLE]

From the equivariance and the density formula (12), (13) for the family $\mu_{x}$ we get

[TABLE]

because $b_{p\eta}(x_{R},px_{R})\approx d(x,px)-2R$, by Lemma 5.3.

Now, $\mu_{x_{R}}({\cal S}(\infty))\asymp e^{-\delta_{\Gamma}R}\mu_{x}({\cal S}(\infty))\asymp e^{-\delta_{\Gamma}R}$, since $X(\infty)=P{\cal S}(\infty)\cup\{\xi\}$ and the mass of $\mu_{x_{R}}$ is not reduced to one atom, so $\mu_{x}({\cal S}(\infty))>0$.

Therefore, from (23) we deduce that, for $\Delta$ large enough, we have

[TABLE]

as $v^{\Delta}_{P}(x,2R)\asymp v_{P}(x,2R)$ by Proposition 2.3, if $\Delta\geq\Delta_{0}$. The estimate (22) follows in this case, since $\zeta=\xi$ and $z_{R}=x_{R}$, so $r=b_{\xi}(x,z_{R})=b_{\xi}(x,x_{R})=R$.

$b)$ Assume now that $\zeta=\gamma\xi$ for some $\gamma\neq 1$.

We then set $\xi^{\prime}=\gamma\xi=\zeta$, $x^{\prime}=\gamma x$, $H_{\xi^{\prime}}=\gamma H_{\xi}$, $x^{\prime}_{t}=\gamma x_{t}$ and $R^{\prime}:=b_{\xi^{\prime}}(x^{\prime},z_{R})$.

Notice that, without loss of generality, we can assume that $x^{\prime}$ lies at distance less than ${\rm diam}(\mathcal{K})$ from the geodesic ray $x\xi^{\prime}$ (actually, as $P$ acts cocompactly on $\partial H_{\xi}$, we can replace $\gamma$ by $\gamma p$ for some suitable $p\in P$), so $d(z_{R-R^{\prime}},x^{\prime})$ and $d(z_{R},x^{\prime}_{R^{\prime}})$ are both bounded by $2{\rm diam}(\mathcal{K})$.

By Lemma 5.2(i), there exists $C>0$ such that

[TABLE]

and then (22) follows from a), by applying the inequality (24) to $\xi^{\prime},x^{\prime}$ and $R^{\prime}$. Actually, as $d(z_{R},x^{\prime}_{R^{\prime}})$ is bounded, we have $d\mu_{x^{\prime}_{R^{\prime}}}/d\mu_{z_{R}}\asymp 1$ and we get from (24)

[TABLE]

which gives (22), since in this case $\xi^{\prime}=\zeta$ and $r=b_{\xi}(x,\gamma^{-1}z_{R})=b_{\xi^{\prime}}(x^{\prime},z_{R})=R^{\prime}$.

Case 2: $\zeta\not\in\Gamma\xi$.

$a)$ Assume first that $\gamma\!=\!1$, so $z_{R}\in H_{\xi}$.

Let $C$ be the constant in Lemma 5.2. If $\xi\in V_{\zeta}(x,R-2C)$ we call $S=R-4C$, so that $\xi\in V_{\zeta}(x,S+2C)$ and we have $V_{\zeta}(x,S)\supset V_{\xi}(x,S+C)$ by Lemma 5.2(ii). Notice that we have $d(z_{R},x_{S})\leq 5C$ by Lemma 5.1. Therefore, applying again (24) to $\xi,x$ and $S$, we get

[TABLE]

and the estimate (22) follows, since here $r=b_{\xi}(x,z_{R})\approx b_{\xi}(x,x_{S})=S$.

On the other hand, if $\xi\not\in V_{\zeta}(x,R-2C)$, let $\bar{\zeta}$ be the point at infinity of the geodesic supporting $]\zeta,x]$, different from $\zeta$, and let $\bar{x}$ be the point of $]\zeta,x]\cap\partial H_{\xi}$ closest to $\zeta$. Moreover, let $\bar{R}:=d(\bar{x},z_{R})$. Notice that $z_{R}=\bar{x}\bar{\zeta}(\bar{R})$ and that, setting $\bar{x}_{\bar{R}}:=\bar{x}\xi(\bar{R})$, we have $d(z_{R},\bar{x}_{\bar{R}})<C$, always by Lemma 5.1, so $d\mu_{\bar{x}_{\bar{R}}}/d\mu_{z_{R}}\asymp 1$.

