Counting closed geodesics in a compact rank one locally CAT(0) space

TL;DR

This paper establishes the asymptotic growth rate of the number of parallel classes of closed geodesics in a compact rank one locally CAT(0) space, linking it to the entropy of the geodesic flow.

Contribution

It proves a precise asymptotic formula for counting closed geodesics in a broad class of non-positively curved spaces with rank one axes.

Findings

The number of parallel classes of closed geodesics grows exponentially with rate determined by entropy.

The asymptotic count matches the exponential growth rate e^{ht}/(ht).

The result extends classical geodesic counting to non-manifold CAT(0) spaces.

Abstract

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. Assume $X$ is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length $\le t$; then $\lim\limits_{t \to \infty} P_t / \frac{e^{ht}}{ht} = 1$, where $h$ is the entropy of the geodesic flow on the space $SX$ of parametrized unit-speed geodesics in $X$.

Equations

$$\mathfrak{v}(v_{0},\varepsilon,\delta)=\pi_{p}^{-1}\Big{(}\overline{\operatorname{B_{\rho}}}(v_{0}^{-},\delta)\times\overline{\operatorname{B_{\rho}}}(v_{0}^{+},\delta)\times[0,\varepsilon]\Big{)}.$$

$${\mathfrak{w}}^{\gamma }(v_0,\epsilon ,\delta ,t)=\mathfrak{v}(v_0,\epsilon ,\delta )\cap \gamma g^{-t}\mathfrak{v}(v_0,\epsilon ,\delta ).$$

$$w$$

$$=(g^t\mathfrak{v}\cap g^{t^{\prime }}\mathfrak{v})\cap (\gamma \mathfrak{v}\cap {\gamma }^{\prime }{g}^r\mathfrak{v}).$$

$${\gamma }^{-1}w$$

$$\subset g^{-\epsilon }{\mathfrak{v}}_{2{\epsilon }_0,{\delta }_0}\cap \varphi g^{-\epsilon }{\mathfrak{v}}_{2{\epsilon }_0,{\delta }_0},$$

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

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

$${\pi }_p(v)=(v^{-},v^{+},b_{v^{-}}(v(0),p))$$

$$s(v)-s({\gamma }^{-1}{g}^{t}v)$$

$$=b_{v^{-}}(v(0),p)-\left[b_{v^{-}}(v(0),\gamma p)+t\right]$$

$$=b_{v^{-}}(\gamma p,p)-t$$

$$={\tau }_{\gamma }(v).\text{qed}$$

$${\int }_{E(\mathfrak{w})}fd\mu ={\int }_{\mathfrak{w}}\frac{f\circ E}{\ell }dm.$$

$${\int }_{E(\mathfrak{w})}f\circ \hat{\tau }d\mu ={\int }_{\mathfrak{w}}\frac{f\circ \tau }{\ell }dm.$$

$$\underset{t\to \infty }{\lim }(\psi \times {\psi }_t{)}_{\ast }{m}_{\Gamma }(C\times D)=({\psi }_{\ast }{m}_{\Gamma }\times {\psi }_{\ast }{m}_{\Gamma })(C\times D)$$

$$\underset{t\to \infty }{\lim }{\int }_{\mathfrak{w}}f(s(v),\sigma (v))dm(v)=\frac{m(\mathfrak{v}{)}^2}{{\epsilon }^2}{\int }_0^{\epsilon }{\int }_0^{\epsilon }f(x,y)dxdy.$$

$$\int {\phi }_{n}d(s\times \sigma {)}_{\ast }m\le \int fd(s\times \sigma {)}_{\ast }m\le \int {\psi }_{n}d(s\times \sigma {)}_{\ast }m$$

$$\frac{m(\mathfrak{v}{)}^2}{{\epsilon }^2}{\int }_0^{\epsilon }{\int }_0^{\epsilon }{\phi }_n\le \underset{t\to \infty }{\mathrm{liminf}}\int fd(s\times \sigma {)}_{\ast }m$$

$$\le \underset{t\to \infty }{\mathrm{limsup}}\int fd(s\times \sigma {)}_{\ast }m\le \frac{m(\mathfrak{v}{)}^2}{{\epsilon }^2}{\int }_0^{\epsilon }{\int }_0^{\epsilon }{\psi }_n.$$

$$\underset{t\to \infty }{\lim }\int fd(s\times \sigma {)}_{\ast }m=\frac{m(\mathfrak{v}{)}^2}{{\epsilon }^2}{\int }_0^{\epsilon }{\int }_0^{\epsilon }f.\text{qed}$$

$${\int }_0^{\epsilon }{\int }_0^{\epsilon }\tilde{f}(x,y)dxdy={\int }_{-\epsilon }^{\epsilon }f(z)dz.$$

$${\int }_0^{\epsilon }{\int }_0^{\epsilon }\tilde{f}(x,y)dxdy={\int }_{-\epsilon }^{\epsilon }\frac{1}{2}{\int }_{-\epsilon +|z|}^{\epsilon -|z|}\frac{f(z)}{\epsilon -\left|z\right|}dwdz={\int }_{-\epsilon }^{\epsilon }f(z)dz.\text{qed}$$

$$\underset{t\to \infty }{\lim }{\int }_{E(\mathfrak{w})}f\circ \hat{\tau }d\mu =\underset{t\to \infty }{\lim }{\int }_{\mathfrak{w}}\frac{f\circ \tau }{\ell }dm=\frac{m(\mathfrak{v}{)}^2}{{\epsilon }^2}{\int }_0^{\epsilon }{\int }_0^{\epsilon }\tilde{f}=\frac{m(\mathfrak{v}{)}^2}{{\epsilon }^2}{\int }_{-\epsilon }^{\epsilon }f.$$

$$\underset{t\to \infty }{\lim }\frac{1}{\mu (E(\mathfrak{w}))}{\int }_{E(\mathfrak{w})}\hat{\ell }d\mu =\frac{\epsilon }{2}\text{and}\underset{t\to \infty }{\lim }\frac{1}{m(\mathfrak{w})}{\int }_{\mathfrak{w}}\ell dm=\frac{2\epsilon }{3}.$$

$$e^{-2hC}\le \frac{{\int }_{U\times \gamma V}f(\xi ,\eta )d\mu (\xi ,\eta )}{{\int }_{U\times V}f(\xi ,\gamma {\eta }^{\prime })e^{-h{b}_{\gamma {\eta }^{\prime }}(p,\gamma p)}d\mu (\xi ,{\eta }^{\prime })}\le e^{2hC}.$$

$${\int }_{U\times \gamma V}f(\xi ,\eta )d\mu (\xi ,\eta )$$

$$={\int }_{U\times V}f(\xi ,\gamma {\eta }^{\prime })e^{-h{\beta }_p(\xi ,\gamma {\eta }^{\prime })}d{\mu }_p(\xi )d{\mu }_{{\gamma }^{-1}p}({\eta }^{\prime })$$

$$={\int }_{U\times V}f(\xi ,\gamma {\eta }^{\prime })e^{-h{\beta }_p(\xi ,\gamma {\eta }^{\prime })}d{\mu }_p(\xi )e^{-h{b}_{{\eta }^{\prime }}({\gamma }^{-1}p,p)}d{\mu }_p({\eta }^{\prime })$$

$$={\int }_{U\times V}f(\xi ,\gamma {\eta }^{\prime })e^{-h{b}_{\gamma {\eta }^{\prime }}(p,\gamma p)}{e}^{-h{\beta }_p(\xi ,\gamma {\eta }^{\prime })}d{\mu }_p(\xi )d{\mu }_p({\eta }^{\prime })$$

$$={\int }_{U\times V}f(\xi ,\gamma {\eta }^{\prime })e^{-h{b}_{\gamma {\eta }^{\prime }}(p,\gamma p)}{e}^{-h[{\beta }_p(\xi ,\gamma {\eta }^{\prime })-{\beta }_p(\xi ,{\eta }^{\prime })]}d\mu (\xi ,{\eta }^{\prime }).$$

$$e^{-6h\mathrm{diam}(\mathfrak{v})}\le \frac{e^{ht}{\int }_{E({\mathfrak{w}}^{\gamma })}f(\xi ,\eta )d\mu (\xi ,\eta )}{{\int }_{E(\mathfrak{v})}f(\xi ,\gamma {\eta }^{\prime })d\mu (\xi ,{\eta }^{\prime })}\le e^{6h\mathrm{diam}(\mathfrak{v})}.$$

$$e^{-4h\mathrm{diam}(\mathfrak{v})}\le \frac{e^{ht}{\int }_{{\mathfrak{v}}^{-}\times \gamma {\mathfrak{v}}^{+}}f(\xi ,\eta )d\mu (\xi ,\eta )}{{\int }_{{\mathfrak{v}}^{-}\times {\mathfrak{v}}^{+}}f(\xi ,\gamma {\eta }^{\prime })e^{h[t-b_{\gamma {\eta }^{\prime }}(p,\gamma p)]}d\mu (\xi ,{\eta }^{\prime })}\le e^{4h\mathrm{diam}(\mathfrak{v})}.$$

$$\left|b_{\gamma {\eta }^{\prime }}(p,\gamma p)-t\right|$$

$$\le \left|b_{w^{+}}(w(0),\gamma \cdot {\gamma }^{-1}{g}^{t}w(0))-t\right|+2\mathrm{diam}(\pi (\mathfrak{v}))$$

$$=2\mathrm{diam}\pi (\mathfrak{v})\le 2\mathrm{diam}(\mathfrak{v}).$$

Full text

1. Introduction

Given a locally geodesic space, it is natural to consider the number $P_{t}$ of closed geodesics of length at most $t>0$. In general, $P_{t}$ may be infinite for all $t$ above a certain threshold $T\geq 0$, but under certain geometric conditions one finds it is finite for all $t$ and can obtain asymptotic information about the growth rate of $P_{t}$.

The classic example of this situation is a theorem of Margulis [10]: If $M$ is a closed, negatively-curved Riemannian manifold, then $\lim\limits_{t\to\infty}{P_{t}}/{\frac{e^{ht}}{ht}}=1$, where $h$ is the entropy of the geodesic flow on the unit tangent bundle $SM$. Margulis also proved that the number $Q_{t}$ of geodesic arcs of length $\leq t$ starting at $x\in M$ and ending at $y\in M$, satisfies $\lim\limits_{t\to\infty}{Q_{t}}/{e^{ht}}=C$, where $C$ depends only on $x,y$.

In nonpositive curvature (instead of strictly negative curvature), there are often parallel geodesics, which can make the number $P_{t}$ as defined above infinite for large $t$. However, if one refines the definition of $P_{t}$ to be the number of parallel classes of closed geodesics of length $\leq t$, it becomes meaningful again in this case, while staying the same in the case of negative curvature. Knieper [6] proved that when $M$ is a closed, rank one nonpositively-curved Riemannian manifold, there exists $C>0$ such that $\frac{1}{C}\leq\liminf{P_{t}}/{\frac{e^{ht}}{ht}}$ and $\limsup{P_{t}}/{e^{ht}}\leq C$. Knieper later improved his bounds [7] to $\frac{1}{C}\leq\liminf{P_{t}}/{\frac{e^{ht}}{ht}}\leq\limsup{P_{t}}/{\frac{e^{ht}}{ht}}\leq C$. (This type of inequality occurs frequently enough in this paper that we will use the notation $\operatorname*{\widetilde{\lim}}$ when the inequality holds for both $\liminf$ and $\limsup$. In this notation, the last inequalities become $\frac{1}{C}\leq\operatorname*{\widetilde{\lim}}{P_{t}}/{\frac{e^{ht}}{ht}}\leq C$.) Knieper’s original bounds were recently proved by different means by Burns, Climenhaga, Fisher, and Thompson [3]. A recent preprint [9] generalizes this beyond nonpositive curvature to the case of closed Riemannian manifolds without focal points.

