The unique measure of maximal entropy for a compact rank one locally CAT(0) space

TL;DR

This paper proves the uniqueness of the measure of maximal entropy for geodesic flow on certain compact rank one locally CAT(0) spaces, linking it to the critical exponent of the Poincaré series.

Contribution

It establishes the uniqueness of the Bowen-Margulis measure as the measure of maximal entropy in this geometric setting.

Findings

Bowen-Margulis measure is unique for the geodesic flow.

Topological entropy equals the critical exponent of the Poincaré series.

Results extend understanding of entropy in non-positive curvature spaces.

Abstract

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincar\'e series.

Equations

$$d_{SX}(v,w)=\underset{t\in \mathbb{R}}{\sup }{e}^{-|t|}{d}_X(v(t),w(t)).$$

$$f(t_0)-f(r)>\delta (e^{t_0}-e^t)=\delta {\int }_r^{t_0}{e}^{t}dt>\delta e^r(t_0-r)\ge L(t_0-r),$$

$$h_{\mathrm{top}}(f,Y)=\underset{\epsilon \to 0^{+}}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log r_n(f,Y,\epsilon )$$

$$\text{and}{h}_{\mathrm{top}}(f)=\sup \left\{h_{\mathrm{top}}(f,K):K\subseteq Z\text{ compact}\right\}.$$

$$h_{\mathrm{top}}(f)=\sup \left\{h_{\mathrm{top}}(f,K):K\subseteq Z\text{ compact and }\mathrm{diam}K<\delta \right\}.$$

$$r_n(f,\epsilon ,Y)\le s_n(f,\epsilon ,Y)\le r_n(f,\frac{\epsilon }{2},Y)$$

$$h_{\mathrm{top}}(f,Y):=\underset{\epsilon \to 0^{+}}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log r_n(f,Y,\epsilon )=\underset{\epsilon \to 0^{+}}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log s_n(f,Y,\epsilon ).$$

$$\mathrm{injrad}(X,\Gamma )=\underset{x\in X}{\inf }\sup \{r\ge 0:{\pi }_{\Gamma \backslash X}^X{|}_{B(x,r)}\text{ is injective}\}.$$

$$h_{\mathrm{top}}(g^t)=h_{\mathrm{top}}(g_{\Gamma }^t).$$

$${\delta }_{\Gamma }=\inf \{s\ge 0:\sum _{\gamma \in \Gamma }{e}^{-sd(p,\gamma q)}<\infty \}$$

$$h_{\mathrm{top}}({\varphi }_{\Gamma })=h_{\mathrm{top}}(\varphi )\ge {\delta }_{\Gamma }.$$

$$\#\left\{\gamma \in \Gamma :\gamma p\in D_1\right\}\le e^{({\delta }_{\Gamma }+\alpha )(a_{\epsilon }+2\mathrm{diam}K)}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log r_n(\varphi ,K^{\prime },\epsilon )\le \underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log \#A_n=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log \#Q_2={\delta }_{\Gamma }+\alpha .$$

$$b_{\xi }(p,q)=\underset{t\to \infty }{\lim }\left[d([q,\xi )(t),p)-t\right].$$

$$d\mu (\xi ,\eta )=e^{-{\delta }_{\Gamma }{\beta }_p(\xi ,\eta )}d{\mu }_p(\xi )d{\mu }_p(\eta )$$

$${\mathrm{pr}}_x(p)=\left\{v^{+}\in \partial X:v\in {\pi }_{\text{fp}}^{-1}(p)\text{ satisfies }v(-d(p,x))=x\right\}$$

$$\text{and}{\mathrm{pr}}_{\xi }(p)=\left\{v^{+}\in \partial X:v\in {\pi }_{\text{fp}}^{-1}(p)\text{ satisfies }{v}^{-}=\xi \right\}.$$

$${\mu }_p({\mathrm{pr}}_x(B(p,R_0)))\ge {\mu }_p(V_i)\ge \ell .$$

$${\mu }_p({\mathrm{pr}}_{x}B(p,R_0))={\mu }_{\gamma p}(\gamma {\mathrm{pr}}_{x}B(p,R_0))={\mu }_{\gamma p}({\mathrm{pr}}_{\gamma x}B(\gamma p,R_0))\ge \ell .\text{qed}$$

$$\frac{1}{b}{e}^{-{\delta }_{\Gamma }d(p,x)}\le {\mu }_p({\mathrm{pr}}_{\xi }(B(x,R_0)))\le b{e}^{-{\delta }_{\Gamma }d(p,x)},$$

$${\mu }_p({\mathrm{pr}}_{\xi }(B(x,R_0)))={\int }_{{\mathrm{pr}}_{\xi }(B(x,R_0))}{e}^{-{\delta }_{\Gamma }{b}_{\eta }(p,x)}d{\mu }_x(\eta )$$

$$\ell e^{-2{\delta }_{\Gamma }{R}_0}\le e^{-2{\delta }_{\Gamma }{R}_0}{\mu }_x({\mathrm{pr}}_{\xi }(B(x,R_0)))$$

$$\le e^{2{\delta }_{\Gamma }{R}_0}{\mu }_x({\mathrm{pr}}_{\xi }(B(x,R_0)))\le e^{2{\delta }_{\Gamma }{R}_0}\Vert {\mu }_x\Vert ,$$

$$\ell e^{-2{\delta }_{\Gamma }{R}_0}{e}^{-{\delta }_{\Gamma }d(p,x)}\le {\mu }_p({\mathrm{pr}}_{\xi }(B(x,R_0)))\le \Vert {\mu }_x\Vert e^{2{\delta }_{\Gamma }{R}_0}{e}^{-{\delta }_{\Gamma }d(p,x)}.$$

$$H_{\nu }(\mathcal{A})=\sum _{i=1}^m-\nu (A_i)\log \nu (A_i).$$

$${\mathcal{A}}_T^{(n)}:=\left\{A_{j_1}\cap T^{-1}{A}_{j_2}\cap \cdots \cap T^{-(n-1)}{A}_{j_{n-1}}:1\le j_1,j_2,\dots ,j_{n-1}\le m\right\},$$

$$h_{\nu }(T,\mathcal{A})=\underset{n\to \infty }{\lim }\frac{1}{n}{H}_{\nu }({\mathcal{A}}_T^{(n)})=\underset{n\in \mathbb{N}}{\inf }\frac{1}{n}{H}_{\nu }({\mathcal{A}}_T^{(n)}),$$

$$h_{\nu }(T)=\underset{\mathcal{A}}{\sup }{h}_{\nu }(T,\mathcal{A}).$$

$$h_{\mathrm{top}}(T)=\underset{\nu \in \mathcal{M}(T)}{\sup }{h}_{\nu }(T).$$

$$d_{\Gamma \backslash X}({\pi }_{\Gamma \backslash X}^{X}x,{\pi }_{\Gamma \backslash X}^{X}y)=\underset{\gamma \in \Gamma }{\inf }{d}_X(x,\gamma y),$$

$$d_{\Gamma \backslash SX}({\pi }_{\Gamma \backslash SX}^{SX}v,{\pi }_{\Gamma \backslash SX}^{SX}w)=\underset{\gamma \in \Gamma }{\inf }{d}_{SX}(v,\gamma w).$$

$$d_k(v,w)=\underset{0\le t\le k}{\sup }d(g^{t}v,g^{t}w),$$

Full text

The author was partially supported by NSF RTG 1045119.

1. Introduction

Let $\Gamma$ be a group acting geometrically (that is: properly discontinuously, cocompactly, and isometrically) and freely on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. The space of unit-speed parametrized geodesics, $SX$, has a natural geodesic flow. The quotient under $\Gamma$ admits a Borel probability measure of full support called the Bowen-Margulis measure [9]. In this paper we show that the geodesic flow has topological entropy equal to the critical exponent of the Poincaré series, and that the Bowen-Margulis measure is (up to rescaling) the unique measure of maximal entropy for the geodesic flow.

Theorem A**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$. The geodesic flow on $\Gamma\backslash SX$ has topological entropy equal to the critical exponent of the Poincaré series.

Theorem B**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. The Bowen-Margulis measure on $\Gamma\backslash SX$ is (up to rescaling) the unique measure of maximal entropy for the geodesic flow on $\Gamma\backslash SX$.

These two theorems generalize well-known results of Manning [8] and Knieper [6], respectively, for nonpositively-curved Riemannian manifolds. The techniques used here are essentially those of the original authors, but the details of the adaptations are finicky enough that we felt they merited being written up properly. This paper provides those details.

We mention two additional points of interest in the current work. First, Manning proved an inequality for the topological entropy for all compact Riemannian manifolds; we extend this to all proper, geodesically complete, geodesic metric spaces (Lemma 5). Second, in order to adapt Knieper’s proof of the uniqueness of the measure of maximal entropy, we show that $\Gamma\backslash SX$ is finite dimensional; this follows from showing that $X$ is finite dimensional. In particular, every closed ball in a proper, geodesically complete $\operatorname{CAT}(\kappa)$ space ($\kappa\in\mathbb{R}$) is finite dimensional (Lemma 7). 111Lemma 7 also follows from independent work by Lytchak and Nagano.

