On the Hausdorff dimension of geodesics that diverge on average

In 1984 Sullivan [S] established several groundbreaking results regarding the action of Kleinian groups on hyperbolic space. He proved that the Hausdorff dimension of the limit set of a geometrically finite group is equal to its critical exponent, and that the critical exponent of a convex cocompact group is equal to the topological entropy of the geodesic flow on the quotient manifold. These results highlight the close relationship among the Hausdorff dimension of limit sets, entropy theory and critical exponents. Since then, several generalizations of Sullivan’s results have been obtained. Bishop and Jones [BJ] proved that the Hausdorff dimension of the radial limit set of any non-elementary group coincides with its critical exponent. Shortly after, Paulin [Pau] generalized this result to Kleinian groups acting on Hadamard manifolds of any dimension. Otal and Peigné [OP] proved that the critical exponent of a non-elementary Kleinian group equals the topological entropy of the quotient manifold, where the compactness assumption is removed and the topological entropy is defined by means of the variational principle. In this article we study the relation between the Hausdorff dimension of a dynamically defined subset of the limit set, the diverging on average radial limit set, and the entropy at infinity of the geodesic flow. The entropy at infinity of a dynamical system measures the chaoticity of the system at the ends of phase space and plays a important role in the study of the ergodic theory and thermodynamic formalism of non-compact dynamical systems, for instance see [IRV, RV, ST, Ve, ITV, GST].

Consider a flow on a non-compact topological space. Given a point, we say that it is divergent if its trajectory eventually leaves any compact set, and it is said to be divergent on average if the time the trajectory spends in any compact set is asymptotically zero. The interest in the study of the set of divergent and divergent on average points, or directions in the case of the geodesic flow, is not new and has been a prominent subject of study in homogeneous dynamics and for the Teichmüller geodesic flow. For instance, Masur [Ma] proved that for a given quadratic differential $q$ on a genus $g$ surface with $n$ punctures, the Hausdorff dimension of the set of angles $\theta$, for which $e^{i\theta}q$ diverges on average for the Teichmüller geodesic flow on the corresponding Teichmuller space is bounded from above by $1/2$. This is also motivated by the fact that if the measured foliation associated to $e^{i\theta}q$ is minimal and not uniquely ergodic, then the quadratic differential is divergent [Ma, Theorem 1.1]. It was later proved by Apisa and Masur [AM] that the set of angles for which $e^{i\theta}q$ diverges on average is in fact equal to $1/2$. In the context of homogeneous dynamics, Dani [D] studied the existence and classification of divergent orbits for diagonalizable flows, establishing a correspondence between divergent orbits and singular vectors and matrices. Notably, Cheung [Ch] calculated the exact Hausdorff dimension of divergent orbits for the action of diag($e^{t},e^{t},e^{-2t}$) on $\text{SL}(3,{\mathbb{R}})/\text{SL}(3,{\mathbb{Z}})$. In the real rank one situation, Kadyrov and Pohl [KP] obtained both upper and lower bounds for the Hausdorff dimension of the set of points that diverge on average, see also [EKP]. In the higher rank case, for the action of $g_{t}=\text{diag}(e^{nt},\ldots,e^{nt},e^{-mt},\ldots,e^{-mt})$ on $\text{SL}(m+n,{\mathbb{R}})/\text{SL}(m+n,{\mathbb{Z}})$ an upper bound was obtained by Kadyrov, Kleinbock, Lindenstrauss and Margulis [KKLM]. More recently, Das, Fishman, Simmons and Urbański [DFSU] proved that the upper bound obtained in [KKLM] is also a lower bound, and therefore establishing the equality. For countable Markov shifts an upper bound for the Hausdorff dimension of the set of recurrent and diverging on average points has been obtained in [ITV].

In a joint work with G. Iommi [IRV], the authors defined the entropy at infinity of a topological flow on a non-compact manifold as the supremum of limits of measure-theoretic entropies over sequences of measures that exhibit complete lose of mass. In [IRV] we proved that the entropy at infinity of the geodesic flow on extended Schottky manifolds coincides with the maximal critical exponent among parabolic subgroups of the fundamental group. This result was later generalized to any geometrically finite manifold in [RV]. For general pinched negatively curved manifolds the entropy at infinity has been studied from the topological and measure theoretic point of view in [ST, Ve, GST]. For symbolic dynamical system, related notions were already considered by Gurevich and Zargaryan [GZ], Ruette [Ru] and Buzzi [Bu].