Another way to generalize the setting of Margulis’ theorem is to allow the spaces to admit singularities. In fact, locally $\operatorname{CAT}(-1)$ spaces are a generalization of negatively-curved manifolds which allow branching and other singularities. They are locally geodesic spaces in which all sufficiently small geodesic triangles are “thinner” than their respective comparison triangles in the hyperbolic plane $\mathbb{H}^{2}$. Roblin proved [13] that if the Bowen–Margulis measure of a proper, locally $\operatorname{CAT}(-1)$ space is finite, then $\lim\limits_{t\to\infty}{Q_{t}}/{e^{ht}}=C$, where $C$ depends only on $x,y$. A recent preprint by Link [8] generalizes this statement from $\operatorname{CAT}(-1)$ to rank one $\operatorname{CAT}(0)$. Locally $\operatorname{CAT}(0)$ spaces generalize nonpositively-curved manifolds by allowing singularities; the definition uses comparison triangles in the Euclidean plane $\mathbb{R}^{2}$ instead of $\mathbb{H}^{2}$. Roblin also proved [13] that if the Bowen–Margulis measure of a proper, locally $\operatorname{CAT}(-1)$ space $X$ is finite and mixing, and $X$ is geometrically finite, then $\lim\limits_{t\to\infty}{P_{t}}/{\frac{e^{ht}}{ht}}=1$. 111Technically, Roblin and Link do not address the question of entropy. The constant $h$ used here is actually $\delta_{\Gamma}$ the critical exponent of the Poincaré series for $\Gamma$ (see Section 5.1). At least in the case where $\Gamma$ acts cocompactly, $\delta_{\Gamma}$ equals the topological entropy $h$.

In this paper, we focus on the case of proper, rank one, locally $\operatorname{CAT}(0)$ spaces. We assume throughout the paper (with the exception of Section 3) that $\Gamma$ is a group acting freely, properly discontinuously, non-elementarily, and by isometries on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. We also assume the geodesic flow is mixing and the Bowen-Margulis measure (constructed in [12]) is finite and mixing under the geodesic flow. When $\Gamma$ acts cocompactly, it is well-known to also act non-elementarily unless $X$ is isometric to the real line; in [12] it was shown that cocompactness also implies the Bowen-Margulis measure is always finite and mixing unless $X$ is homothetic to a tree with integer edge lengths. We prove the following.

Theorem A**.**

Let $\Gamma$ be a group acting freely, geometrically (that is, properly discontinuously, cocompactly, and by isometries) on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Assume $X$ is not homothetic to a tree with integer edge lengths. Let $P_{t}$ be the number of parallel classes of oriented closed geodesics of length $\leq t$ in $\Gamma\backslash X$; then $\lim\limits_{t\to\infty}P_{t}/\frac{e^{ht}}{ht}=1$, where $h$ is the entropy of the geodesic flow on the space $SX$ of parametrized unit-speed geodesics in $X$.

We remark that if $X$ is homothetic to a tree with integer edge lengths, then the limit of $P_{t}/\frac{e^{ht}}{ht}$ does not exist. Also, the closed geodesics which bound a half flat in the universal cover (called the singular geodesics) grow at a strictly smaller exponential rate.

We note that a recent preprint [4] generalizes Knieper’s bounds $\frac{1}{C}\leq\operatorname*{\widetilde{\lim}}{P_{t}}/{\frac{e^{ht}}{ht}}\leq C$ to the proper, rank one, locally $\operatorname{CAT}(0)$ case. We prove the exact limit. We also note that an unpublished paper from 2007 by Roland Gunesch [5] claims our result for compact, rank one, nonpositely-curved manifolds. Indeed, many of the ideas in Gunesch’s work are good and inspired the current paper.

We proceed as follows in the paper. First, after establishing notation and standard facts about rank one $\operatorname{CAT}(0)$ spaces, we use Papasoglu and Swenson’s $\pi$-convergence theorem to prove a statement about local uniform expansion along unstable horospheres. Next, we construct product boxes (which behave better than standard flow boxes for measuring lengths of intersection for orbits), and use mixing to prove a result about the total measure of intersections under the flow for these product boxes. We use this to count the number of intersections coming from periodic orbits. Then we construct measures equally-weighted along periodic orbits. We adapt Knieper’s proof of an equidistribution result to prove Theorem A.

2. Preliminaries

A geodesic in a metric space $X$ is an isometric embedding of the real line $\mathbb{R}$ into $X$. A geodesic segment is an isometric embedding of a compact interval, and a geodesic ray is an isometric embedding of $[0,\infty)$.

A metric space $X$ is called uniquely geodesic if for every pair of distinct $x,y\in X$ there is a unique geodesic segment $u\colon[a,b]\to X$ such that $u(a)=x$ and $u(b)=y$. The space $X$ is geodesically complete (or, $X$ has the geodesic extension property) if every geodesic segment in $X$ extends to a full geodesic in $X$.

A CAT(0) space is a uniquely geodesic space such that for every triple of distinct points $x,y,z\in X$, the geodesic triangle is no fatter than the corresponding comparison triangle in Euclidean $\mathbb{R}^{2}$ (the triangle with the same edge lengths). A detailed account of $\operatorname{CAT}(0)$ spaces is found in [1] or [2].

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. If $X$ is proper (meaning all closed balls are compact), then both $\partial X$ and $\overline{X}=X\cup\partial X$ are compact metrizable spaces.

STANDING HYPOTHESIS: From now on, $X$ will always be a proper, geodesically complete $\operatorname{CAT}(0)$ space.

Denote by $SX$ the space of all geodesics $\mathbb{R}\to X$, where $SX$ is endowed with the compact-open topology. Then $SX$ is naturally a proper metric space, and there is a canonical footpoint projection map $\pi\colon SX\to X$ given by $\pi(v)=v(0)$; this map is proper. There is also a canonical endpoint projection map $\operatorname{E}\colon SX\to\partial X\times\partial X$ defined by $\operatorname{E}(v)=(v^{-},v^{+}):=(\lim_{t\to-\infty}v(t),\lim_{t\to+\infty}v(t))$. And $w\in SX$ is parallel to $v\in SX$ if and only if $\operatorname{E}(w)=\operatorname{E}(v)$.

The geodesic flow $g^{t}$ on $SX$ is defined by the formula $(g^{t}v)(r)=v(t+r)$.

A geodesic $v$ in $X$ is called higher rank if it can be extended to an isometric embedding of the half-flat $\mathbb{R}\times[0,\infty)\subseteq\mathbb{R}^{2}$ into $X$. A geodesic which is not higher rank is called rank one. Let $\mathcal{R}\subseteq SX$ denote the set of rank one geodesics. The following lemma describes an important aspect of the geometry of rank one geodesics in a $\operatorname{CAT}(0)$ space.

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

Let $w\colon\mathbb{R}\to X$ be a geodesic which does not bound a flat strip of width $R>0$. Then there are neighborhoods $U$ and $V$ in $\bar{X}$ of the endpoints of $w$ such that for any $\xi\in U$ and $\eta\in V$, there is a geodesic joining $\xi$ to $\eta$. For any such geodesic $v$, we have $d(v,w(0))<R$; in particular, $v$ does not bound a flat strip of width $2R$.

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 $v$.

Now let $\Gamma$ be a group acting properly discontinuously, by isometries on $X$. The $\Gamma$-action on $X$ naturally induces an action by homeomorphisms on $\overline{X}$ (and therefore on $\partial X$). The limit set of $\Gamma$ is $\operatorname{\Lambda}=\overline{\Gamma x}\cap\partial X$, for some $x\in X$. The limit set is closed and invariant, and it does not depend on choice of $x$. The action is called elementary if either $\operatorname{\Lambda}$ contains at most two points, or $\Gamma$ fixes a point in $\partial X$.

The $\Gamma$-action on $X$ also induces a properly discontinuous, isometric action on $SX$. Denote by $g_{\Gamma}^{t}$ the induced flow on the quotient $\Gamma\backslash SX$, and let $\operatorname{pr}\colon SX\to\Gamma\backslash SX$ be the canonical projection map.

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 $t>0$. If some rank one geodesic $v\in\mathcal{R}$ is an axis for $\gamma\in\operatorname{Isom}X$, we call $\gamma$ rank one. We call the $\Gamma$-action rank one if some $\gamma\in\Gamma$ is rank one.

STANDING HYPOTHESIS: $\Gamma$ is a group acting properly discontinuously, by isometries on $X$. Except in Section 3, we further assume the action is rank one, non-elementary, and free (that is, no nontrivial $\gamma\in\Gamma$ fixes a point of $x\in X$).

3. Locally Uniform Expansion along Unstable Horospheres

There is a topology on $\partial X$, finer than the visual topology, that comes from the Tits metric $\operatorname{d_{T}}$ on $\partial X$. The Tits metric is complete $\operatorname{CAT}(1)$, and measures the asymptotic angle between geodesic rays in $X$. In fact, a geodesic $v\in SX$ is rank one if and only if $\operatorname{d_{T}}(v^{-},v^{+})>\pi$. Write $\operatorname{B_{T}}(\xi,r)$ for the open Tits ball of $\operatorname{d_{T}}$-radius $r$ about $\xi$ in $\partial X$ and $\overline{\operatorname{B_{T}}}(\xi,r)$ for the closed ball.

Papasoglu and Swenson’s $\pi$-convergence:

Theorem 2** (Lemma 18 of [11]).**

Let $X$ be a proper $\operatorname{CAT}(0)$ space and $G$ a group acting by isometries on $X$. Let $x\in X$, $\theta\in[0,\pi]$, and $(g_{i})\subset G$ such that $g_{i}(x)\to p\in\partial X$ and $g_{i}^{-1}(x)\to n\in\partial X$. For any compact set $K\subset\partial X\smallsetminus\overline{\operatorname{B_{T}}}(n,\theta)$, $g_{i}(K)\to\overline{\operatorname{B_{T}}}(p,\pi-\theta),$ (in the sense that for any open $U\supset\overline{\operatorname{B_{T}}}(p,\pi-\theta),g_{i}(K)\subset U$ for all $i$ sufficiently large).

From Theorem 2 we prove that the geodesic flow expands unstable horospheres locally uniformly (Theorem 6).

Lemma 3**.**

The evaluation map $\mathrm{ev}\colon SX\times(-\infty,\infty)\to X$ given by $\mathrm{ev}(v,t)=v(t)$ extends continuously to a map $SX\times[-\infty,\infty]\to\overline{X}$.