2. Topological entropy of the geodesic flow

2.1. The space of geodesics

Let $X$ be a proper metric space (i.e.  all closed metric balls are compact). A geodesic in $X$ is an isometric embedding $v\colon\mathbb{R}\to X$. A geodesic segment is an isometric embedding of a compact interval, and a geodesic ray is an isometric embedding of $[0,\infty)$. We call the space $X$ geodesic if for every pair of distinct $x,y\in X$ there is a geodesic segment in $X$ with endpoints $x$ and $y$. We call $X$ geodesically complete if every geodesic segment in $X$ can be extended to a geodesic $v\colon\mathbb{R}\to X$.

Denote by $SX$ the space of all geodesics $\mathbb{R}\to X$, where $SX$ is endowed with the compact-open topology. This space is metrizable with metric

[TABLE]

Under this metric, $SX$ is a proper metric space; moreover, the footpoint projection $\pi_{\textit{fp}}\colon SX\to X$ given by $\pi_{\textit{fp}}(v)=v(0)$ is $1$-Lipschitz. Note that $\pi_{\textit{fp}}$ is a proper map by the Arzelà-Ascoli Theorem since $X$ is proper. The geodesic flow $g^{t}$ on $SX$, defined by $(g^{t}v)(r)=v(t+r)$, is $e^{\left|t\right|}$-Lipschitz for all $t\in\mathbb{R}$.

The following lemma gives uniform control over the compact-open topology.

Lemma 1**.**

For every $\delta>0$ there exists $r\geq 0$ such that for every $L$-Lipschitz map $f\colon\mathbb{R}\to\mathbb{R}$, if $e^{-\left|t\right|}f(t)\leq\delta$ for all $t\in[-r,r]$, then $e^{-\left|t\right|}f(t)\leq\delta$ for all $t\in\mathbb{R}$.

Proof.

Let $r=\max\left\{0,\log(\frac{L}{\delta})\right\}$. Suppose that $t_{0}>r$ satisfies $f(t_{0})>\delta e^{\left|t_{0}\right|}$ but $f(t)\leq\delta e^{\left|t\right|}$ for all $t\in[-r,r]$. Then

[TABLE]

contradicting our hypothesis that $f$ is $L$-Lipschitz. Similarly for $t_{0}<-r$. ∎

Corollary 2**.**

For every $\delta>0$ there is some $r>0$ such that for all $v,w\in SX$, if $d_{X}(v(t),w(t))<\delta$ for all $t\in[-r,r]$, then $d_{SX}(v,w)<\delta$.

Now let $\Gamma$ be a group acting properly discontinuously, by isometries on $X$. The $\Gamma$-action on $X$ naturally induces a properly discontinuous, isometric action on $SX$, which is cocompact if and only if the action on $X$ is. Since the $\Gamma$-action on $SX$ commutes with the geodesic flow $g^{t}$ on $SX$, $g^{t}$ descends to a flow $g^{t}_{\Gamma}$ on the quotient $\Gamma\backslash SX$. We will write $\pi_{\Gamma\backslash SX}^{SX}\colon SX\to\Gamma\backslash SX$ for the canonical projection map.

2.2. Topological entropy

We now define topological entropy and mention some basic properties. A good reference is [11]. Let $X$ be a metric space and $f\colon X\to X$ be continuous.

For $Y\subseteq X$, $n\in\mathbb{N}$, and $\varepsilon>0$, we say a subset $A\subseteq Y$ $(n,\varepsilon)$-spans $Y$ (with respect to $f$) if for all $y\in Y$ there exists $a\in A$ such that for all $i$ with $0\leq i<n$, we have $d(f^{i}(y),f^{i}(a))\leq\varepsilon$. Let $r_{n}(f,Y,\varepsilon)$ be the minimum number of elements in an $(n,\varepsilon)$-spanning set of $Y$ under $f$, and define the topological entropy

[TABLE]

It is a standard fact (see e.g.  [11, Corollary 7.5.1]) that when $f$ is uniformly continuous, we may use arbitrarily small compact sets $K$ when calculating topological entropy; that is, for all $\delta>0$ we have

[TABLE]

It is often useful to use the notion of a separated set, which is in some sense dual to the notion of a spanning set. For $n\in\mathbb{N}$ and $\varepsilon>0$, we say a subset $A\subseteq X$ $(n,\varepsilon)$-separated (with respect to $f$) if for all distinct $x,y\in A$, some $i$ with $0\leq i<n$ satisfies $d(f^{i}(x),f^{i}(y))>\varepsilon$. We call a set $A\subseteq X$ $\varepsilon$-separated if it is $(1,\varepsilon)$-separated, i.e.  $d(x,y)>\varepsilon$ for all distinct $x,y\in A$; likewise a set $A\subseteq Y$ $\varepsilon$-spans $Y$ if $d(y,A)\leq\varepsilon$ for all $y\in Y$. Write $s_{n}(f,Y,\varepsilon)$ for the maximum number of elements in an $(n,\varepsilon)$-separated subset of $Y$ under $f$. Then

[TABLE]

and therefore

[TABLE]

Thus we could use either spanning sets or separated sets in the definition of $h_{\mathrm{top}}(f)$.

It is well known (see e.g.  [11, Theorem 7.10]) that when $f\colon X\to X$ is uniformly continuous, then $h_{\mathrm{top}}(f^{k})=kh_{\mathrm{top}}(f^{k})$ for all $k>0$. Thus, for a continuous flow $\phi=(\phi^{t})_{t\in\mathbb{R}}\colon X\to X$, we define the topological entropy of $\phi$ to be the topological entropy of the time-one map $\phi^{1}$, i.e.  $h_{\mathrm{top}}(\phi):=h_{\mathrm{top}}(\phi^{1})$. Then $h_{\mathrm{top}}(\phi^{r})=rh_{\mathrm{top}}(\phi)$ for all $r>0$. We remark that although in general $h_{\mathrm{top}}(\phi^{-1})\neq h_{\mathrm{top}}(\phi)$, when $\phi$ is the geodesic flow on some $SX$ then $h_{\mathrm{top}}(\phi^{r})=\left|r\right|h_{\mathrm{top}}(\phi^{r})$ for all $r\in\mathbb{R}$.

Definition 3**.**

Let $\Gamma$ be a group acting properly discontinuously, isometrically on a proper metric space $X$. Let $\pi_{\Gamma\backslash X}^{X}\colon X\to\Gamma\backslash X$ be the canonical projection. Define

[TABLE]

Note if $x,y\in X$ have $d(x,y)<\operatorname{injrad}(X,\Gamma)$, then $d(\pi_{\Gamma\backslash SX}^{SX}(x),\pi_{\Gamma\backslash SX}^{SX}(y))=d(x,y)$.

The next lemma follows from [11, Theorem 8.12].

Lemma 4**.**

Let $\Gamma$ act isometrically on a proper metric space $X$, and let $f\colon X\to X$ be a $\Gamma$-equivariant and uniformly continuous map. If $\operatorname{injrad}(X,\Gamma)>0$, then the factor map $f_{\Gamma}\colon\Gamma\backslash X\to\Gamma\backslash X$ satisfies $h_{\mathrm{top}}(f_{\Gamma})=h_{\mathrm{top}}(f)$.

Thus the geodesic flow $g^{t}$ on $SX$ for a proper, geodesically complete, geodesic metric space $X$, under an isometric action by $\Gamma$ with $\operatorname{injrad}(X,\Gamma)>0$, must satisfy

[TABLE]

2.3. A general inequality

Let $\Gamma$ be a group acting properly discontinuously and isometrically on a proper metric space $X$. The critical exponent

[TABLE]

of the Poincaré series for $\Gamma$ does not depend on choice of $p$ or $q$. We remark that $\delta_{\Gamma}<\infty$ whenever $\Gamma$ is finitely generated.

Lemma 5** (cf.  Theorem 1 of [8]).**

Let $\Gamma$ be a group acting properly discontinuously and isometrically on a proper, geodesically complete, geodesic metric space $X$. Then the geodesic flow $\phi=(\phi^{t})_{t\in\mathbb{R}}=(g^{t})_{t\in\mathbb{R}}$ on $SX$ satisfies $h_{\mathrm{top}}(\phi)\geq\delta_{\Gamma}$.

In particular, if $\operatorname{injrad}(X,\Gamma)>0$ then the factor flow $\phi_{\Gamma}$ on $\Gamma\backslash SX$ satisfies

[TABLE]

Proof.