In order to make precise statements of our results we start by introducing some notation. For further details we refer the reader to Section 2. Let $\widetilde{\mathcal{M}}$ be a complete, simply connected, pinched negatively curved manifold, and let $\Gamma$ be a discrete, torsion free, non-elementary group of isometries of $\widetilde{\mathcal{M}}$. Consider the quotient manifold $\mathcal{M}=\widetilde{\mathcal{M}}/\Gamma$ and let $p:T^{1}\widetilde{\mathcal{M}}\to T^{1}\mathcal{M}$ be the projection map between unit tangent spaces. The geodesic flow on $\mathcal{M}$ is denoted by $(g_{t})_{t\in{\mathbb{R}}}$, where $g_{t}:T^{1}\mathcal{M}\to T^{1}\mathcal{M}$ is the time $t$-map. Let $\chi_{A}$ be the characteristic function of a subset $A$.

A vector in $T^{1}\widetilde{\mathcal{M}}$ is said to be divergent on average, divergent or recurrent depending on whether its projection to $T^{1}\mathcal{M}$ under the map $p$ has the corresponding property. Given $x$ in $\mathcal{M}$ or $\widetilde{\mathcal{M}}$, we use $\mathcal{R}(x)$ to denote the set of recurrent vectors based at $x$, and $\mathcal{DA}(x)$ to denote the set of vectors that diverge on average based at $x$. We define

Fix a reference point $z\in\widetilde{\mathcal{M}}$ and denote by $\Theta_{z}:T^{1}_{z}\widetilde{\mathcal{M}}\to\partial_{\infty}\widetilde{\mathcal{M}},$ the canonical identification that sends $v\in T^{1}_{z}\widetilde{\mathcal{M}}$ to the geodesic ray that starts at $z$ with direction $v$. Observe that $\Lambda_{\Gamma}^{rad}=\Theta_{z}(\mathcal{R}(z))$, where $\Lambda_{\Gamma}^{rad}$ is the radial limit set of $\Gamma$. Define the diverging on average radial limit set of $\Gamma$ by

Since two asymptotic rays approach exponentially fast, the sets $\Theta_{z}(\mathcal{R}(z))$ and $\Theta_{z}(\mathcal{DA}(z))$ are independent of the reference point $z$. The entropy at infinity of the geodesic flow on $\mathcal{M}$ is denoted by $\delta_{\Gamma}^{\infty}$. The main result of this article is the following.

It worth noticing that if $\mathcal{M}={\mathbb{H}}^{n}/\Gamma$ is hyperbolic, then the map $\Theta_{z}$ is bi-Lipschitz considering $T^{1}_{z}{\mathbb{H}}^{n}$ endowed with the hyperbolic metric. In particular, Theorem 1.2 can be stated as

where the Hausdorff dimension is computed with respect to the hyperbolic metric and $x\in\mathcal{M}$ is any point.

In the case that $\mathcal{M}$ is geometrically finite with parabolic, then it is known that $\delta^{\infty}_{\Gamma}=\max\{\delta_{\mathcal{P}}:{\mathcal{P}}\subset\Gamma\textrm{ parabolic}\}$, see [RV, ST]. In this case we obtain a formula for the Hausdorff dimension of the diverging on average radial limit set. In particular, if $\mathcal{M}$ is non-compact, finite volume and hyperbolic, then $\text{HD}(\Lambda_{\Gamma}^{\infty,rad})=\frac{n-1}{2}$, where $n=\dim\mathcal{M}$. More generally, if $\mathcal{M}$ is hyperbolic and geometrically finite with cusps, then the Hausdorff dimension is half the maximal rank of parabolic subgroups of $\Gamma$.

Following Ledrappier [Led] we consider the entropy of infinite, $\sigma$-finite, ergodic and conservative measures. We denote by $h(m)$ the entropy of the measure $m$. As a consequence of Theorem 1.2 we obtain the following result.

Let $M_{\infty}(T^{1}\mathcal{M},g)$ be the space of geodesic flow invariant $\sigma$-finite Borel measures on $T^{1}\mathcal{M}$ that are infinite, ergodic and conservative. Theorem 1.3 can be stated as

Establishing whether the aforementioned inequality equates to an equality is a interesting problem, this would indicate a variational principle for the entropy of infinite measures.