Lemma 4**.**

Let $\Gamma$ be a group acting properly isometrically on a proper $\operatorname{CAT}(0)$ space $X$. Let $\mathfrak{v}\subset SX$ be compact. Let $\mathfrak{v}^{-}=\left\{v^{-}:v\in\mathfrak{v}\right\}$ and $\mathfrak{v}^{+}=\left\{v^{+}:v\in\mathfrak{v}\right\}$. Let $(\gamma_{i})$ be a sequence in $\Gamma$ such that $\gamma_{i}x\to\xi\in\partial X$ for some (hence any) $x\in X$ and $\mathfrak{v}\cap\gamma_{i}g^{-t_{i}}\mathfrak{v}\neq\varnothing$ for some sequence $(t_{i})$ in $[0,\infty)$. Then $\xi\in\mathfrak{v}^{+}$. Let $K\subset\partial X$ be compact such that $\operatorname{d_{T}}(\mathfrak{v}^{-},K)>\pi-c$ for some $c\in[0,\pi]$. If $U\subseteq\partial X$ is an open set such that $\overline{\operatorname{B_{T}}}(\xi,c)\subseteq U$, then $\gamma_{i}(K)\subseteq U$ for all $i$ sufficiently large.

Proof.

First observe that the sets $\pi(g^{[0,\infty]}\mathfrak{v})=\mathfrak{v}^{+}\cup\left\{v(t):v\in\mathfrak{v}\text{ and }t\geq 0\right\}$ and $\pi(g^{[-\infty,0]}\mathfrak{v})=\mathfrak{v}^{-}\cup\left\{v(t):v\in\mathfrak{v}\text{ and }t\leq 0\right\}$ are closed in $\overline{X}$ because $\mathfrak{v}$ is compact.

For each $i\in\mathbb{N}$, let $v_{i}\in\mathfrak{v}\cap\gamma_{i}g^{-t_{i}}\mathfrak{v}$. Passing to a subsequence if necessary, we may assume the sequence $(v_{i})$ converges to some $v_{0}\in\mathfrak{v}$, and $(\gamma_{i}^{-1}g^{t_{i}}v_{i})$ converges to some $w_{0}\in\mathfrak{v}$. Let $x_{0}=v_{0}(0)$ and $y_{0}=w_{0}(0)$. Recall that $\gamma_{i}y_{0}\to\xi\in\partial X$. We may assume the sequence $(\gamma_{i}^{-1}x_{0})$ converges to some $\eta\in\partial X$.

We know $d(\gamma_{i}w_{0},g^{t_{i}}v_{i})\to 0$, so $d(\gamma_{i}y_{0},v_{i}(t_{i}))\to 0$. Since $\pi(g^{[0,\infty]}\mathfrak{v})$ is closed, we may conclude $\xi=\lim v_{i}(t_{i})\in\mathfrak{v}^{+}$. Now for each $i\in\mathbb{N}$ let $w_{i}=\gamma_{i}^{-1}g^{t_{i}}v_{i}$. Then $d(\gamma_{i}^{-1}v_{0},g^{-t_{i}}w_{i})=d(\gamma_{i}^{-1}v_{0},\gamma_{i}^{-1}v_{i})\to 0$, and so $d(\gamma_{i}^{-1}x_{0},w_{i}(-t_{i}))\to 0$. Since each $w_{i}\in\mathfrak{v}$ and $\pi(g^{[-\infty,0]}\mathfrak{v})$ is closed, we see that $\eta=\lim w_{i}(-t_{i})\in\mathfrak{v}^{-}$.

Thus $\gamma_{i}x_{0}\to\xi\in\mathfrak{v}^{+}$ and $\gamma_{i}^{-1}x_{0}\to\eta\in\mathfrak{v}^{-}$. Apply Theorem 2. ∎

Theorem 5**.**

Let $X$ be a proper $\operatorname{CAT}(0)$ space and $\Gamma$ a group acting properly isometrically on $X$. Let $\mathfrak{v}\subset SX$ be compact. Let $\mathfrak{v}^{-}=\left\{v^{-}:v\in\mathfrak{v}\right\}$ and $\mathfrak{v}^{+}=\left\{v^{+}:v\in\mathfrak{v}\right\}$. Let $c\in[0,\pi]$ and let $\left\{U_{\lambda}\right\}$ be an open cover of $\mathfrak{v}^{+}$ such that for every $\xi\in\mathfrak{v}^{+}$, there is some $\lambda$ such that $\overline{\operatorname{B_{T}}}(\xi,c)\subseteq U_{\lambda}$. For any compact set $K\subset\partial X$ such that $\operatorname{d_{T}}(\mathfrak{v}^{-},K)>\pi-c$, there is some $t_{0}\geq 0$ such that for all $t\geq t_{0}$ and $\gamma\in\Gamma$, if $\mathfrak{v}\cap\gamma g^{-t}\mathfrak{v}\neq\varnothing$ then $\gamma K\subseteq U_{\lambda}$ for some $\lambda$.

Proof.

Suppose not. Then for each $i\in\mathbb{N}$ there exist $\gamma_{i}\in\Gamma$ and $t_{i}\to\infty$ such that $v_{i}\in\mathfrak{v}\cap\gamma_{i}g^{-t_{i}}\mathfrak{v}$ but $\gamma_{i}\mathfrak{v}^{+}\nsubseteq U_{\lambda}$ for all $i,\lambda$. Since $(\gamma_{i})$ escapes to infinity, we may assume $\gamma_{i}x\to\xi\in\partial X$ for some $\xi\in\partial X$ and $x\in X$. This contradicts Lemma 4. Therefore, the theorem must hold. ∎

Putting $c=0$ into Theorem 5, we obtain the following.

Corollary 6**.**

Let $X$ be a proper $\operatorname{CAT}(0)$ space and $\Gamma$ a group acting properly isometrically on $X$. Let $\mathfrak{v}\subset SX$ be compact, let $\mathfrak{v}^{-}=\left\{v^{-}:v\in\mathfrak{v}\right\}$ and $\mathfrak{v}^{+}=\left\{v^{+}:v\in\mathfrak{v}\right\}$, and let $\left\{U_{\lambda}\right\}$ be an open cover of $\mathfrak{v}^{+}$. For any compact set $K\subset\partial X$ such that $\operatorname{d_{T}}(\mathfrak{v}^{-},K)>\pi$, there is some $t_{0}\geq 0$ such that for all $t\geq t_{0}$ and $\gamma\in\Gamma$, if $\mathfrak{v}\cap\gamma g^{-t}\mathfrak{v}\neq\varnothing$ then $\gamma K\subseteq U_{\lambda}$ for some $\lambda$.

4. Quasi-Product Neighborhoods

Fix a metric $\rho$ on $\partial X$ (with the cone topology). Let $v_{0}\in\mathcal{R}$, let $p=v_{0}(0)$, and let $\varepsilon\geq 0$. For each $\delta>0$, let

[TABLE]

We may abbreviate $\mathfrak{v}(v_{0},\varepsilon,\delta)=\mathfrak{v}_{\varepsilon,\delta}=\mathfrak{v}_{\delta}=\mathfrak{v}$. Since $v_{0}\in\mathcal{R}$, by Lemma 1 we know $\mathfrak{v}_{\delta}$ is always compact for $\delta$ sufficiently small. In fact, we have the following.

Lemma 7**.**

Let $v_{0}\in\mathcal{R}$. For all $\varepsilon\geq 0$ we have $\lim\limits_{\delta\to 0}\operatorname{diam}\mathfrak{v}_{\varepsilon,\delta}\leq 2\varepsilon+\operatorname{diam}CS(v_{0})$.

Proof.

Suppose, by way of contradiction, there exist $\alpha>0$ and sequences of $\delta_{n}>0$ and $v_{n},w_{n}\in\mathfrak{v}_{\varepsilon,\delta_{n}}$ such that $\delta_{n}\to 0$ but $d(v_{n},w_{n})\geq 2\varepsilon+\operatorname{diam}CS(v_{0})+\alpha$ for all $n$. For each $n$ find $s_{n},t_{n}\in[0,\varepsilon]$ such that $g^{-s_{n}}v_{n},g^{-t_{n}}w_{n}\in\mathfrak{v}_{0,\delta}$. By the triangle inequality $d(g^{-s_{n}}v_{n},g^{-t_{n}}w_{n})\geq\operatorname{diam}CS(v_{0})+\alpha$ for all $n$. We may assume $g^{-s_{n}}v_{n}\to v$ and $g^{-t_{n}}w_{n}\to w$ for some $v,w\in\bigcap_{\delta>0}\mathfrak{v}_{0,\delta}$. Thus $v,w\in CS(v_{0})$, hence $d(v,w)\leq\operatorname{diam}CS(v_{0})$, contradicting $g^{-s_{n}}v_{n}\to v$ and $g^{-t_{n}}w_{n}\to w$. Therefore, the statement of the lemma must hold. ∎

Let $\varepsilon,\delta>0$. For each $t\in\mathbb{R}$ and $\gamma\in\Gamma$, let

[TABLE]

We may abbreviate $B_{\gamma}(v_{0},\varepsilon,\delta,t)=B_{\gamma}^{\varepsilon,\delta,t}=\mathfrak{w}^{\gamma}_{\delta,t}=\mathfrak{w}^{\gamma}$.

Lemma 8**.**

Let $v_{0}\in\mathcal{R}$ have zero width. Assume $\Gamma$ acts freely, properly discontinuously, by isometries on $X$. There exist $\varepsilon_{0}>0$ and $\delta_{0}>0$ such that for all $\varepsilon\in[0,\varepsilon_{0}]$, $\delta\in(0,\delta_{0}]$, and $t\in\mathbb{R}$, the sets $\operatorname{E}(\mathfrak{w}^{\gamma})=\operatorname{E}(\mathfrak{w}^{\gamma}_{\varepsilon,\delta,t})$ are pairwise disjoint.

Proof.

Let $p=v_{0}(0)$. Because $\Gamma$ acts freely and properly discontinuously on $X$, there is some $r_{0}>0$ such that $d(p,\gamma p)\geq r_{0}$ for all nontrivial $\gamma\in\Gamma$. Let $\varepsilon_{0}={r_{0}}/{12}$, and let $\delta_{0}>0$ be small enough that $\operatorname{diam}\mathfrak{v}_{2\varepsilon_{0},\delta_{0}}<6\varepsilon_{0}$. This implies $\mathfrak{v}_{2\varepsilon_{0},\delta_{0}}\cap\gamma\mathfrak{v}_{2\varepsilon_{0},\delta_{0}}=\varnothing$ for all $\gamma\neq\operatorname{id}$ by the triangle inequality.

Now let $\varepsilon\in[0,\varepsilon_{0}]$ and $\delta\in(0,\delta_{0}]$. Let $\gamma,\gamma^{\prime}\in\Gamma$ be such that $\operatorname{E}(\mathfrak{w}^{\gamma})\cap\operatorname{E}(\mathfrak{w}^{\gamma^{\prime}})$ is nonempty. By definition of $\mathfrak{v}$, there is exist $t^{\prime}\in\mathbb{R}$ and $w\in SX$ such that $w\in g^{t}\mathfrak{w}^{\gamma}\cap g^{t^{\prime}}\mathfrak{w}^{\gamma^{\prime}}$. Let $r=t^{\prime}-t$ and $\phi=\gamma^{-1}\gamma^{\prime}$. Then

[TABLE]

So $w\in g^{t}\mathfrak{v}\cap g^{t^{\prime}}\mathfrak{v}$, hence $\left|r\right|\leq\varepsilon$ by definition of $\mathfrak{v}$. Then also

[TABLE]

which is empty by the previous paragraph unless $\phi=\operatorname{id}$. Therefore $\gamma=\gamma^{\prime}$. ∎

Corollary 9**.**

All the $\mathfrak{w}^{\gamma}$ are disjoint.