The case $\delta_{\Gamma}=0$ is trivial, so assume $\delta_{\Gamma}>0$. Fix $x\in X$, and find $\varepsilon>0$ small enough that for all $\gamma\in\Gamma$, either $\gamma x=x$ or $d(x,\gamma x)>3\varepsilon$. Let $\alpha>0$ be arbitrary. For each $r>0$, let $S(r)=\operatorname{B}(x,r+\varepsilon)\smallsetminus\operatorname{B}(x,r)$ and $V_{S}(r)=\#{\left\{\gamma\in\Gamma:\gamma x\in S(r)\right\}}$. It follows from the definition of $\delta_{\Gamma}$ that there is a sequence $r_{k}\to\infty$ such that $V_{S}(r_{k})\geq e^{(\delta_{\Gamma}-\alpha)r_{k}}$ for all $k$; we may assume each $r_{k}=\varepsilon n_{k}$ for some $n_{k}\in\mathbb{N}$. Now for each point $y\in S(r_{k})\cap\Gamma x$ choose $v_{x,y}\in SX$ such that $v_{x,y}(0)=x$ and $v_{x,y}(d(x,y))=y$. Then the set $A_{r_{k}}:=\left\{v_{x,y}:y\in S(r_{k})\cap\Gamma x\right\}$ is $(r_{k},\varepsilon)$-separated because each pair of distinct $v_{x,y},v_{x,y^{\prime}}\in A_{r_{k}}$ has $d(g^{r_{k}}v_{x,y},g^{r_{k}}v_{x,y^{\prime}})\geq d(v_{x,y}(r_{k}),v_{x,y^{\prime}}(r_{k}))\geq d(y,y^{\prime})-d(y,v_{x,y}(r_{k}))-d(y^{\prime},v_{x,y^{\prime}}(r_{k}))>3\varepsilon-\varepsilon-\varepsilon=\varepsilon$. Also, $A_{r_{k}}\subseteq\pi_{\textit{fp}}^{-1}\left\{x\right\}$, which is compact because $X$ is proper. Thus $h_{\mathrm{top}}(\phi)\geq\limsup_{r_{k}\to\infty}\frac{1}{r_{k}}\log\#{A_{r_{k}}}=\limsup_{k\to\infty}r_{k}^{-1}\log V_{S}(r_{k})\geq\delta_{\Gamma}-\alpha$. Since $\alpha>0$ was arbitrary, $h_{\mathrm{top}}(\phi)\geq\delta_{\Gamma}$. ∎

2.4. Nonpositive curvature

A metric space $X$ is called CAT(0) if $X$ is geodesic and satisfies the CAT(0) inequality: for every triple of distinct points $x,y,z\in X$, points on the geodesic triangle $\triangle(x,y,z)$ are no further apart than the corresponding points on a triangle in Euclidean $\mathbb{R}^{2}$ with the same edge lengths. This generalizes nonpositive curvature from Riemannian manifolds. For more on $\operatorname{CAT}(0)$ spaces, see [1] or [3]. An important consequence of the $\operatorname{CAT}(0)$ inequality is that every $\operatorname{CAT}(0)$ metric is convex—that is, for every pair of geodesics (or even geodesic segments) $v,w$ in $X$, the function $t\mapsto d_{X}(v(t),w(t))$ is convex.

Lemma 6** (cf.  Theorem 2 of [8]).**

Let $\Gamma$ be a group acting geometrically on a proper, geodesically complete, convex (e.g.  $\operatorname{CAT}(0)$) geodesic metric space $X$. Then the geodesic flow $\phi=(\phi^{t})_{t\in\mathbb{R}}=(g^{t})_{t\in\mathbb{R}}$ on $SX$ satisfies $h_{\mathrm{top}}(\phi)\leq\delta_{\Gamma}$.

Proof.

Choose a basepoint $p\in X$, and find $R>0$ such that the compact set $K_{1}:=\overline{\operatorname{B}}(p,R)$ in $X$ satisfies $\Gamma K_{1}:=\bigcup_{\gamma\in\Gamma}\gamma K_{1}=X$. Let $L\subset SX$ be an arbitrary compact set. Now $\pi_{\textit{fp}}(L)$ is compact in $X$, so there is some $c\geq 1$ such that $\pi_{\textit{fp}}(L)$ is a subset of the compact set $K:=\overline{\operatorname{B}}(p,cR)$. Hence $L$ is a subset of the compact set $K^{\prime}:=\pi_{\textit{fp}}^{-1}K$. Since every $(n,\varepsilon)$-separated subset of $L$ is an $(n,\varepsilon)$-separated subset of $K^{\prime}$, we see that $h_{\mathrm{top}}(\phi,L)\leq h_{\mathrm{top}}(\phi,K^{\prime})$. We will prove $h_{\mathrm{top}}(\phi,K^{\prime})\leq\delta_{\Gamma}$.

Let $\alpha,\varepsilon,n>0$ be arbitrary. It follows from the definition of $\delta_{\Gamma}$ that there is some $r_{0}>0$ such that for all $r\geq r_{0}$, we have $\#{\left\{\gamma\in\Gamma:\gamma p\in\operatorname{B}(p,r)\right\}}\leq e^{(\delta_{\Gamma}+\alpha)r}$. Choose $a_{\varepsilon}\geq r_{0}$ such that for all $v,w\in SX$, if $d(v(t),w(t))\leq\varepsilon$ for all $t\in[-a_{\varepsilon},a_{\varepsilon}]$, then $d(v,w)\leq\varepsilon$.

Choose a minimal $\frac{\varepsilon}{3}$-spanning set $Q\subset K$ for $K$. Let $B_{1}=\operatorname{B}(p,a_{\varepsilon}+\operatorname{diam}K)$. By assumption on $K$, the set $B_{1}$ is covered by $\Gamma$-translates of $K$. Every such $\Gamma$-translate of $K$ is contained in $D_{1}=\operatorname{B}(p,a_{\varepsilon}+2\operatorname{diam}K)$. Since $a_{\varepsilon}\geq r_{0}$, we find

[TABLE]

and therefore $B_{1}$ is covered by at most $e^{(\delta_{\Gamma}+\alpha)(a_{\varepsilon}+2\operatorname{diam}K)}$ $\Gamma$-translates of $K$. Thus $B_{1}$ contains an $\frac{\varepsilon}{3}$-spanning set $Q_{1}$ (with points in $\Gamma Q$) of cardinality at most $e^{(\delta_{\Gamma}+\alpha)(a_{\varepsilon}+2\operatorname{diam}K)}\#{Q}$. Similarly, let $B_{2}=\operatorname{B}(p,a_{\varepsilon}+n+\operatorname{diam}K)$; the above argument shows $B_{2}$ contains an $\frac{\varepsilon}{3}$-spanning set $Q_{2}$ of cardinality at most $e^{(\delta_{\Gamma}+\alpha)(a_{\varepsilon}+n+2\operatorname{diam}K)}\#{Q}$.

Note that $K^{\prime}:=\pi_{\textit{fp}}^{-1}(K)\subset SX$ is compact by properness of $\pi_{\textit{fp}}$. For each $x\in Q_{1}$ and $y\in Q_{2}$, choose some $v_{x,y}\in SX$ such that $v_{x,y}(-a_{\varepsilon})=x$ and $v_{x,y}(d(x,y)-a_{\varepsilon})=y$. We claim the set $A_{n}=\left\{v_{x,y}:x\in Q_{1}\text{ and }y\in Q_{2}\right\}$ is an $(n,\varepsilon)$-spanning set for $K^{\prime}$. Let $w\in K^{\prime}$ be arbitrary. Since $w(0)\in K$, we know $w(-a_{\varepsilon})\in B_{1}$ and $w(n+a_{\varepsilon})\in B_{2}$. Thus there exist $x\in Q_{1}$ and $y\in Q_{2}$ such that $d(x,w(-a_{\varepsilon}))\leq\frac{\varepsilon}{3}$ and $d(y,w(n+a_{\varepsilon}))\leq\frac{\varepsilon}{3}$. Then $d(y,v_{x,y}(n+a_{\varepsilon}))=\left|d(x,y)-(n+2a_{\varepsilon})\right|=\left|d(x,y)-d(w(-a_{\varepsilon}),w(n+a_{\varepsilon}))\right|\leq\frac{2\varepsilon}{3}$ by the triangle inequality, and we conclude $d(v_{x,y}(n+a_{\varepsilon}),w(n+a_{\varepsilon}))\leq\varepsilon$. By convexity of the metric on $X$, $d(v_{x,y}(t),w(t))\leq\varepsilon$ for all $t\in[-a_{\varepsilon},n+a_{\varepsilon}]$. Therefore, $d(g^{t}v_{x,y},g^{t}v_{x^{\prime},y^{\prime}})\leq\varepsilon$ for all $t\in[0,n]$ by choice of $a_{\varepsilon}$. This proves $A_{n}$ is an $(n,\varepsilon)$-spanning set for $K^{\prime}$, as claimed.

Thus we see that, since $\#{Q}$ and $\#{Q_{1}}$ do not depend on $n$,

[TABLE]

Since $\alpha,\varepsilon>0$ were arbitrary, $h_{\mathrm{top}}(\phi,K^{\prime})\leq\delta_{\Gamma}$. Thus $h_{\mathrm{top}}(\phi,L)\leq h_{\mathrm{top}}(\phi,K^{\prime})\leq\delta_{\Gamma}$. Since $L$ was arbitrary, $h_{\mathrm{top}}(\phi)\leq\delta_{\Gamma}$. ∎

Combining Lemmas 4, 5, and 6 gives us Theorem A.