Now, we have $V_{\zeta}(x,R)=X(\infty)\smallsetminus V_{\bar{\zeta}}(\bar{x},\bar{R})$ and $X(\infty)\smallsetminus V_{\zeta}(x,R-2C)=V_{\bar{\zeta}}(\bar{x},\bar{R}+2C)$; as $\xi\not\in V_{\zeta}(x,R-2C)$ we deduce that $V_{\zeta}(x,R)\supset X(\infty)\smallsetminus V_{\xi}(\bar{x},\bar{R}-C)$ by Lemma 5.2(ii).

Hence,

[TABLE]

Similarly to case 1, we can estimate this by applying Lemma 5.3 to $\bar{x}$ and $\xi$ :

[TABLE]

as $\mu_{\bar{x}_{\bar{R}}}(p{\cal S}(\infty))\asymp\mu_{\bar{x}_{\bar{R}}}({\cal S}(\infty))$ because, by Lemma 5.3 $(ii)$,

[TABLE]

if $d(\bar{x},p\bar{x})\leq 2(\bar{R}-2C)$. Moreover, $d(z_{R},\bar{x}_{\bar{R}})$ is bounded, so we deduce that $\mu_{\bar{x}_{\bar{R}}}({\cal S}(\infty))\asymp\mu_{z_{R}}({\cal S}(\infty))\asymp e^{-\delta_{\Gamma}R}$ and that

[TABLE]

so, combining (26) and (27) we obtain

[TABLE]

(since $\bar{x}$ is at bounded distance from the orbit of $x$).

$b)$ Assume now that $\gamma\neq 1$.

We set $\xi^{\prime}=\gamma\xi$, $x^{\prime}=\gamma x$, $H_{\xi^{\prime}}=\gamma H_{\xi}$, $x^{\prime}_{R}=\gamma x_{R}$, with $d(x^{\prime},[x,\xi[)\leq diam({\cal K})$, and we proceed as above, according to the cases $\xi^{\prime}\in V_{\zeta}(x,R-2C)$ or $\xi^{\prime}\not\in V_{\zeta}(x,R-2C)$.

In the first case, we call $S:=R-4C$, $S^{\prime}=b_{\xi^{\prime}}(x^{\prime},z_{S})$, so $V_{\zeta}(x,S)\supset V_{\xi^{\prime}}(x,S+C)$ and we have $d(z_{R},x^{\prime}_{S^{\prime}})\leq 6C+2diam({\cal K})$; then, using Lemma 5.2, we deduce similarly to (25), that

[TABLE]

which yields (22), as here $r=b_{\xi}(x,\gamma^{-1}z_{R})\approx b_{\xi^{\prime}}(x^{\prime},z_{S})=S^{\prime}$.

In the second case, we call again $\bar{\zeta}$ the point at infinity opposite to $\zeta$ with respect to $x$, $\bar{x}$ the point of $]\zeta,x]\cap\partial H_{\xi^{\prime}}$ closest to $\zeta$, and we set $\bar{R}:=d(\bar{x},z_{R})$, $\bar{x}_{\bar{R}}:=\bar{x}\xi^{\prime}(\bar{R})$. So, $z_{R}=\bar{x}\bar{\zeta}(\bar{R})$, $d(z_{R},\bar{x}_{\bar{R}})<2C$ and $d\mu_{\bar{x}_{\bar{R}}}/d\mu_{z_{R}}\asymp 1$. Then, we deduce as before that $V_{\zeta}(x,R)\supset X(\infty)\smallsetminus V_{\xi^{\prime}}(\bar{x},\bar{R}+C)$ and we obtain, analogously to (28), that

[TABLE]

which concludes the proof as, in this case, $\bar{R}=b_{\xi^{\prime}}(\bar{x},\bar{x}_{\bar{R}})\approx b_{\xi^{\prime}}(x^{\prime},z_{R})=r\;.\Box$