Lemma 10**.**

Fix a zero-width geodesic $v_{0}\in SX$. Assume $\Gamma$ acts freely, properly discontinuously, by isometries on $X$. There exist $\varepsilon_{0}>0$ and $\delta_{0}>0$ such that for every $\delta\in(0,\delta_{0}]$ and $\varepsilon\in[0,\varepsilon_{0}]$, the set $\mathfrak{v}=\mathfrak{v}(v_{0},\varepsilon,\delta)$ satisfies all the following:

1. (1)

If $\varepsilon>0$ then $\mathfrak{v}$ contains an open neighborhood of $g^{{\varepsilon}/{2}}v_{0}$ in $SX$. 2. (2)

$\mathfrak{v}$* is compact.* 3. (3)

For all $v\in\mathfrak{v}$, $g^{t}v\in\mathfrak{v}$ if and only if $0\leq s(g^{t}v)\leq\varepsilon$. 4. (4)

$\operatorname{d_{T}}(\mathfrak{v}_{\delta}^{-},\mathfrak{v}_{\delta}^{+})>\pi$. 5. (5)

The sets $\operatorname{E}(\mathfrak{w}^{\gamma})=\operatorname{E}(\mathfrak{w}^{\gamma}_{\varepsilon,\delta,t})$ are pairwise disjoint for all $t\in\mathbb{R}$.

Proof.

Property (1) follows from continuity of $\pi_{p}$, (2) and (4) Lemma 1, (3) the definitions, and (5) Lemma 8. ∎

Remark*.*

Only property (5) requires $v_{0}$ zero-width and $\Gamma$ acting freely. The others require only $v_{0}$ rank-one and $\Gamma$ acting properly isometrically.

5. Mixing Calculations

5.1. Measures

We recall the measures constructed in [12]. 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 $\gamma\in\operatorname{Isom}X$.

The critical exponent $\delta_{\Gamma}=\inf\left\{s\geq 0:\sum_{\gamma\in\Gamma}e^{-sd(p,\gamma q)}<\infty\right\}$ of the Poincaré series for $\Gamma$ does not depend on choice of $p$ or $q$. We shall always assume $\delta_{\Gamma}<\infty$ (which holds whenever $\Gamma$ is finitely generated, for instance). Then 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 11**.**

A conformal density of dimension $\delta$ is a family $\left(\mu_{p}\right)_{p\in X}$ of equivalent finite Borel measures on $\partial X$, supported on $\operatorname{\Lambda}$, 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 $(v^{-},v^{+})\in\operatorname{E}(SX)$, let $\beta_{p}\colon\operatorname{E}(SX)\to\mathbb{R}$ 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 $m$, a Radon measure on $SX$ that is invariant under both $g^{t}$ and $\Gamma$, is constructed as follows: The measure $\mu\times\lambda$ on $\partial X\times\partial X\times\mathbb{R}$ ($\lambda$ is Lebesgue measure) is supported on $\operatorname{E}(\mathcal{Z})\times\mathbb{R}$, where $\mathcal{Z}\subseteq SX$ is the set of zero-width geodesics in $X$. Then the map $\pi_{p}\colon SX\to\partial X\times\partial X\times\mathbb{R}$ given by

[TABLE]

is seen to be a homeomorphism from $\mathcal{Z}$ to $\operatorname{E}(\mathcal{Z})$, hence $m=\mu\times\lambda$ may be viewed as a Borel measure on $SX$. Moreover, from [12] we have the following.

Proposition 12**.**

The zero-width geodesics are dense in $SX$.

(However, the zero-width geodesics do not in general form an open set in $SX$.)

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 by saying that whenever $A\subset SX$ is a Borel set on which $\operatorname{pr}$ is injective, $m_{\Gamma}(\operatorname{pr}A)=m(A)$.

One can adapt the methods of Knieper’s proof [6] that the Bowen-Margulis measure is the unique measure of maximal entropy to the locally $\operatorname{CAT}(0)$ case. One thus obtains the following theorem.

Theorem 13**.**

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 (up to rescaling) of maximal entropy for the geodesic flow, which has entropy $h=\delta_{\Gamma}$.

To simplify notation, we write $h:=\delta_{\Gamma}$, even if $\Gamma$ does not act cocompactly.

The $\Gamma$-action on $X$ is said to have arithmetic length spectrum if the translation lengths of axes are all contained in some discrete subgroup $c\mathbb{Z}$ of $\mathbb{R}$. In [12], we showed that when $\operatorname{\Lambda}=\partial X$, $X$ is geodesically complete, and $m_{\Gamma}$ is finite, the only examples of arithmetic length spectrum are when $X$ is a tree with integer edge lengths, up to homothety. Moreover, when the $\Gamma$-action on $X$ does not have arithmetic length spectrum, the measure $m_{\Gamma}$ is mixing under the geodesic flow $g_{\Gamma}^{t}$.

STANDING HYPOTHESIS: We assume throughout that $m_{\Gamma}$ is finite, and thus we may normalize the measure by assuming $m_{\Gamma}(\Gamma\backslash SX)=1$. We also assume non-arithmetic length spectrum, so $m_{\Gamma}$ is mixing.

5.2. Averaging

Fix a zero-width geodesic $v_{0}\in SX$, and let $p=v_{0}(0)$. Let $\varepsilon\in(0,\varepsilon_{0}]$ and $\delta\in(0,\delta_{0}]$.

Our goal in this section is to prove Corollary 26, which describes the total measure of intersections $\mathfrak{v}\cap\Gamma g^{t}(\mathfrak{v})$ for large $t$. It is easy to see by mixing that $\lim_{t\to\infty}m(\mathfrak{w})=m(\mathfrak{v})^{2}$. Less obvious, however, is that $\lim_{t\to\infty}\mu(\operatorname{E}(\mathfrak{w}))=\frac{2}{\varepsilon}m(\mathfrak{v})^{2}$.

Definition 14**.**

Define $s\colon SX\to\mathbb{R}$ by $s(v)=b_{v^{-}}(v(0),p)$, and $\tau_{\gamma}\colon SX\to\mathbb{R}$ by $\tau_{\gamma}(v)=b_{v^{-}}(\gamma p,p)-t$.

Lemma 15**.**

$\tau_{\gamma}(v)=s(v)-s(\gamma^{-1}g^{t}v)$.

Proof.

We compute

[TABLE]

Let $\mathfrak{w}=\bigcup_{\gamma\in\Gamma}\mathfrak{w}^{\gamma}$.

Define $\phi\colon\mathfrak{w}\to\Gamma$ by putting $\phi(v)$ equal to the unique $\gamma\in\Gamma$ such that $v\in\mathfrak{w}^{\gamma}$.

Define $\tau\colon\mathfrak{w}\to\mathbb{R}$ by $\tau(v)=\tau_{\phi(v)}(v)$ and $\ell\colon\mathfrak{w}\to\mathbb{R}$ by $\ell(v)=\varepsilon-\left|\tau(v)\right|$.

Lemma 16**.**

$\ell(v)$* is the length of the geodesic segment $g^{\mathbb{R}}(v)\cap\mathfrak{w}$.*

Proof.

This follows from Lemma 15, by (3) and (5) of Lemma 10. ∎

Corollary 17**.**

For all $f\in L^{1}(\mu)$,

[TABLE]

By Lemma 10 (5), the map $\phi\colon\mathfrak{w}\to\Gamma$ factors as $\phi=\hat{\phi}\circ\operatorname{E}$ for some $\hat{\phi}\colon\operatorname{E}(\mathfrak{w})\to\Gamma$. Similarly, $\tau\colon\mathfrak{w}\to\mathbb{R}$ factors as $\tau=\hat{\tau}\circ\operatorname{E}$ for some $\hat{\tau}\colon\operatorname{E}(\mathfrak{w})\to\mathbb{R}$, and $\ell=\hat{\ell}\circ\operatorname{E}$.

Corollary 18**.**

For all $f\in L^{1}(\mathbb{R})$,

[TABLE]

Define $\sigma\colon\mathfrak{w}\to\mathbb{R}$ by $\sigma(v)=s(\phi(v)^{-1}g^{t}v)$.

Lemma 19**.**

$\sigma$* is continuous.*

Proof.

The restriction of $\sigma$ to each $\mathfrak{w}^{\gamma}$ is $s\circ\gamma^{-1}\circ g^{t}$, and $\mathfrak{w}$ is the disjoint union of finitely many (closed) $\mathfrak{w}^{\gamma}$. ∎

Fact 20**.**

$\tau=s-\sigma$.

Fact 21**.**

Both $s(v),\sigma(v)\in[0,\varepsilon]$ for all $v\in\mathfrak{w}$.

Lemma 22**.**

Let $\psi\colon\Gamma\backslash SX\to\mathbb{R}$ be measurable, and let $\psi_{t}=\psi\circ g^{t}_{\Gamma}$. Then

[TABLE]

for every measurable $C\times D\subseteq\mathbb{R}^{2}$.

Proof.

By mixing, $\lim\limits_{t\to\infty}m_{\Gamma}(\psi^{-1}(C)\cap\psi_{t}^{-1}(D))=m_{\Gamma}(\psi^{-1}(C))\cdot m_{\Gamma}(\psi^{-1}(D))$. ∎

Lemma 23**.**

If $f\colon[0,\varepsilon]\times[0,\varepsilon]\to\mathbb{R}$ is Riemann integrable, then

[TABLE]

Thus $(s\times\sigma)_{*}m$ converges weakly to $\frac{m(\mathfrak{v})^{2}}{\varepsilon^{2}}$ times Lebesgue measure on $[0,\varepsilon]^{2}$.

Proof.

Since $s_{*}m$ is $\frac{m(\mathfrak{v})}{\varepsilon}$ times Lebesgue measure on $[0,\varepsilon]$, by Lemma 22 the conclusion of the theorem holds whenever $f$ is the characteristic function of a measurable product set $C\times D\subseteq[0,\varepsilon]^{2}$. This easily extends to all finite linear combinations of characteristic functions of measurable product sets.

Now if $f$ is Riemann integrable, there exist step functions $\varphi_{n}\leq f\leq\psi_{n}$ satisfying $\lim_{n}\int_{0}^{\varepsilon}\!\int_{0}^{\varepsilon}\varphi_{n}=\lim_{n}\int_{0}^{\varepsilon}\!\int_{0}^{\varepsilon}\psi_{n}=\int_{0}^{\varepsilon}\!\int_{0}^{\varepsilon}f$. Then

[TABLE]

and so letting $t\to\infty$ we obtain

[TABLE]

Letting $n\to\infty$ we find

[TABLE]

Lemma 24**.**

If $f\colon(-\varepsilon,\varepsilon)\to\mathbb{R}$ is Riemann integrable, then the function $\tilde{f}\colon(0,\varepsilon)\times(0,\varepsilon)\to\mathbb{R}$ given by $\tilde{f}(x,y)=\frac{1}{\varepsilon-\left|x-y\right|}f(x-y)$ is Riemann integrable, and

[TABLE]

Proof.