3. Finite Dimension

We now show $\Gamma\backslash SX$ is finite dimensional; we will use this to prove Corollary 21.

Every complete $\operatorname{CAT}(0)$ space $X$ has an ideal boundary, written $\partial X$, obtained by taking equivalence classes of asymptotic geodesic rays. The compact-open topology on the set of rays induces a topology on $\partial X$, called the cone or visual topology. When $X$ is proper, both $\partial X$ and $\overline{X}=X\cup\partial X$ are compact metrizable spaces.

By classical theory (see, e.g.  [4, Theorem 1.7.7]) for a separable metric space, the Lebesgue covering dimension, small inductive dimension, and large induction dimension are all equal. Moreover, Kleiner remarks [5, p. 412] that the geometric dimension of a separable $\operatorname{CAT}(0)$ space $X$ equals the Lebesgue covering dimension.

Throughout this section $X$ will be a proper $\operatorname{CAT}(0)$ space, and thus all the spaces $X$, $SX$, $\Gamma\backslash X$, $\Gamma\backslash SX$, $\partial X$, and $\overline{X}$ are separable metric spaces. Therefore, we will simply write finite dimensional to mean any of the equivalent definitions.

Lemma 7**.**

Let $X$ be a proper, geodesically complete $\operatorname{CAT}(0)$ space which admits a cocompact action by isometries. Then $X$ is finite dimensional.

Proof.

Suppose, by way of contradiction, that $X$ has infinite geometric dimension. By Kleiner [5, Theorem A], this means for every $k\in\mathbb{N}$ there exist $p_{k}\in X$ and sequences $R_{j}\to 0,S_{j}\subset X$ such that $d(S_{j},p_{k})\to 0$, and $\frac{1}{R_{j}}S_{j}$ converges to the unit ball $\operatorname{B}(1)\subset\mathbb{R}^{k}$ in the Gromov–Hausdorff topology. By cocompactness we find some $p\in X$ such that for every $k\in\mathbb{N}$ there exist sequences $R_{j}\to 0,S_{j}\subset X$ such that $d(S_{j},p)\to 0$, and $\frac{1}{R_{j}}S_{j}$ converges to the unit ball $\operatorname{B}(1)\subset\mathbb{R}^{k}$ in the Gromov–Hausdorff topology. Thus by [9, Theorem 9.4] we find round spheres of dimension $k$ in the link of $X$ at $p$, for all $k$. Each such $k$-dimensional sphere contains $2k$ points that are $\frac{\pi}{2}$-separated, and by geodesic completeness we can extend these directions to obtain a $\frac{\pi}{2}$-separated subset of $2k$ points in the metric sphere of radius $1$ about $p$ in $X$. This holds for all $k$, violating properness of the metric on $X$. Therefore, $X$ must have finite geometric dimension. ∎

Remark*.*

The above proof works, with minor modification, when $X$ is assumed to be a proper, geodesically complete $\operatorname{CAT}(\kappa)$ space which admits a cocompact action by isometries, for any $\kappa\in\mathbb{R}$. Additional minor modifications show $X$, even when not cocompact, is still locally finite dimensional.

The following is due to Eric Swenson [10].

Theorem 8** (Theorem 12 of [10]).**

If $X$ is a proper $\operatorname{CAT}(0)$ space which admits a cocompact action by isometries, then $\partial X$ is finite dimensional.

Lemma 9**.**

Let $X$ be a proper $\operatorname{CAT}(0)$ space. The map $SX\to X\times\partial X\times\partial X$ given by $v\mapsto(\pi(v),\operatorname{E}(v))=(v(0),v^{-},v^{+})$ is a topological embedding.

Lemma 10**.**

Let $X$ be a proper, geodesically complete $\operatorname{CAT}(0)$ space which admits a cocompact action by isometries. Then $SX$ is finite dimensional.

Proof.

The product $X\times\partial X\times\partial X$ of finite-dimensional spaces is finite dimensional [4, Theorem 1.5.16]. Since $SX$ embeds into $X\times\partial X\times\partial X$, it is also finite dimensional [4, Theorem 1.1.1]. ∎

Corollary 11**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$. Then $\Gamma\backslash X$ and $\Gamma\backslash SX$ are finite dimensional.

Proof.

The projections $X\to\Gamma\backslash X$ and $SX\to\Gamma\backslash SX$ are open with discrete fibers, so $\dim(X)=\dim(\Gamma\backslash X)$ and $\dim(SX)=\dim(\Gamma\backslash SX)$ by [4, Theorem 1.12.7]. ∎

4. Review of Bowen-Margulis measures

STANDING HYPOTHESIS: For the remainder of the paper, let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete, $\operatorname{CAT}(0)$ space $X$ with a rank one axis. To simplify the exposition, we will exclude the trivial case $X=\mathbb{R}$ by always assuming $\partial X$ has at least three points.

4.1. More on $\operatorname{CAT}(0)$ spaces

We have a natural endpoint projection $\operatorname{E}\colon SX\to\partial X\times\partial X$ defined by $\operatorname{E}(v)=(v^{-},v^{+}):=(\lim_{t\to-\infty}v(t),\lim_{t\to+\infty}v(t))$. And in fact $v\in SX$ is parallel to $w\in SX$ if and only if $\operatorname{E}(v)=\operatorname{E}(w)$. We will also use the map $\pi_{p}\colon SX\to\partial X\times\partial X\times\mathbb{R}$ given by $\pi_{p}(v)=(v^{-},v^{+},b_{v^{-}}(v(0),p))$. Define the cross section of $v\in SX$ to be $CS(v)=\pi_{p}^{-1}\left\{\pi_{p}(v)\right\}$, and the width of a geodesic $v\in SX$ to be $\operatorname{width}(v)=\operatorname{diam}CS(v)$. The width of $v$ is in fact the maximum width of a flat strip $\mathbb{R}\times[0,R]$ in $X$ parallel to $v$.

We call a geodesic in a $\operatorname{CAT}(0)$ space rank one if $\operatorname{width}(v)<\infty$, and zero width if $\operatorname{width}(v)=0$. Write $\mathcal{R}\subseteq SX$ for the set of rank one geodesics, and $\mathcal{Z}\subseteq\mathcal{R}$ for the set of zero-width geodesics. The following lemma describes an important aspect of the geometry of rank one geodesics in a $\operatorname{CAT}(0)$ space.

Lemma 12** (Lemma III.3.1 in [1]).**

Let $v\in SX$ have $\operatorname{width}(v)<R$ for some $R>0$. There are neighborhoods $U$ and $V$ in $\overline{X}$ of $v^{-}$ and $v^{+}$ such that for any $\xi\in U$ and $\eta\in V$, there is a geodesic $w$ joining $\xi$ to $\eta$. For any such $w$, we have $d(v(0),\operatorname{im}(w))<R$; in particular, we may assume $\operatorname{width}(w)<R$ for all such $w$.

The Tits metric $\operatorname{d_{T}}$ on $\partial X$ induces a topology that is finer (usually strictly finer) than the visual topology. The Tits metric is complete $\operatorname{CAT}(1)$, and measures the asymptotic angle between geodesic rays in $X$. In fact, $(\xi,\eta)\in\operatorname{E}(\mathcal{R})$ if and only if $\operatorname{d_{T}}(\xi,\eta)>\pi$.

A geodesic $v\in SX$ is axis of an isometry $\gamma\in\operatorname{Isom}X$ if $\gamma$ translates along $v$, i.e., $\gamma v=g^{t}v$ for some $t>0$. We note the $\Gamma$-action on $X$ also naturally extends to an action by homeomorphisms on $\overline{X}$ (and therefore on $\partial X$). It is well-known that if $X$ has a rank one axis, then the $\Gamma$-action on $\partial X$ is minimal.

4.2. Measures

We recall the measures constructed in [9].

For $\xi\in\partial X$ and $p,q\in X$, let $b_{\xi}(p,q)$ be the Busemann cocycle

[TABLE]

These functions are $1$-Lipschitz in both variables and satisfy the cocycle property $b_{\xi}(x,y)+b_{\xi}(y,z)=b_{\xi}(x,z)$. Furthermore, $b_{\gamma\xi}(\gamma x,\gamma y)=b_{\xi}(x,y)$ for all $\gamma\in\operatorname{Isom}X$.

Since $\Gamma$ acts geometrically on $X$, $\Gamma$ is finitely generated and therefore $\delta_{\Gamma}<\infty$. Thus Patterson’s construction yields a conformal density $\left(\mu_{p}\right)_{p\in X}$ of dimension $\delta_{\Gamma}$ on $\partial X$, called the Patterson-Sullivan measure.