By change of variables (putting $z=x-y$ and $w=x+y$),

[TABLE]

Remark*.*

In the notation of Lemma 24, $\frac{f\circ\tau}{\ell}=\tilde{f}\circ(s\times\sigma)$.

Proposition 25**.**

Let $X$ be a proper $\operatorname{CAT}(0)$ space. Assume $\Gamma$ acts freely, properly discontinuously, and by isometries on $X$, and that $m_{\Gamma}$ is finite and mixing. If $f\colon(-\varepsilon,\varepsilon)\to\mathbb{R}$ is Riemann integrable and $\tilde{f}$ is as in Lemma 24, then

[TABLE]

Proof.

The first equality follows from Corollary 18, the last equality from Lemma 24, and the middle asymptotic from Lemma 23 because $\frac{f\circ\tau}{\ell}=\tilde{f}\circ(s\times\sigma)$. ∎

Corollary 26**.**

$\lim_{t\to\infty}\mu(\operatorname{E}(\mathfrak{w}))=\frac{2}{\varepsilon}m(\mathfrak{v})^{2}=\lim_{t\to\infty}\frac{2}{\varepsilon}m(\mathfrak{w})$.

Proof.

Putting $f=1$ in Proposition 25, we obtain $\lim_{t\to\infty}\mu(\operatorname{E}(\mathfrak{w}))=\frac{m(\mathfrak{v})^{2}}{\varepsilon^{2}}\int_{-\varepsilon}^{\varepsilon}1=\frac{2}{\varepsilon}m(\mathfrak{v})^{2}$. Putting $\tilde{f}=1$, we find $\frac{2}{\varepsilon}m(\mathfrak{v})^{2}=\lim_{t\to\infty}\frac{2}{\varepsilon}m(\mathfrak{w})$ because $\frac{f\circ\tau}{\ell}=\tilde{f}\circ(s\times\sigma)$. ∎

Remark*.*

In terms of averages, we find $\lim_{t\to\infty}\frac{1}{\mu(\operatorname{E}(\mathfrak{w}))}\int_{\operatorname{E}(\mathfrak{w})}f\circ\hat{\tau}\;d\mu=\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon}f$. In particular,

[TABLE]

6. Product Estimates

For this section, fix $v_{0}\in SX$ and $\varepsilon,\delta>0$, and let $t\in\mathbb{R}$.

Lemma 27**.**

Let $U,V\subseteq\partial X$ be Borel sets and let $\gamma\in\Gamma$. Assume $\gamma V\subseteq V$ and $\left|\beta_{p}\right|\leq C$ on $U\times V$. Then

[TABLE]

Proof.

By the properties of conformal densities and the definition of $\mu$,

[TABLE]

The conclusion of the lemma follows immediately. ∎

We will use Lemma 27 with $U\times V=\mathfrak{v}_{\delta}^{-}\times\mathfrak{v}_{\delta}^{+}$. By Lemma 5.3 of [12], $\beta_{p}$ is continuous on $\mathcal{R}^{E}$. Thus $\lim\limits_{\delta\to 0}\max\limits_{v\in\mathfrak{v}_{\delta}}\left|\beta_{p}(v)\right|=0$. However, for simplicity we will just use the bound $\max\limits_{v\in\mathfrak{v}_{\delta}}\left|\beta_{p}(v)\right|\leq 2\operatorname{diam}\pi(\mathfrak{v}_{0,\delta})\leq 2\operatorname{diam}(\mathfrak{v}_{\varepsilon,\delta})$.

Definition 28**.**

Let $I=I(v_{0},\varepsilon,\delta,t)$ be the set of nontrivial $\gamma\in\Gamma$ such that $\mathfrak{w}^{\gamma}=\mathfrak{v}\cap g^{-t}\gamma\mathfrak{v}$ is not empty. Call $\gamma\in I$ unclipped if $\gamma\mathfrak{v}^{+}\subseteq\mathfrak{v}^{+}$ and $\mathfrak{v}^{-}\subseteq\gamma\mathfrak{v}^{-}$. Equivalently, $\operatorname{E}(\mathfrak{w}^{\gamma})=\mathfrak{v}^{-}\times\gamma\mathfrak{v}^{+}$. Let $I^{\rm unclipped}$ be the set of unclipped $\gamma\in I$.

Lemma 29**.**

Let $\gamma\in\Gamma$ be unclipped. Then

[TABLE]

Proof.

By Lemma 27,

[TABLE]

If $\eta^{\prime}\in\mathfrak{v}^{+}$ then $\gamma\eta^{\prime}=w^{+}$ for some $w\in\mathfrak{w}^{\gamma}$ because $\gamma$ is unclipped. So both $w$ and $\gamma^{-1}g^{t}w$ are in $\mathfrak{v}$. Hence

[TABLE]

Therefore,

[TABLE]

Definition 30**.**

To simplify future statements, we write $C_{\varepsilon,\delta}=e^{6h\operatorname{diam}(\mathfrak{v}_{\varepsilon,\delta})}$.

Notice that for $\varepsilon>0$ fixed, $C_{\varepsilon,\delta}$ is an upper semicontinuous increasing function of $\delta$. And for $\delta>0$ fixed, $C_{\varepsilon,\delta}$ is a continuous increasing function of $\varepsilon$.

Corollary 31**.**

Let $\gamma\in I_{\varepsilon,\delta,t}^{\rm unclipped}$. Then

[TABLE]

7. Counting Unclipped Intersections

Fix a zero-width geodesic $v_{0}\in SX$. Let $N=N(v_{0},\varepsilon,\delta,t)=\#{I(v_{0},\varepsilon,\delta,t)}$ and $N^{\rm unclipped}=N^{\rm unclipped}(v_{0},\varepsilon,\delta,t)=\#{I^{\rm unclipped}}(v_{0},\varepsilon,\delta,t)$.

Lemma 32**.**

Assume $\varepsilon\in(0,\varepsilon_{0}]$ and $\delta\in(0,\delta_{0}]$. Then

[TABLE]

Proof.

Start with the identity

[TABLE]

By Corollary 31,

[TABLE]

for $\gamma$ unclipped, so

[TABLE]

8. Jiggling Near Rank One Geodesics

Clearly the inclusions $I^{\rm unclipped}_{\delta,t}\subseteq I_{\delta,t}$ and $\mathfrak{w}^{\rm unclipped}_{\delta,t}\subseteq\mathfrak{w}_{\delta,t}$ always hold. We now prove inclusions when we allow $\delta>0$ to vary.

Lemma 33**.**

Let $v_{0}\in\mathcal{R}$ and $0<r<\delta\leq\delta_{0}$. There exists $t_{0}\geq 0$ such that

[TABLE]

for all $t\geq t_{0}$ and $\varepsilon\in(0,\varepsilon_{0}]$.

Proof.

Let $\alpha=\delta-r>0$. By Corollary 6, there exists $t_{1}\geq 0$ such that for all $t\geq t_{1}$ and $\gamma\in I_{r,t}$ (i.e. $\mathfrak{v}_{r}\cap\gamma g^{-t}\mathfrak{v}_{r}\neq\varnothing$), $\gamma\mathfrak{v}_{\delta}^{+}\subseteq\operatorname{B_{\rho}}(\mathfrak{v}_{r}^{+},\alpha)=\mathfrak{v}_{\delta}^{+}$. Similarly, there exists $t_{2}\geq 0$ such that for all $t\geq t_{2}$ and $\gamma\in I_{r,t}$ (i.e. $\mathfrak{v}_{r}\cap\gamma^{-1}g^{t}\mathfrak{v}_{r}\neq\varnothing$), $\gamma^{-1}\mathfrak{v}_{\delta}^{-}\subseteq\operatorname{B_{\rho}}(\mathfrak{v}_{r}^{-},\alpha)=\mathfrak{v}_{\delta}^{-}$. So for all $t\geq t_{0}=\max\left\{t_{1},t_{2}\right\}$, if $\gamma\in I_{r,t}$ then $\gamma\in I^{unclipped}_{\delta,t}$. ∎

Corollary 34**.**

Let $v_{0}\in\mathcal{R}$ and $0<r<\delta\leq\delta_{0}$. There exists $t_{0}\geq 0$ such that

[TABLE]

for all $t\geq t_{0}$ and $\varepsilon\in(0,\varepsilon_{0}]$.

In the sequel, we shall often want to state things for both $\liminf$ and $\limsup$. The following definition makes this more convenient: Write $a\leq\operatorname*{\widetilde{\lim}}_{t\to\infty}f(t)\leq b$ if for every $\varepsilon>0$ there exists $t_{0}\in\mathbb{R}$ such that $a-\varepsilon\leq f(t)\leq b+\varepsilon$ for all $t\geq t_{0}$. In other words, $\liminf_{t\to\infty}f(t)\geq a$ and $\limsup_{t\to\infty}f(t)\leq b$.

Lemma 35**.**

Let $v_{0}\in SX$ be zero-width and $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0}]$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

Proof.

By Corollary 26, $\lim_{t\to\infty}\mu(\operatorname{E}(\mathfrak{w}_{r,t}))=\frac{2}{\varepsilon}m(\mathfrak{v}_{r})^{2}$ for all $r\in(0,\delta_{0}]$. Hence $\delta$ is a point of continuity of the function $f(r)=\lim_{t\to\infty}\mu(\operatorname{E}(\mathfrak{w}_{r,t}))$. So by Corollary 34,

[TABLE]

But now

[TABLE]

by Lemma 32 because $\mu(\operatorname{E}(\mathfrak{v}))=\frac{m(\mathfrak{v})}{\varepsilon}$. ∎

Remark*.*

The points of continuity of $r\mapsto m(\mathfrak{v}_{r})=\varepsilon\cdot\mu(\mathfrak{v}_{r}^{-}\times\mathfrak{v}_{r}^{+})$ do not depend on $\varepsilon$. Also, for such $r$ we find that $\mathfrak{v}_{r}$ is a continuity set for $m$ (that is, the topological frontier $\partial\mathfrak{v}_{r}$ of $\mathfrak{v}_{r}$ has $m(\partial\mathfrak{v}_{r})=0$); this is easy to see because the projection $SX\to\partial X\times\partial X\times\mathbb{R}$ is continuous, and therefore $\partial\mathfrak{v}_{r}\subseteq\partial\operatorname{E}(\mathfrak{v}_{r})\times\left\{0,\varepsilon\right\}$.

Lemma 36**.**

Let $v_{0}\in SX$ be zero-width and $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0})$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

Proof.

Whenever $\delta^{\prime}\in(\delta,\delta_{0}]$, we find $N^{\rm unclipped}_{\delta,t}\leq N_{\delta,t}\leq N^{\rm unclipped}_{\delta^{\prime}\!,t}\leq N_{\delta^{\prime}\!,t}$ for all $t$ sufficiently large by Lemma 33, hence

[TABLE]

satisfy $\phi(\delta)\leq\psi(\delta)\leq\phi(\delta^{\prime})\leq\psi(\delta^{\prime})$. Taking a decreasing sequence $\delta^{\prime}_{n}\to\delta$ such that each $\delta^{\prime}_{n}>\delta$ is a point of continuity of $r\mapsto m(\mathfrak{v}_{r})$, we find by Lemma 35 that

[TABLE]

and

[TABLE]

9. Counting Periodic Intersections

Definition 37**.**

Let $v_{0}\in\mathcal{R}$ and $\varepsilon,\delta>0$. Define

[TABLE]

and $N_{\varepsilon,\delta,t}^{\rm periodic}=\#{I_{\varepsilon,\delta,t}^{\rm periodic}}$.