Definition 13**.**

A conformal density of dimension $\delta$ is a family $\left(\mu_{p}\right)_{p\in X}$ of equivalent finite Borel measures on $\partial X$, such that for all $p,q\in X$ and $\gamma\in\Gamma$:

1. (1)

the pushforward $\gamma_{*}\mu_{p}=\mu_{\gamma p}$ and 2. (2)

the Radon-Nikodym derivative $\frac{d\mu_{q}}{d\mu_{p}}(\xi)=e^{-\delta b_{\xi}(q,p)}$.

Now fix $p\in X$. For $(v^{-},v^{+})\in\operatorname{E}(SX)$, define $\beta_{p}\colon\operatorname{E}(SX)\to\mathbb{R}$ by $\beta_{p}(v^{-},v^{+})=(b_{\xi}+b_{\eta})(v(0),p)$; this does not depend on choice of $v\in\operatorname{E}^{-1}(v^{-},v^{+})$. The measure $\mu$ on $\partial X\times\partial X$ defined by

[TABLE]

is $\Gamma$-invariant and does not depend on choice of $p\in X$; it is called a geodesic current.

The Bowen-Margulis measure $m$ is a Radon measure on $SX$ that is invariant under both $g^{t}$ and $\Gamma$, constructed as follows: The measure $\mu\times\lambda$ on $\partial X\times\partial X\times\mathbb{R}$, where $\lambda$ is Lebesgue measure, is supported on $\operatorname{E}(\mathcal{Z})\times\mathbb{R}$. Then $\pi_{p}\colon SX\to\partial X\times\partial X\times\mathbb{R}$ is seen to restrict a homeomorphism from $\mathcal{Z}$ to $\operatorname{E}(\mathcal{Z})\times\mathbb{R}$, hence $m=\mu\times\lambda$ may be viewed as a Borel measure on $SX$.

The Bowen-Margulis measure $m$ has a quotient measure $m_{\Gamma}$ on $\Gamma\backslash SX$. Since we assume $\Gamma$ acts freely on $X$ (and therefore on $SX$), $m_{\Gamma}$ can be described as follows: Whenever $A\subset SX$ is a Borel set on which $\pi_{\Gamma\backslash SX}^{SX}$ is injective, $m_{\Gamma}(\pi_{\Gamma\backslash SX}^{SX}A)=m(A)$. By proper normalization of $\mu_{p}$, we may (and will) assume $\left\|m_{\Gamma}\right\|:=m_{\Gamma}(\Gamma\backslash SX)=1$.

5. Shadows

Define shadows in $\overline{X}$ as follows: Let $p\in X$. For $x\in X$ and $\xi\in\partial X$, write

[TABLE]

We will need some estimates on the measure of these shadows. We present here an adapted argument from [6, Proposition 2.3].

Lemma 14**.**

For each $\xi\in\overline{X}$ and $p\in X$ there exist $R>0$, $\eta\in\partial X$, and an open neighborhood $U\times V$ of $(\xi,\eta)$ in $\overline{X}\times\partial X$ such that $V\subseteq\operatorname{pr}_{\xi^{\prime}}(\operatorname{B}(p,R))$ for all $\xi^{\prime}\in U$.

Proof.

Let $p\in X$ and $\xi\in\overline{X}$. If $\xi\in X$ then $\operatorname{pr}_{\xi}(\operatorname{B}(p,R))$ is open (and nonempty) for all $R>0$ by definition of the cone topology. And for all $r>0$, the open set $U:=\operatorname{B}(\xi,r)\subset X$ satisfies $\bigcap_{y\in U}\operatorname{pr}_{y}(\operatorname{B}(p,R+r))\supset\operatorname{pr}_{\xi}(\operatorname{B}(p,R))$ by convexity. So let $\xi\in\partial X$. Choose a Tits-isolated point $\eta\neq\xi$ in $\partial X$. Then there exists $v\in\mathcal{R}$ with $(v^{-},v^{+})=(\xi,\eta)$. By Lemma 12 there exists an open neighborhood $U\times V$ of $(v^{-},v^{+})$ in $\partial X\times\partial X$ such that for all $(\xi^{\prime},\eta^{\prime})\in U\times V$, the set $\mathcal{R}\cap\operatorname{E}^{-1}(\xi^{\prime},\eta^{\prime})$ is not empty, and there is some $r>0$ (not depending on $(\xi^{\prime},\eta^{\prime})$) such that every $w\in\operatorname{E}^{-1}(\xi^{\prime},\eta^{\prime})$ passes through $\operatorname{B}(v(0),r)$. Putting $R=r+d(p,v(0))$, we see that $V\subseteq\operatorname{pr}_{\xi^{\prime}}(\operatorname{B}(p,R))$ by the triangle inequality for all $\xi^{\prime}\in U$. ∎

Lemma 15**.**

There are constants $R_{0},\ell>0$ such that $\mu_{p}(\operatorname{pr}_{x}(\operatorname{B}(p,R_{0})))\geq\ell$ for all $x\in\overline{X}$ and $p\in X$.

Proof.

For each $p\in X$, by compactness of $\overline{X}$ we obtain by Lemma 14 a finite collection of nonempty open sets $V_{i}\subset\partial X$ and constants $R_{i}>0$ such that for every $x\in\overline{X}$, some $V_{i}\subseteq\operatorname{pr}_{x}(\operatorname{B}(p,R_{i}))$. Now fix a compact set $K\subset X$ such that $X=\Gamma K:=\bigcup_{\gamma\in\Gamma}\gamma K$, and fix $p_{0}\in K$. Set $r_{0}=\max R_{i}$, $R_{0}=\operatorname{diam}K+r_{0}$, and $\ell=e^{-\delta_{\Gamma}\operatorname{diam}K}\min\mu_{p_{0}}(V_{i})$. Since the $\Gamma$-action on $\partial X$ is minimal, $\operatorname{supp}\mu_{p_{0}}=\partial X$. Thus $\ell>0$. We now have $\operatorname{pr}_{x}(\operatorname{B}(p,R_{0}))\supseteq\operatorname{pr}_{x}(\operatorname{B}(p_{0},r_{0}))$ for all $x\in\overline{X}$, and thus

[TABLE]

Now let $x\in\overline{X}$ and $p\in X$ be arbitrary. Find $\gamma\in\Gamma$ such that $\gamma p\in K$. Then

[TABLE]

Lemma 16**.**

There is a constant $b>1$ such that for all $p,x\in X$ and $\xi\in\operatorname{pr}_{x}(p)$,

[TABLE]

where $R_{0}>0$ is the constant from Lemma 15.

Proof.

Let $p,x\in X$ and $\xi\in\operatorname{pr}_{x}(p)$. Then (reversing the usual geodesic) there is some $v\in SX$ such that $v(0)=p$, $v(r)=x$, and $v^{-}=\xi$, where $r=d(p,x)$.

Let $\eta\in\operatorname{pr}_{\xi}(\operatorname{B}(x,R_{0}))$, and find $w\in SX$ such that $(w^{-},w^{+})=(\xi,\eta)$ and $w(r)\in\operatorname{B}(x,R_{0})$. Write $y=w(r)$ and $q=w(0)$. By convexity of $t\mapsto d(v(t),w(t))$, we find $d(p,q)\leq d(x,y)<R_{0}$. By $1$-Lipschitzness of $b_{\eta}$ in each variable, we see from $b_{\eta}(q,y)=r$ that $b_{\eta}(p,x)\in(r-2R_{0},r+2R_{0})$ for all $\eta\in\operatorname{pr}_{\xi}(\operatorname{B}(x,R_{0}))$.

Hence

[TABLE]

lies between $e^{-2\delta_{\Gamma}R_{0}}e^{-\delta_{\Gamma}r}\mu_{x}(\operatorname{pr}_{\xi}(\operatorname{B}(x,R_{0})))$ and $e^{2\delta_{\Gamma}R_{0}}e^{-\delta_{\Gamma}r}\mu_{x}(\operatorname{pr}_{\xi}(\operatorname{B}(x,R_{0})))$. This gives us the bounds

[TABLE]

and therefore

[TABLE]

Notice $\left\|\mu_{x}\right\|\leq e^{2\delta_{\Gamma}\operatorname{diam}(\Gamma\backslash X)}\left\|\mu_{q}\right\|$ for all $q\in X$ by Definition 13. ∎

6. Maximal entropy

Let $\nu$ be a probability measure on a space $Z$. The entropy of a measurable partition $\mathcal{A}=\left\{A_{1},\dotsc,A_{m}\right\}$ of $Z$ is

[TABLE]

Let $T\colon Z\to Z$ be a measure-preserving transformation. For the partitions

[TABLE]

$n\mapsto\frac{1}{n}H_{\nu}(\mathcal{A}_{T}^{(n)})$ is a subadditive function. Hence $\frac{1}{n}H_{\nu}(\mathcal{A}_{T}^{(n)})$ decreases to a limit

[TABLE]

called the entropy of $T$ with respect to $\mathcal{A}$. The measure-theoretic entropy of $T$ is

[TABLE]

As with topological entropy, $h_{\nu}(T^{k})=kh_{\nu}(T^{k})$ for all $k>0$, and in fact if $T$ is invertible then $h_{\nu}(T^{k})=\left|k\right|h_{\nu}(T^{k})$ for all $k\in\mathbb{Z}$. Thus, for a measure-preserving flow $\phi=(\phi^{t})_{t\in\mathbb{R}}\colon Z\to Z$, we define $h_{\nu}(\phi)=h_{\nu}(\phi^{1})$. And then $h_{\nu}(\phi^{r})=\left|r\right|h_{\nu}(\phi)$ for all $r\in\mathbb{R}$.