Clearly the inclusion $I_{\varepsilon,\delta,t}^{\rm periodic}\subseteq I_{\varepsilon,\delta,t}$ always holds.

Lemma 38**.**

Let $v_{0}\in SX$ be zero-width, and let $\varepsilon\in(0,\varepsilon_{0}]$ and $\delta\in(0,\delta_{0}]$. Then $I_{\varepsilon,\delta,t}^{\rm unclipped}\subseteq I_{\varepsilon,\delta,t}^{\rm periodic}$ for all $t>0$.

Proof.

Let $\gamma\in I_{\varepsilon,\delta,t}^{\rm unclipped}$. Since $\gamma\mathfrak{v}^{+}\subseteq\mathfrak{v}^{+}$, the nested intersection $\bigcap_{n\in\mathbb{N}}\gamma^{n}\mathfrak{v}^{+}$ of compact sets must contain a point $\xi\in\partial X$. Similarly the nested intersection $\bigcap_{n\in\mathbb{N}}\gamma^{-n}\mathfrak{v}^{-}$ must contain a point $\eta\in\partial X$. Then $\xi\in\mathfrak{v}^{+}$ and $\eta\in\mathfrak{v}^{-}$ must be the endpoints of an axis for $\gamma$. Because $\operatorname{E}(\mathfrak{v})=\mathfrak{v}^{-}\times\mathfrak{v}^{+}$, $\mathfrak{v}$ contains an axis for $\gamma$. ∎

Proposition 39**.**

Let $X$ be a proper $\operatorname{CAT}(0)$ space. Assume $\Gamma$ acts freely, properly discontinuously, and by isometries on $X$, and that $m_{\Gamma}$ is finite and mixing. Let $v_{0}\in SX$ be zero-width, and let $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0})$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

Proof.

By Lemma 38, $N_{\varepsilon,\delta,t}^{\rm unclipped}\leq N_{\varepsilon,\delta,t}^{\rm periodic}\leq N_{\varepsilon,\delta,t}$ for all sufficiently large $t$, hence

[TABLE]

Now apply the bounds from Lemma 35 and Lemma 36. ∎

10. Conjugacy Classes and Intersection Segments

For this section, we assume $\Gamma$ acts freely, properly discontinuously, by isometries on $X$. A non-identity element $\gamma\in\Gamma$ is called axial if there exist $v\in SX$ and $t>0$ such that $\gamma v=g^{t}v$.

10.1. Conjugacy Classes

Let $\mathfrak{C}(\Gamma)$ be the set of axial conjugacy classes $[\gamma]$ of $\Gamma$. Call a function $\operatorname{\mathfrak{a}}\colon\mathfrak{C}(\Gamma)\to SX$ a choice of axis if every $\operatorname{\mathfrak{a}}[\gamma]$ is an axis for some $\gamma^{\prime}\in[\gamma]$. In other words, for every axial $\gamma\in\Gamma$ there exists $\phi\in\Gamma$ such that $\phi\operatorname{\mathfrak{a}}[\gamma]$ is an axis for $\gamma$.

Call a conjugacy class $[\gamma]\in\mathfrak{C}(\Gamma)$ primitive if $\gamma=\phi^{n}$ for some $\phi\in\Gamma$ and $n>1$; note this does not depend on choice of representative $\gamma$ for $[\gamma]$. Let $\mathfrak{C}^{\rm prime}(\Gamma)\subset\mathfrak{C}(\Gamma)$ be the set of conjugacy classes which are not primitive.

For any subset $U\subseteq SX$, write $\mathfrak{C}^{U}(\Gamma)\subseteq\mathfrak{C}(\Gamma)$ for the set of conjugacy classes $[\gamma]$ such that $\gamma$ has an axis in $\Gamma U$; this also does not depend on choice of representative $\gamma$ for $[\gamma]$. Also define $\mathfrak{C}^{{\rm prime},U}(\Gamma)=\mathfrak{C}^{\rm prime}(\Gamma)\cap\mathfrak{C}^{U}(\Gamma)$.

For $v\in SX$, let $\left|v\right|$ be the length of the smallest period under $g^{t}_{\Gamma}$ of the projection $\operatorname{pr}(v)\in\Gamma\backslash SX$, with $\left|v\right|=\infty$ if $\operatorname{pr}(v)$ is not periodic.

For $\gamma\in\Gamma$, let $\left|\gamma\right|$ be the translation length of $\gamma$.

For $t\geq t^{\prime}\geq 0$, let $\mathfrak{C}_{\Gamma}(t^{\prime},t)=\left\{[\gamma]\in\mathfrak{C}(\Gamma):t^{\prime}\leq\left|\gamma\right|\leq t\right\}$. Similarly define $\mathfrak{C}_{\Gamma}^{\rm prime}(t^{\prime},t)$, $\mathfrak{C}_{\Gamma}^{U}(t^{\prime},t)$, and $\mathfrak{C}_{\Gamma}^{{\rm prime},U}(t^{\prime},t)$ for $U\subseteq SX$. Let ${\rm Conj}_{\Gamma}(t^{\prime},t)=\#{\mathfrak{C}_{\Gamma}(t^{\prime},t)}$, and similarly define ${\rm Conj}_{\Gamma}^{\rm prime}(t^{\prime},t)$, ${\rm Conj}_{\Gamma}^{U}(t^{\prime},t)$, and ${\rm Conj}_{\Gamma}^{{\rm prime},U}(t^{\prime},t)$.

10.2. Intersection Segments

Let $v_{0}\in SX$, $\varepsilon\in(0,\varepsilon_{0}]$, and $\delta\in(0,\delta_{0}]$.

For every $v\in SX$, the intersection of $\Gamma\mathfrak{v}_{\varepsilon,\delta}$ with $g^{\mathbb{R}}v$ is the disjoint union of orbit segments of length $\varepsilon$. Call these segments intersection segments for $v$ with $\mathfrak{v}_{\varepsilon,\delta}$; call two segments equivalent if there is an isometry $\gamma\in\Gamma$ carrying one to the other.

Let $\operatorname{\mathfrak{S}^{\mathfrak{v}_{\varepsilon,\delta}}}(v)$ be the collection of equivalence classes of intersection segments for $v$ with $\mathfrak{v}_{\varepsilon,\delta}$, and let $\operatorname{S}^{\mathfrak{v}_{\varepsilon,\delta}}(v)=\#{\operatorname{\mathfrak{S}^{\mathfrak{v}_{\varepsilon,\delta}}}(v)}$. Notice that $\operatorname{\mathfrak{S}^{\mathfrak{v}_{\varepsilon,\delta}}}(v)$ is in natural bijection with the collection of disjoint orbit segments (length $\varepsilon$) arising as intersections of $\mathfrak{v}_{\varepsilon,\delta}$ with $\Gamma g^{\mathbb{R}}v$. Immediately we deduce the following.

Lemma 40**.**

For all $U$ satisfying $\mathfrak{v}_{\varepsilon,\delta}\subseteq U\subseteq SX$, we have

[TABLE]

Proof.

By construction of $\mathfrak{v}_{\varepsilon,\delta}$, if $v\in\mathfrak{v}_{\varepsilon,\delta}$ and $w\parallel v$ then $g^{t}w\in\mathfrak{v}_{\varepsilon,\delta}$ for some $t\in\mathbb{R}$. So $N_{\varepsilon,\delta,t}^{\rm periodic}$ is the number of $\gamma\in\Gamma$ with an axis in $\mathfrak{v}_{\varepsilon,\delta}$ such that $\left|\gamma\right|\in[t-\varepsilon,t+\varepsilon]$, while on the other hand $\mathfrak{C}_{\Gamma}^{U}(t-\varepsilon,t+\varepsilon)$ is the set of $[\gamma]\in\mathfrak{C}(\Gamma)$ such that $\gamma$ has a conjugate with an axis in $U$ and $\left|\gamma\right|\in[t-\varepsilon,t+\varepsilon]$, and $\operatorname{S}^{\mathfrak{v}_{\varepsilon,\delta}}(\operatorname{\mathfrak{a}}[\gamma])$ is the number of conjugates of $\gamma$ with an axis in $\mathfrak{v}_{\varepsilon,\delta}$. ∎

Lemma 41**.**

Let $v_{0}\in SX$ be zero-width, and let $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0})$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

Proof.

Since $\operatorname{S}^{\mathfrak{v}_{\varepsilon,\delta}}(\operatorname{\mathfrak{a}}[\gamma])\geq 1$ for all $[\gamma]\in\mathfrak{C}_{\Gamma}^{\mathfrak{v}_{\varepsilon,\delta}}$, we have ${\rm Conj}_{\Gamma}^{\mathfrak{v}_{\varepsilon,\delta}}(t-\varepsilon,t+\varepsilon)\leq N_{\varepsilon,\delta,t}^{\rm periodic}$ by Lemma 40. Apply the bounds from Proposition 39. ∎

11. Measuring along Periodic Orbits

For each $v\in SX$, let $\lambda^{v}$ be Lebesgue measure on $g^{\mathbb{R}}v$. Notice the quotient measure $\lambda_{\Gamma}^{v}$ on $\Gamma\backslash SX$ has $\left\|\lambda_{\Gamma}^{v}\right\|=\left|v\right|$.

Lemma 42**.**

Let $v_{0}\in SX$, $\varepsilon\in(0,\varepsilon_{0}]$, and $\delta\in(0,\delta_{0}]$. For all $v\in SX$, there are $\frac{1}{\varepsilon}\lambda_{\Gamma}^{v}(\operatorname{pr}\mathfrak{v}_{\varepsilon,\delta})$ equivalence classes of intersection segments for $v$ with $\mathfrak{v}_{\varepsilon,\delta}$; that is,

[TABLE]

Proof.

The intersection segments for $v$ with $\mathfrak{v}_{\varepsilon,\delta}$ are each of length $\varepsilon$, and they are pairwise disjoint. Hence $\lambda_{\Gamma}^{v}(\operatorname{pr}\mathfrak{v}_{\varepsilon,\delta})=\varepsilon\cdot\operatorname{S}^{\mathfrak{v}_{\varepsilon,\delta}}(v)$. ∎

For any $U\subseteq SX$ and $t,\alpha>0$, define

[TABLE]

Lemma 43**.**

For all $U$ satisfying $\mathfrak{v}_{\varepsilon,\delta}\subseteq U\subseteq SX$, we have

[TABLE]

Proof.

Combining Lemma 40 and Lemma 42, we obtain

[TABLE]

From Proposition 39 and Lemma 43 we obtain the following.

Corollary 44**.**

Let $v_{0}\in SX$ be zero-width, and let $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0})$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

whenever $\mathfrak{v}_{\varepsilon,\delta}\subseteq U\subseteq SX$.

The measures $\tilde{\lambda}_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm mult},U}$ and $\lambda_{\operatorname{\mathfrak{a}},t+\alpha,2\alpha}^{{\rm mult},U}$ have the same weak limits. In fact, one easily checks the following, directly from the definitions.