Now let $Z$ be a metric space and $T\colon Z\to Z$ be continuous. Write $\mathcal{M}(T)$ for the set of $T$-invariant Borel probability measures on $Z$. If $Z$ is compact then

[TABLE]

(This result is called the variational principle.) A measure of maximal entropy is a measure $\nu\in\mathcal{M}(T)$ such that $h_{\nu}(T)=h_{\mathrm{top}}(T)$, i.e. $h_{\nu}(T)$ realizes the supremum.

Convention**.**

From now on, we will write $\phi=g_{\Gamma}^{1}$, the time-one map of the geodesic flow on $\Gamma\backslash SX$. On occasion, we will write $\tilde{\phi}$ for $g^{1}$ on $SX$.

Recall the metric $d_{SX}$ on $SX$ is $d_{SX}(v,w)=\sup_{t\in\mathbb{R}}e^{-\left|t\right|}d_{X}(v(t),w(t))$. The quotient metric $d_{\Gamma\backslash X}$ on $\Gamma\backslash X$ is

[TABLE]

and the quotient metric $d_{\Gamma\backslash SX}$ on $\Gamma\backslash SX$ is

[TABLE]

We shall write $d$ for all these metrics—$d_{X}$, $d_{SX}$, $d_{\Gamma\backslash X}$, and $d_{\Gamma\backslash SX}$—as long as the context is clear.

For $k\geq 0$, we also define the metric $d_{k}$ on $SX$ by

[TABLE]

and write $d_{k}$ for the corresponding quotient metric on $\Gamma\backslash SX$. (Since $\Gamma\backslash SX$ is compact, $h_{\mathrm{top}}(\phi)$ is the same whether computed using the metric $d$ or $d_{k}$.)

Lemma 17** (cf.  Lemma 2.5 of [6]).**

Let $0<\varepsilon<\min\left\{R_{0},\operatorname{injrad}(X,\Gamma)\right\}$, where $R_{0}$ is the constant from Lemma 15. Let $\mathcal{A}=\left\{A_{1},\dotsc,A_{m}\right\}$ be a measurable partition of $\Gamma\backslash SX$ such that $\operatorname{diam}_{d_{1}}\mathcal{A}=\max_{A_{i}\in\mathcal{A}}\operatorname{diam}_{d_{1}}A_{i}<\varepsilon$. Then there is some constant $a>0$ such that

[TABLE]

for all $\alpha\in\mathcal{A}_{\phi}^{(n)}$.

Proof.

Let $\alpha\in\mathcal{A}_{\phi}^{(n)}$ be arbitrary. Since $\varepsilon<\operatorname{injrad}(X,\Gamma)$, we may lift $\alpha$ to a measurable set $\tilde{\alpha}\subset SX$ with $\operatorname{diam}_{d_{n}}(\tilde{\alpha})<\varepsilon$. Choose $v\in\tilde{\alpha}$, and let $p=v(0)$ and $x=v(n)$. Let $w\in\tilde{\alpha}$ be arbitrary, and put $\xi=w^{-}$. Find $u\in SX$ such that $u(n)=x$ and $u^{-}=\xi$. Let $q=u(0)$. Since $d(v(0),w(0))<\varepsilon$ and $d(v(n),w(n))<\varepsilon$, we obtain

[TABLE]

by convexity. Thus $d(q,x)\geq d(x,p)-d(p,q)>n-2\varepsilon$, and therefore

[TABLE]

where $b>0$ is the constant from Lemma 16.

Since $w$ was arbitrary, it follows from $d(q,p)<2\varepsilon$ that $w^{-}=\xi\in\operatorname{pr}_{x}(\operatorname{B}(p,2\varepsilon))$ for all $w\in\tilde{\alpha}$. Thus

[TABLE]

Since no $w\in\tilde{\alpha}$ can spend more than $\varepsilon$ time in $\tilde{\alpha}$, we find therefore that

[TABLE]

where each $w\in\operatorname{E}^{-1}(\xi,\eta)$. But for all $w\in\tilde{\alpha}$, we know $w(0)\in\operatorname{B}(p,\varepsilon)$, hence $(b_{\xi}+b_{\eta})(w(0),p)<2\varepsilon$ for all $w\in\tilde{\alpha}$. Thus

[TABLE]

and therefore $m_{\Gamma}(\alpha)=m(\tilde{\alpha})\leq\varepsilon e^{2\delta_{\Gamma}(2\varepsilon+R_{0})}b\left\|\mu_{p}\right\|$. Fixing an arbitrary $p_{0}\in X$, we have $\left\|\mu_{p}\right\|\leq e^{2\delta_{\Gamma}\operatorname{diam}(\Gamma\backslash X)}\left\|\mu_{p_{0}}\right\|$, which proves the lemma. ∎

Theorem 18** (cf. Theorem 2.6 of [6]).**

The Bowen-Margulis measure $m_{\Gamma}$ is a measure of maximal entropy for the geodesic flow on $\Gamma\backslash SX$, i.e.,

[TABLE]

Proof.

Let $\mathcal{A}$ be a measurable partition of $\Gamma\backslash SX$ with $\operatorname{diam}_{d_{1}}\mathcal{A}<\varepsilon$. By Lemma 17,

[TABLE]

Thus $h(\phi,\mathcal{A})\geq\delta_{\Gamma}$. By Theorem A, we now have $\delta_{\Gamma}=h_{\mathrm{top}}(\phi)\geq h_{m_{\Gamma}}(\phi)\geq h_{m_{\Gamma}}(\phi,\mathcal{A})\geq\delta_{\Gamma}$, proving Theorem 18. ∎

7. Entropy expansive

Let $T\colon V\to V$ be a self-homeomorphism of a metric space $V$. For $x\in X$ and $\varepsilon>0$, define the Bowen ball

[TABLE]

We say $T$ is $h$-expansive (or, entropy expansive) if there is some $\varepsilon>0$ (called an $h$-expansivity constant for $T$) such that

[TABLE]

The following result is due to Rufus Bowen [2, Theorem 3.5].

Theorem 19** (Bowen).**

Let $V$ be a finite-dimensional compact metric space and $\mathcal{A}=\left\{A_{1},\dotsc,A_{m}\right\}$ a Borel partition with $\operatorname{diam}\mathcal{A}<\varepsilon$. Let $\nu$ be a $T$-invariant Borel probability measure on $V$. If $T$ is $h$-expansive with expansivity constant $\varepsilon$, then

[TABLE]

Lemma 20** (cf.  Proposition 3.3 of [6]).**

The time-$k$ map $\phi^{k}=g^{k}_{\Gamma}$ of the geodesic flow on $\Gamma\backslash SX$ is $h$-expansive with $h$-expansivity constant $\varepsilon=\frac{1}{3}\operatorname{injrad}(X,\Gamma)$.

Proof.

Let $v\in SX$ and $\bar{v}=\pi_{\Gamma\backslash SX}^{SX}{v}$. Since $2\varepsilon<\operatorname{injrad}(X,\Gamma)$, the Bowen ball

[TABLE]

lifts to the Bowen ball

[TABLE]

which is a subset of the set $P(v)$ of geodesics parallel to $v$, and therefore has $h_{\mathrm{top}}(\tilde{\phi},Z_{\tilde{\phi}^{k},\varepsilon}(v))=0$ by convexity of the $\operatorname{CAT}(0)$ metric. Apply Lemma 4. ∎

It follows that the entropy map $\nu\mapsto h_{\nu}(\phi)$ is upper semicontinuous on $\mathcal{M}(\phi)$. Moreover, we have the following from Theorem 19 and Corollary 11.

Corollary 21** (cf.  Corollary 3.4 of [6]).**

Let $\mathcal{A}$ be a Borel partition of $\Gamma\backslash SX$ such that $\operatorname{diam}_{d_{k}}\mathcal{A}\leq\frac{1}{3}\operatorname{injrad}(X,\Gamma)$, and let $\nu\in\mathcal{M}(\phi)$. Then

[TABLE]

From now on, we will write $h=\delta_{\Gamma}$.

8. Uniqueness

We turn to the proof that the Bowen-Margulis measure $m_{\Gamma}$ on $\Gamma\backslash SX$ is the unique measure of maximal entropy.

STANDING HYPOTHESIS: Fix a compact set $K\subset X$ such that $X=\Gamma K:=\bigcup_{\gamma\in\Gamma}\gamma K$. Also fix $p\in K$, and let $R\geq R_{0}$, the constant from Lemma 15.

For each $R^{\prime}\geq 2R+\frac{1}{2}$ such that $K\subseteq\operatorname{B}(p,R^{\prime})$, and each $x\in X$, define

[TABLE]

Lemma 22**.**

For each $n$, there exist points $x_{1},\dotsc x_{k(n)}\in X$ and a partition $\mathcal{F}^{n}=\{F_{1}^{n},\dotsc,F_{k(n)}^{n}\}$ of $\pi_{\textit{fp}}^{-1}(\operatorname{B}(p,R^{\prime}))$ such that

[TABLE]

and $d(p,x_{i})=2n+2R+R^{\prime}$ for all $i$.

Proof.

Set $r(n)=2n+2R+R^{\prime}$, and let $P=\left\{x_{1},\dotsc,x_{k(n)}\right\}$ be a maximal weakly $2R$-separated set—that is, $d(x_{i},x_{j})\geq 2R$ for all $i\neq j$ and $P$ is maximal for this property—on the sphere $S(p,r(n))$ in $X$ of radius $r(n)$ about $p$. Since $P$ is weakly $2R$-separated, the balls $\operatorname{B}(x_{i},R)$ are pairwise disjoint; since $P$ is maximal, the balls $\operatorname{B}(x_{i},2R)$ cover $S(p,r(n))$. Thus the sets $D(x_{i},R^{\prime},R)$ are pairwise disjoint, and the sets $D(x_{i},R^{\prime},2R)$ cover $\pi_{\textit{fp}}^{-1}(\operatorname{B}(p,R^{\prime}))$. A partition $\mathcal{F}^{n}$ with the desired property is now straightforward to construct. ∎

For each $n$, fix a partition $\mathcal{F}^{n}$ guaranteed by Lemma 22, and let $L_{i}^{n}=\pi_{\Gamma\backslash SX}^{SX}(F_{i}^{n})$. Then $\mathcal{L}^{n}=\{L_{i}^{n}\}$ covers $\Gamma\backslash SX$ because $K\subseteq\operatorname{B}(p,R^{\prime})$ and $\pi_{\textit{fp}}$ is onto.

Lemma 23** (cf.  Lemma 5.1 of [6]).**

Let $R^{\prime}\geq 2R+\frac{1}{2}$ such that $K\subseteq\operatorname{B}(p,R^{\prime})$. There exists a constant $c^{\prime}>0$ such that for all $x\in X$ satisfying $d(p,x)\geq R^{\prime}+R$,

[TABLE]

Proof.

Let $\ell>0$ be the constant from Lemma 15, and let $x\in X$ satisfy $d(p,x)\geq R^{\prime}+R$. Let $\eta\in\operatorname{pr}_{x}(\operatorname{B}(p,R))$, and choose $w\in\pi_{\textit{fp}}^{-1}(x)$ such that $w^{-}=\eta$. By choice of $\eta$, we have $q:=w(t_{0})\in\operatorname{B}(p,R)$ for some $t_{0}<0$. We claim $(\eta,\zeta)\in\operatorname{E}(D(x,R^{\prime},R))$ for every $\zeta\in\operatorname{pr}_{\eta}(\operatorname{B}(x,R))$. So let $\zeta\in\operatorname{pr}_{\eta}(\operatorname{B}(x,R))$. By choice of $\zeta$, there exists $v\in\operatorname{E}^{-1}(\eta,\zeta)\cap\pi_{\textit{fp}}^{-1}(\operatorname{B}(x,R))$. Since $w(t_{0})\in\operatorname{B}(p,R)$, we obtain $v(t_{0})\in\operatorname{B}(p,2R)$ by convexity of the metric. Thus $g^{t_{\phantom{}{}_{0}}}v\in D(x,R^{\prime},R)$, which proves our claim. Hence

[TABLE]

by Lemma 16. Now for each $\zeta\in\operatorname{pr}_{\eta}(\operatorname{B}(x,R))$, if $v\in D(x,R^{\prime},R)\cap\operatorname{E}^{-1}(\eta,\zeta)$ is chosen such that $v(0)$ is the closest point on $v$ to $p$, then we have at least $v([-\frac{1}{2},\frac{1}{2}])\subset D(x,R^{\prime},R)$. Therefore,

[TABLE]

Lemma 24** (cf.  Lemma 5.3 of [6]).**

There exist constants $r,c>0$ such that for all $n\in\mathbb{N}$, no $\bar{v}\in\Gamma\backslash SX$ lies in more than $r$ sets $L_{i}^{n}$, and $m_{\Gamma}(L_{i}^{n})\geq ce^{-2hn}$ for all $i$.

Proof.

The number $r$ can be taken to be the number of $\gamma\in\Gamma$ such that $\gamma K\cap\operatorname{B}(p,R^{\prime})$ is nonempty. Hence by Lemma 23,

[TABLE]

for all $n\in\mathbb{N}$ and $i\in\left\{1,2,\dotsc,k(n)\right\}$. Thus we may take $c=\frac{c^{\prime}}{r}e^{-h(2R+R^{\prime})}$. ∎

We say a set $A$ is $(d_{n},\varepsilon)$-separated if it is $(n,\varepsilon)$-separated (under $\phi$) with respect to the metric $d_{n}$.

Lemma 25** (cf.  Lemma 5.2 of [6]).**

Fix $n\in\mathbb{N}$, and let $q\in X$ satisfy $d(q,p)\geq n+R+R^{\prime}$. Then the cardinality of a $(d_{n},\varepsilon)$-separated set of $D(q,R^{\prime},R)$ is bounded from above by a constant $a=a(\varepsilon,R^{\prime},R)$ depending only on $\varepsilon$, $R^{\prime}$, and $R$.

Proof.

Fix $\varepsilon>0$. Let $R^{\prime\prime}$ be the $r$ guaranteed by Corollary 2 for $\frac{\varepsilon}{2}$. Choose maximal $\frac{\varepsilon}{2}$-separated subsets $Q_{1}\subset\operatorname{B}(p,R^{\prime}+R^{\prime\prime})$ and $Q_{2}\subset\operatorname{B}(q,R+R^{\prime\prime})$. For each $x\in Q_{1}$ and $y\in Q_{2}$, choose $v_{x,y}\in SX$ such that $v_{x,y}(-R^{\prime\prime})=x$ and $v_{x,y}(d(x,y)-R^{\prime\prime})=y$. We claim the set $A_{n}=\left\{v_{x,y}:x\in Q_{1}\text{ and }y\in Q_{2}\right\}$ is an $(n,\frac{\varepsilon}{2})$-separated set for $D(q,R^{\prime},R)$. So let $w\in D(q,R^{\prime},R)$ be arbitrary. Then $w(0)\in\operatorname{B}(p,R^{\prime\prime})$ and $w(t_{0})\in\operatorname{B}(q,R)$, hence $w(-R^{\prime\prime})$ and $w(t_{0}+R^{\prime\prime})$ lie within distance $\frac{\varepsilon}{2}$ of some $x\in Q_{1}$ and $y\in Q_{2}$, respectively. By convexity, $d_{X}(w(t),v_{x,y}(t))\leq\frac{\varepsilon}{2}$ for all $t\in[-R^{\prime\prime},n+R^{\prime\prime}]$. Thus $d_{SX}(g^{t}w,g^{t}v_{x,y})\leq\frac{\varepsilon}{2}$ for all $t\in[0,n]$, proving the claim. Therefore, $\bigcup_{v_{x,y}\in A_{n}}\overline{\operatorname{B}}_{d_{n}}(v_{x,y},\frac{\varepsilon}{2})\supseteq D(q,R^{\prime},R)$. Now any $\frac{\varepsilon}{2}$-separated set in $\operatorname{B}(p,R^{\prime}+R^{\prime\prime})$ is, up to an isometry $\gamma\in\Gamma$, a $\frac{\varepsilon}{2}$-separated set in $\operatorname{B}(K,R^{\prime}+R^{\prime\prime})$; similarly for $\operatorname{B}(q,R+R^{\prime\prime})$. Hence $\#{A_{n}}\leq\#{Q_{1}}\cdot\#{Q_{2}}\leq s_{1}(\phi,\operatorname{B}(K,R^{\prime}+R^{\prime\prime}),\frac{\varepsilon}{2})\cdot s_{1}(\phi,\operatorname{B}(K,R+R^{\prime\prime}),\frac{\varepsilon}{2})$, which depends only on $\varepsilon$, $R$, and $R^{\prime}$. Since any $\varepsilon$-separated set in $D(q,R^{\prime},R)$ can have at most $\#{A_{n}}$ elements, we have proved the lemma. ∎

Lemma 26** (cf.  Lemma 5.4 of [6]).**

Let $P=\left\{\bar{v}_{1},\dotsc,\bar{v}_{\ell}\right\}$ be a maximal $(d_{2n},\varepsilon)$-separated set in $\Gamma\backslash SX$, and let $\mathcal{B}$ be a partition of $\Gamma\backslash SX$ such that for each $B\in\mathcal{B}$ there is some $\bar{v}_{j}\in P$ such that

[TABLE]

Then for each $L_{i}^{n}\in\mathcal{L}^{n}$,

[TABLE]

where $a(\varepsilon,R^{\prime}+\varepsilon,2R+\varepsilon)$ is the constant from Lemma 25.