Lemma 45**.**

Let $U\subseteq SX$ be such that $\mathfrak{v}_{\varepsilon,\delta}\subseteq U$. For any fixed $\alpha>0$ and choice of axis $\operatorname{\mathfrak{a}}$, $\lim\limits_{t\to\infty}\left\|\tilde{\lambda}_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm mult},U}-\lambda_{\operatorname{\mathfrak{a}},t+\alpha,2\alpha}^{{\rm mult},U}\right\|=0$.

Corollary 46**.**

Let $v_{0}\in SX$ be zero-width, and let $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0})$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

Proof.

By Corollary 44 and Lemma 45,

[TABLE]

The fact that $\lambda_{\operatorname{\mathfrak{a}},t+\varepsilon,2\varepsilon}^{{\rm mult},\mathfrak{v}_{\varepsilon,\delta}}$ is a probability measure gives us the desired inequality. ∎

Combining Lemma 41 and Corollary 46, we obtain the following result.

Proposition 47**.**

Let $X$ be a proper $\operatorname{CAT}(0)$ space. Assume $\Gamma$ acts freely, properly discontinuously, and by isometries on $X$, and that $m_{\Gamma}$ is finite and mixing. Let $v_{0}\in SX$ be zero-width, and let $\varepsilon\in(0,\varepsilon_{0}]$. Let $\delta\in(0,\delta_{0})$ be a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then for every $\alpha>0$ there exists $t_{0}>0$ such that for all $t\geq t_{0}$,

[TABLE]

Lemma 48**.**

Let $U\subseteq SX$ contain a nonempty open set, and let $\alpha>0$. There exist $C>0$ and $t_{0}>0$ such that for all $t\geq t_{0}$ and

[TABLE]

Proof.

Let $V\subseteq U$ be a nonempty open set. By Proposition 12, there is some zero-width $v_{0}\in V$. By Lemma 1, there exist $\delta>0$ and $\varepsilon>0$ such that $\mathfrak{v}_{\varepsilon,\delta}=\mathfrak{v}(v_{0},\varepsilon,\delta)$ is completely contained in $\mathcal{R}\cap V$. We may assume $\varepsilon\leq\min\left\{\alpha,\varepsilon_{0}\right\}$ and that $\delta\in(0,\delta_{0}]$ is chosen such that

[TABLE]

by Corollary 46. Thus there exist $C>0$ and $t_{0}>0$ such that for all $t\geq t_{0}$,

[TABLE]

It is easy to see that Lemma 48 is equivalent to the following statement, where we replace ${\rm Conj}_{\Gamma}^{U}(t-\alpha,t+\alpha)$ by ${\rm Conj}_{\Gamma}^{U}(t-\alpha,t)$.

Corollary 49**.**

Let $U\subseteq SX$ contain a nonempty open set, and let $\alpha>0$. Then there exist $C>0$ and $t_{0}>0$ such that for all $t\geq t_{0}$ and

[TABLE]

12. Counting Multiplicities

We start with a simple upper bound on the number of conjugacy classes, coming from the construction of the Patterson–Sullivan measures.

Lemma 50**.**

If $K\subset SX$ is compact, then $\lim\limits_{t\to\infty}e^{-h^{\prime}t}{\rm Conj}_{\Gamma}^{K}(0,t)=0$ for all $h^{\prime}>h$.

Proof.

Consider that for $\gamma\in\Gamma$ with an axis in $K$, we know $d(\gamma p,p)\leq\left|\gamma\right|+2\operatorname{diam}\pi(K)$, and therefore for all $h^{\prime}>h$,

[TABLE]

converges because $h$ is the critical exponent of the Poincaré series for Patterson’s construction. It follows that $\lim_{t\to\infty}e^{-h^{\prime}t}{\rm Conj}_{\Gamma}^{K}(0,t)=0$. ∎

Lemma 51**.**

Let $U\subseteq SX$ contain a nonempty open set, and assume $U\subseteq\Gamma K$ for some compact set $K\subseteq SX$. Then for every $\alpha>0$,

[TABLE]

Remark*.*

In particular, if $\Gamma$ acts cocompactly on $X$, then $\lim\limits_{t\to\infty}\frac{{\rm Conj}_{\Gamma}^{\rm prime}(t-\alpha,t)}{{\rm Conj}_{\Gamma}(t-\alpha,t)}=1$.

Proof.

By Corollary 49 and Lemma 50, there exist $C\geq 1$ and $t_{0}>0$ such that

[TABLE]

for all $t\geq t_{0}$, Since every primitive $[\gamma]\in\mathfrak{C}_{\Gamma}^{\mathfrak{v}_{\varepsilon,\delta}}(t-\alpha,t)\smallsetminus\mathfrak{C}_{\Gamma}^{{\rm prime},\mathfrak{v}_{\varepsilon,\delta}}(t-\alpha,t)$ is a multiple of some $[\phi]\in\mathfrak{C}_{\Gamma}^{\mathfrak{v}_{\varepsilon,\delta}}(0,\frac{t}{2})$, we see that

[TABLE]

and therefore

[TABLE]

Since ${\rm Conj}_{\Gamma}^{U}(0,t)$ diverges, we obtain the following corollary.

Corollary 52**.**

Under the hypotheses of Lemma 51,

[TABLE]

It follows from Lemma 51 that the probability measures $\lambda_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm prime},\mathfrak{v}_{\varepsilon,\delta}}$ and $\lambda_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm mult},\mathfrak{v}_{\varepsilon,\delta}}$ have the same weak limits. In fact, we have the following.

Lemma 53**.**

Let $U\subseteq SX$ contain a nonempty open set, and assume $U\subseteq\Gamma K$ for some compact set $K\subseteq SX$. For any fixed $\alpha>0$ and choice of axis $\operatorname{\mathfrak{a}}$,

[TABLE]

Proof.

Let $W$ be a Borel subset of $SX$. By the definitions,

[TABLE]

The outside coefficients are asymptotically equal (and nonzero), and the difference in the sums is at most ${\rm Conj}_{\Gamma}^{U}(t-\alpha,t)-{\rm Conj}_{\Gamma}^{{\rm prime},U}(t-\alpha,t)$, which is asymptotically zero compared to ${\rm Conj}_{\Gamma}^{U}(t-\alpha,t)$ by Lemma 51. ∎

13. Limiting Process

For a fixed interval $[a,b]\subset\mathbb{R}$ and continuous function $f\colon[a,b]\to\mathbb{R}$, the Riemann sums ${\sum_{k=1}^{n}2\varepsilon_{n}f(x_{n})}$ converge to ${\int_{a}^{b}f(x)\;dx}$, for $\varepsilon_{n}=\frac{b-a}{2n}$ and $x_{n}=(2k-1)\varepsilon_{n}$. This also holds whenever $f$ is Riemann integrable, e.g.  $f$ is bounded and nondecreasing. For completeness, we give here a proof of a standard generalization of this fact to asymptotic intervals.

Lemma 54**.**

Let $F\colon\mathbb{R}\to\mathbb{R}$ be eventually positive and nondecreasing. Then

[TABLE]

where $C=\limsup\limits_{x\to\infty}\frac{F(x+\varepsilon)}{F(x)}$.

Proof.

For any fixed $a\in\mathbb{R}$ and $m\in\mathbb{Z}$,

[TABLE]

so without loss of generality we may assume $F$ is positive and nondecreasing on $[0,\infty)$. We may similarly assume, for $\alpha>0$ fixed, that $1\leq\frac{F(x+\varepsilon)}{F(x)}\leq C+\alpha$ for all $x>-2\varepsilon$. Let $t>0$ and put $n=\lfloor\frac{t}{2\varepsilon}\rfloor$. For each $k=0,1,2,\dotsc,n$, we have

[TABLE]

for all $x\in[t-(2k+2)\varepsilon,t-2k\varepsilon]$. Thus

[TABLE]

for each $k=0,1,2,\dotsc,n$, and therefore

[TABLE]

But

[TABLE]

so

[TABLE]

As $\alpha>0$ was arbitrary, we find

[TABLE]

The following is another standard calculation which we include for completeness.

Lemma 55**.**

[TABLE]

where $C=e^{h\varepsilon}$.

Proof.

It is a standard fact that for any fixed $t_{0}>0$,

[TABLE]

This comes from the calculation

[TABLE]

the second term of the last expression tends to zero relative to ${e^{ht}}/{ht}$ because it is constant, the third because \lim\limits_{x\to\infty}{\dfrac{e^{hx}}{hx^{2}}}\bigg{/}{\dfrac{e^{hx}}{x}}=0. On the other hand, for all $\varepsilon>0$, Lemma 54 gives us

[TABLE]

and therefore

[TABLE]

from (56). ∎

14. Entropy and Equidistribution

Knieper also proves an equidistribution result [6, Proposition 6.4]; adapting his proof we obtain a similar result. For clarity, we include a proof.

A significant portion of Knieper’s proof of his Proposition 6.4 is spent proving the following (unstated) general lemma.

Lemma 57**.**

Let $\phi$ be a measurable map of a measurable space to itself. Let $(\mu_{k})$ be a sequence of $\phi$-invariant probability measures, and let $\operatorname{\mathcal{A}}$ be a measurable partition. Then

[TABLE]

for all integers $q>1$ and sequences $(n_{k})$ in $\mathbb{N}$ such that $n_{k}\to\infty$.

Write $\operatorname{injrad}(\Gamma\backslash X)$ for the injectivity radius of $\Gamma\backslash X$.

Lemma 58**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Let $t_{0}>0$ and let $P\subset\mathfrak{C}_{\Gamma}(t_{0}-\alpha,t_{0})$. If $\alpha<\frac{2}{3}\operatorname{injrad}(\Gamma\backslash X)$ then $\operatorname{pr}(\operatorname{\mathfrak{a}}(P))$ is $(\lceil t_{0}\rceil,\alpha)$-separated for any choice of axis $\operatorname{\mathfrak{a}}$.

Proof.

Let $\operatorname{\mathfrak{a}}$ be a choice of axis, and let $0<\alpha<\frac{2}{3}\operatorname{injrad}(\Gamma\backslash X)$. Let $\gamma_{1},\gamma_{2}\in\Gamma$ represent distinct conjugacy classes $[\gamma_{1}],[\gamma_{2}]\in P$. Let $v=\operatorname{\mathfrak{a}}[\gamma_{1}]$ and $w=\operatorname{\mathfrak{a}}[\gamma_{2}]$, and write $\bar{v}=\operatorname{pr}v$ and $\bar{w}=\operatorname{pr}w$. We may assume, replacing $w$ by $\gamma w$ and $\gamma_{2}$ by $\gamma\gamma_{2}\gamma^{-1}$ (for some $\gamma\in\Gamma$) if necessary, that $d(\bar{v},\bar{w})=d(v,w)$.

Suppose, by way of contradiction, $d(g_{\Gamma}^{n}\bar{v},g_{\Gamma}^{n}\bar{w})\leq\alpha$ for all $n=0,1,2,\dotsc,\lceil t_{0}\rceil$. Since $\alpha<\operatorname{injrad}(\Gamma\backslash X)$, we find $d(v(n),w(n))=d(\bar{v}(n),\bar{w}(n))\leq\alpha$ for all such $n$. Thus $d(v(t),w(t))\leq\alpha$ for all $t\in[0,t_{0}]$ by convexity. Find $t_{1},t_{2}\in[t_{0}-\alpha,t_{0}]$ such that $\gamma_{1}v=g^{t_{1}}v$ and $\gamma_{2}w=g^{t_{2}}w$. Then

[TABLE]

Hence $d(\gamma_{2}^{-1}\gamma_{1}v(0),v(0))\leq 3\alpha<2\operatorname{injrad}(\Gamma\backslash X)$, which is only possible if $\gamma_{2}^{-1}\gamma_{1}$ is trivial. This contradicts our hypothesis that $[\gamma_{1}]$ and $[\gamma_{2}]$ are distinct. Therefore, there must be some $n\in\left\{0,1,2,\dotsc,\lceil t_{0}\rceil\right\}$ such that $d(g_{\Gamma}^{n}\bar{v},g_{\Gamma}^{n}\bar{w})>\alpha$, and thus we see that $\operatorname{pr}(\operatorname{\mathfrak{a}}(P))$ is $(\lceil t_{0}\rceil,\alpha)$-separated. ∎

Definition 59**.**

Let $P\subset\mathfrak{C}_{\Gamma}$ be finite. Call a $g^{t}$-invariant probability measure $\nu$ on $\Gamma\backslash SX$ equal-weighted along $\operatorname{\mathfrak{a}}(P)$ if $\nu$ gives measure $\frac{1}{\#{P}}$ to the orbit of $\operatorname{pr}(\operatorname{\mathfrak{a}}[\gamma])$ for each $[\gamma]\in P$, where $\operatorname{pr}\colon SX\to\Gamma\backslash SX$ is the canonical projection map.

Proposition 60**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Let $(\nu_{k})$ be a sequence of $g^{t}$-invariant probability measures on $\Gamma\backslash SX$. Assume there exists $\varepsilon_{0}$ such that $0<\varepsilon_{0}<\frac{2}{3}\operatorname{injrad}(\Gamma\backslash X)$ and each $\nu_{k}$ is equal-weighted along $\operatorname{\mathfrak{a}}(P_{k})$ for some choice of axis $\operatorname{\mathfrak{a}}$ and subset $P_{k}\subset\mathfrak{C}_{\Gamma}(t_{k}-\varepsilon_{0},t_{k})$, where $t_{k}\to\infty$ as $k\to\infty$. If

[TABLE]

then $\nu_{k}\to m_{\Gamma}$ weakly.

Proof.

By compactness of the space of $g^{t}$-invariant Borel probability measures on $\Gamma\backslash SX$ under the weak* topology, every subsequence $(\nu_{k_{j}})$ has at least one weak* accumulation point $\nu$ of $\left\{\nu_{k}\right\}$. By uniqueness of the measure of maximal entropy, it suffice to prove that every such $\nu$ is a measure of maximal entropy for $g_{\Gamma}^{t}$.

Let $\nu$ be a weak* accumulation point of $\left\{\nu_{k}\right\}$; passing to a subsequence if necessary, we may assume $\nu_{k}\to\nu$ in the weak* topology. Fix a measurable partition $\operatorname{\mathcal{A}}=\left\{A_{1},\dotsc,A_{m}\right\}$ of $\Gamma\backslash SX$ such that $\operatorname{diam}\operatorname{\mathcal{A}}\leq\delta<\varepsilon_{0}$ and $\nu(\partial A_{i})=0$. Let $n_{k}=\lceil t_{k}\rceil$. Since the closed geodesics in $\operatorname{pr}(\operatorname{\mathfrak{a}}(P_{k}))$ are $(n_{k},\delta)$-separated by Lemma 58, each $\alpha\in\operatorname{\mathcal{A}}_{\phi}^{(n_{k})}$ touches at most one orbit from $\operatorname{pr}(\operatorname{\mathfrak{a}}(P_{k}))$, and thus $\nu_{k}(\alpha)\leq\frac{1}{\#{P_{k}}}$. Therefore the entropy

[TABLE]

Since $\nu(\partial A_{i})=0$ for all $A_{i}\in\operatorname{\mathcal{A}}$, we have $H_{\nu_{k}}(\operatorname{\mathcal{A}}_{\phi}^{(q)})\to H_{\nu}(\operatorname{\mathcal{A}}_{\phi}^{(q)})$ and thus

[TABLE]

By Lemma 57 and the inequality $H_{\nu_{k}}(\operatorname{\mathcal{A}}_{\phi}^{(n_{k})})\geq\log\#{P_{k}}$ from above,

[TABLE]

Therefore $h_{\nu}(\phi)\geq h$, which shows that $\nu$ is a measure of maximal entropy. ∎

Corollary 61**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Then

[TABLE]

for all $\varepsilon_{0}\in(0,\frac{2}{3}\operatorname{injrad}(\Gamma\backslash SX))$. In particular,

[TABLE]

Proof.

Suppose not. Then we have $\varepsilon_{0}\in(0,\frac{2}{3}\operatorname{injrad}(\Gamma\backslash SX))$ and $t_{k}\to\infty$ such that the sets $P_{k}=\mathfrak{C}_{\Gamma}^{{\rm prime},SX\smallsetminus\mathcal{R}}(t_{k}-\varepsilon_{0},t)$ satisfy $\lim_{k\to\infty}\frac{\log\#{P_{k}}}{t_{k}}=h$. Hence by Proposition 60, $\lambda_{\operatorname{\mathfrak{a}},t_{k},\alpha}^{{\rm prime},SX\smallsetminus\mathcal{R}}\to m_{\Gamma}$ weakly. But $SX\smallsetminus\mathcal{R}$ is closed in $SX$, so $m_{\Gamma}$ must be supported on $\Gamma\backslash(SX\smallsetminus\mathcal{R})$, which contradicts the fact that $m_{\Gamma}$ is supported on $\mathcal{R}$. Therefore, the statement of the corollary must hold. ∎

Theorem 62**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Let $U\subseteq SX$ contain a nonempty open set. For any fixed $\alpha$ with $0<\alpha<\frac{2}{3}\operatorname{injrad}(\Gamma\backslash X)$ and choice of axis $\operatorname{\mathfrak{a}}$, the measures $\lambda_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm prime},U}$ converge weakly to $m_{\Gamma}$.

Proof.

Let $(t_{k})$ be a sequence of positive real numbers such that $t_{k}\to\infty$. Let $P_{k}=\mathfrak{C}_{\Gamma}(t_{k}-\alpha,t_{k})$. By Corollary 49, $\lim_{k\to\infty}\frac{\log\#{P_{k}}}{t_{k}}=h$, and thus $\lambda_{\operatorname{\mathfrak{a}},t_{k},\alpha}^{{\rm prime},U}\to m_{\Gamma}$ weakly by Proposition 60. Since $(t_{k})$ was arbitrary, it follows that the measures $\lambda_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm prime},U}$ converge weakly to $m_{\Gamma}$. ∎

15. Using Equidistribution

From Theorem 62 we obtain the following.

Lemma 63**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Let $U\subseteq SX$ contain a nonempty open set. For any fixed $\alpha$ with $0<\alpha<\frac{1}{3}\operatorname{injrad}(\Gamma\backslash X)$ and choice of axis $\operatorname{\mathfrak{a}}$, the measures $\tilde{\lambda}_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm mult},U}$ converge weakly to $m_{\Gamma}$.

Proof.

By Theorem 62, the measures $\lambda_{\operatorname{\mathfrak{a}},t+\alpha,2\alpha}^{{\rm prime},U}$ converge weakly to $m_{\Gamma}$. Then by Lemma 53 and Lemma 45, the measures $\lambda_{\operatorname{\mathfrak{a}},t+\alpha,2\alpha}^{{\rm mult},U}$ and $\tilde{\lambda}_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm mult},U}$ do likewise. ∎

Lemma 64**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Fix a zero-width geodesic $v_{0}\in SX$. Let $\varepsilon\in(0,\varepsilon_{0}]$ and $\delta\in(0,\delta_{0})$ be small enough that $\varepsilon<\frac{1}{3}\operatorname{injrad}(\Gamma\backslash X)$. Assume $\delta\in(0,\delta_{0})$ is a point of continuity of the nondecreasing function $r\mapsto m(\mathfrak{v}_{r})$. Then

[TABLE]

for all $U$ satisfying $\mathfrak{v}_{\varepsilon,\delta}\subseteq U\subseteq SX$.

Proof.

Since $\mathfrak{v}_{\varepsilon,\delta}$ is a continuity set for $m$ and $\operatorname{diam}\mathfrak{v}_{\varepsilon,\delta}<\operatorname{injrad}(\Gamma\backslash SX)$, by Lemma 63 we see that $\lim_{t\to\infty}\tilde{\lambda}_{\operatorname{\mathfrak{a}},t,\alpha}^{{\rm mult},U}(\operatorname{pr}\mathfrak{v}_{\varepsilon,\delta})=m_{\Gamma}(\operatorname{pr}\mathfrak{v}_{\varepsilon,\delta})=m(\mathfrak{v}_{\varepsilon,\delta})$. Apply Corollary 44. ∎

Putting $F(t)=e^{ht}/t$ in Lemma 54, by Lemma 64 we obtain our desired asymptotics for ${\rm Conj}_{\Gamma}^{U}(0,t)$. But to do so, we need to check the overlaps we get from counting the endpoints of closed intervals are asymptotically small.

Lemma 65**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Fix a zero-width geodesic $v_{0}\in SX$. Let $\varepsilon,\delta>0$ be small enough that $\operatorname{diam}\mathfrak{v}_{\varepsilon,\delta}<\operatorname{injrad}(\Gamma\backslash SX)$ and $\varepsilon<\frac{1}{3}\operatorname{injrad}(\Gamma\backslash X)$. Assume $\delta$ is chosen so that $\mathfrak{v}_{\varepsilon,\delta}$ is a continuity set for $m$. Then

[TABLE]

for all $U\subseteq SX$ such that $\mathfrak{v}_{\varepsilon,\delta}\subseteq U\subseteq SX$.

Proof.

By Lemma 64, for all $\alpha\in(0,\varepsilon]$ and $U$ such that $\mathfrak{v}_{\varepsilon,\delta}\subseteq U\subseteq SX$, we have

[TABLE]

and therefore

[TABLE]

Since for all $\alpha\in(0,\varepsilon)$,

[TABLE]

letting $\alpha\to\varepsilon$ from below gives us

[TABLE]

Thus by Lemma 55,

[TABLE]

where $C=e^{h\varepsilon}$. ∎

Theorem 66**.**

Let $\Gamma$ be a group acting freely geometrically on a proper, geodesically complete $\operatorname{CAT}(0)$ space $X$ with rank one axis. Let $U\subseteq SX$ contain a nonempty open set. Then

[TABLE]

Proof.

Let $v_{0}$ be a zero-width geodesic in the interior of $U$. Then $\lim_{\varepsilon,\delta\to 0}C_{\varepsilon,\delta}=1$, so the second equality holds by Lemma 65. The first equality holds by Corollary 52. ∎

In particular, this holds for $U=\mathcal{R}$ and $U=SX$:

[TABLE]

This proves Theorem A.

16. Finish

Much of the proof of Theorem 66 goes through without assuming cocompactness. In particular, what we used was equidistribution (the conclusion of Lemma 63) for the second equality and Corollary 52 for the first.