Proof.

Lift $P$ to a set $\tilde{P}=\left\{v_{1},\dotsc,v_{\ell}\right\}\subset SX$ such that each $\pi_{\Gamma\backslash SX}^{SX}(v_{j})=\bar{v}_{j}$, and $d_{2n}(v_{j},F_{i}^{n})<\varepsilon$ whenever $d_{2n}(\bar{v}_{j},L_{i}^{n})<\varepsilon$. Then $\tilde{P}$ is $(d_{2n},\varepsilon)$-separated, and

[TABLE]

by hypothesis on $\mathcal{B}$ and choice of $\tilde{P}$. But every geodesic $v\in SX$ that satisfies $d_{2n}(v,F_{i}^{n})<\varepsilon$ lies in $D(x_{i},R^{\prime}+\varepsilon,2R+\varepsilon)$, so by Lemma 25

[TABLE]

Let $\mathcal{A}^{n}=\left\{\phi^{n}(L_{i}^{n})\right\}$ for each $n\in\mathbb{N}$.

Lemma 27** (cf.  Lemma 5.6 of [6]).**

Let $\nu$ be a Borel probability measure on $\Gamma\backslash SX$ and $\Omega\subseteq\Gamma\backslash SX$ be a Borel set such that $\pi_{\Gamma\backslash SX}^{SX}\phantom{}{}^{-1}(\Omega)$ contains all geodesics of nonzero width. Then there exists a union $C_{n}$ of sets $A\in\mathcal{A}^{n}=\left\{\phi^{n}(L_{i}^{n})\right\}$ such that

[TABLE]

Proof.

By construction of $\mathcal{L}^{n}$, for every $n\in\mathbb{N}$ and $v,w\in A\in\mathcal{A}^{n}$, there exist lifts $\tilde{v},\tilde{w}$ of $v,w$ (respectively) such that $d(\tilde{v}(t),\tilde{w}(t))\leq R^{\prime}$ for all $t\in[-n,n]$.

Let $\delta>0$ and choose compact sets $K_{1}^{\delta}\subset\Omega$ and $K_{2}^{\delta}\subset(\Gamma\backslash SX)\smallsetminus\Omega$ satisfying $\nu(\Omega\smallsetminus K_{1}^{\delta})<\delta$ and $\nu((\Gamma\backslash SX)\smallsetminus K_{2}^{\delta})<\delta$. Let $\tilde{K}_{1}^{\delta}=\pi_{\Gamma\backslash SX}^{SX}\phantom{}{}^{-1}(K_{1}^{\delta})$ and $\tilde{K}_{2}^{\delta}=\pi_{\Gamma\backslash SX}^{SX}\phantom{}{}^{-1}(K_{2}^{\delta})$. We claim there exists $n_{\delta}\in\mathbb{N}$ such that for all $v\in\tilde{K}_{1}^{\delta}$ and $w\in\tilde{K}_{2}^{\delta}$, there exists $t\in[-n_{\delta},n_{\delta}]$ such that $d(v(t),w(t))>b$. Otherwise, by compactness there exist $v_{0}\in\tilde{K}_{1}^{\delta}$ and $w_{0}\in\tilde{K}_{2}^{\delta}$ such that $d(v_{0}(t),w_{0}(t))\leq b$ for all $t\in\mathbb{R}$. But then $v_{0}$ and $w_{0}$ bound a flat strip, contradicting the fact that $w_{0}$ has zero width. Thus, when $n\geq n_{\delta}$, we see by our first observation that no $A\in\mathcal{A}^{n}$ intersects both $K_{1}^{\delta}$ and $K_{2}^{\delta}$.

Let $n\geq n_{\delta}$, and let $C_{n}^{\delta}$ be the union of all sets $A\in\mathcal{A}^{n}$ such that $A\cap K_{1}^{\delta}\neq\varnothing$. Notice $K_{1}^{\delta}\subseteq C_{n}^{\delta}$ by construction, and $C_{n}^{\delta}\cap K_{2}^{\delta}=\varnothing$ by the previous paragraph. Thus we see that

[TABLE]

Therefore, for every $\delta>0$ there exists a sequence $(C_{n}^{\delta})$ of measurable sets and $n_{\delta}\in\mathbb{N}$ such that for all $n\geq n_{\delta}$, $\nu(\Omega\Delta C_{n}^{\delta})\leq\delta$. So take a decreasing sequence $\delta_{k}\to 0$. Since $\nu(\Omega\Delta C_{n}^{\delta})\leq\delta$ for all $n\geq n_{\delta}$ (not just $n_{\delta}$ itself), we may assume that $n_{\delta_{k}}$ is strictly increasing in $k$. Let $C_{n}=C_{n}^{\delta_{k_{n}}}$, where $k_{n}=\max\left\{k:n_{\delta_{k}}\leq n\right\}$. This is the desired sequence $(C_{n})$. ∎

Theorem 28** (cf.  Theorem 5.8 of [6]).**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. The Bowen-Margulis measure $m_{\Gamma}$ on $\Gamma\backslash SX$ is the unique measure of maximal entropy (up to rescaling) for the geodesic flow $g_{\Gamma}^{t}$ on $\Gamma\backslash SX$.

Proof.

It remains to show that if $\nu\in\mathcal{M}(\phi)$, then $\nu\neq m_{\Gamma}$ implies $h_{\nu}(\phi)<h_{\mathrm{top}}(\phi)$. So let $\nu\neq m_{\Gamma}$ be an invariant Borel probability measure on $\Gamma\backslash SX$. Since $m_{\Gamma}$ is ergodic, it suffices to consider the case that $\nu$ and $m_{\Gamma}$ are mutually singular.

Let $\varepsilon=\frac{1}{3}\operatorname{injrad}(X,\Gamma)$ and let $P$ be a maximal $(d_{2n},\varepsilon)$-separated set in $\Gamma\backslash SX$. Let $\mathcal{B}^{n}$ be a partition of $\Gamma\backslash SX$ such that for each $B\in\mathcal{B}^{n}$ there exists $\bar{v}_{j}\in P$ such that $\overline{\operatorname{B}}_{d_{2n}}(\bar{v}_{j},\frac{\varepsilon}{2})\subset B\subset\overline{\operatorname{B}}_{d_{2n}}(\bar{v}_{j},\varepsilon)$. By Corollary 21,

[TABLE]

Since $\nu$ and $m_{\Gamma}$ are mutually singular, there is a set $\Omega\subset\Gamma\backslash SX$ such that $m_{\Gamma}(\Omega)=0$ and $\nu(\Omega)=1$. Since $m$ gives zero measure to the set of all geodesics of nonzero width, we may assume $\pi_{\Gamma\backslash SX}^{SX}\phantom{}{}^{-1}(\Omega)$ contains every geodesic of nonzero width. By Lemma 27, there are sets $C_{n}$ which are unions of elements of $\mathcal{A}^{n}$ and which satisfy $(\nu+m_{\Gamma})(C_{n}\Delta\Omega)\to 0$ as $n\to\infty$. In particular,

[TABLE]

It is a standard fact that if $a_{1},\dotsc,a_{k}\geq 0$ satisfy $\sum_{i=1}^{k}a_{i}\leq 1$, then

[TABLE]

Thus, splitting $\sum_{B\in\mathcal{B}^{n}}\nu(B)\log\nu(B)$ into two sums—gathering those $B\in\mathcal{B}^{n}$ for which $\phi^{n}(B)\cap C_{n}$ is nonempty or empty together—and applying (31) to each sum, we find

[TABLE]

by (29), where

[TABLE]

Since $C_{n}$ is a union of sets $A_{i}^{n}=\phi^{n}(L_{i}^{n})$, every $B\in\mathcal{B}_{C}^{n}$ satisfies $B\cap L_{i}^{n}\neq\varnothing$ for at least one $L_{i}^{n}\in\mathcal{L}^{n}$ such that $L_{i}^{n}\subseteq\phi^{-n}(C_{n})$, and therefore

[TABLE]

by Lemma 26. Now by Lemma 24,

[TABLE]

where $c^{\prime\prime}=\frac{r}{c}\cdot a(\varepsilon,R^{\prime}+\varepsilon,2R+\varepsilon)$ is constant. Thus we have shown

[TABLE]

Repeating this argument with $\Gamma\backslash SX$ in place of $C_{n}$, we find

[TABLE]

and therefore by (32),

[TABLE]

In other words, $2nh_{\nu}(\phi)-2hn\leq\log c^{\prime\prime}+b_{n}\log m_{\Gamma}(C_{n})+\frac{2}{e}$. But we know

[TABLE]

as $n\to\infty$ from (30), so $b_{n}\log m_{\Gamma}(C_{n})\to-\infty$ as $n\to\infty$, forcing $h_{\nu}(\phi)<h$. ∎