Consider a complete, simply connected, pinched negatively curved Riemannian manifold $\widetilde{\mathcal{M}}$ of dimension $n$. We assume that the sectional curvatures of the metric lie in the interval $[-b^{2},-1]$, for some $b\geqslant 1$. The group of isometries of $\widetilde{\mathcal{M}}$ is denoted by $\text{Iso}(\widetilde{\mathcal{M}})$ and consists of the set of diffeomorphisms that preserve the Riemannian metric. The Riemannian distance of $\widetilde{\mathcal{M}}$ will be denoted by $d$.

We say that $\phi:[0,T]\to\widetilde{\mathcal{M}}$ is a piecewise geodesic curve, where $T\in{\mathbb{R}}^{+}\cup\{+\infty\}$, if it is a continuous map that parametrizes a collection of countably many geodesic segments. In this article we are mostly interested in the case where $T=+\infty$. In this case, there exists an increasing sequence of real numbers $(t_{n})_{n}$ converging to $+\infty$, where $t_{0}=0$ and such that $\phi|_{[t_{n-1},t_{n}]}$ is a geodesic segment parametrized by arc-length for every $n\in{\mathbb{N}}$. By definition, the distance between $\phi(t_{n-1})$ and $\phi(t_{n})$ is given by $\ell_{n}=t_{n}-t_{n-1}$. The interior angle at $\phi(t_{n})$ is denoted by $\theta_{n}$, where an interior angle is by definition in $[0,\pi]$. Using this notation, we say that $\phi$ has lengths $(\ell_{n})_{n}$ and angles $(\theta_{n})_{n}$.

Let $\alpha\in(0,\pi)$ be fixed. In the proofs we will need to shadow a piecewise geodesic curve with angles at least $\alpha$ and sufficiently large lenghts by a geodesic ray. To do so, we first need to establish that such a curve is a local quasi-geodesic. This is fairly standard but we provide details for completeness. Let $x,y,z\in\widetilde{\mathcal{M}}$ be points such that the interior angle $\theta$ between $[x,y]$ and $[y,z]$ is at least $\alpha$. The manifold $\widetilde{\mathcal{M}}$ is a $\text{CAT}(-1)$ space, since the sectional curvature is bounded above by $-1$. Denote by $d_{\mathbb{H}^{2}}$ the hyperbolic metric in $\mathbb{H}^{2}$. Let $\bar{x},\bar{y},\bar{z}\in\mathbb{H}^{2}$ be points such that $d(x,y)=d_{\mathbb{H}^{2}}(\bar{x},\bar{y})=:b$, $d(y,z)=d_{\mathbb{H}^{2}}(\bar{y},\bar{z})=:c$, and such that the angle at $\bar{y}$ of the geodesic triangle determined by $\bar{x},\bar{y},\bar{z}$ is equal to $\theta$. Then $d(x,z)\geq d_{\mathbb{H}^{2}}(\bar{x},\bar{z})=:a$ (see [BH, Proposition 1.7, Chapter 2]). Moreover, using the hyperbolic law of cosine and that $\theta\geq\alpha$, we get

Observe that the last term is at least $\sin^{2}(\alpha/2)\cosh(b+c)$, therefore

where the last inequality holds provided that $b+c\geq L_{0}$, for $L_{0}=L_{0}(\alpha)$ large enough. Set $\lambda_{0}(\alpha)=1/\sin^{2}(\alpha/2)$. Therefore,

Let us now consider a piecewise geodesic curve $\phi:[0,\infty]\to\widetilde{\mathcal{M}}$ with lengths $(\ell_{n})_{n}$ and angles $(\theta_{n})_{n}$, where $\theta_{n}\geqslant\alpha$ and $\ell_{n}\geqslant L_{0}(\alpha)$, for every $n\in{\mathbb{N}}$. We claim that $\phi$ is a $(\lambda_{0}(\alpha),L_{0}(\alpha))$-local-quasi-geodesic. Note that if $[c,d]\subseteq[0,\infty]$ and

then either $\phi([c,d])=[\phi(c),\phi(d)]$, or there exists $e\in(c,d)$ such that $\phi([c,d])=[\phi(c),\phi(e)]\cup[\phi(e),\phi(d)].$ In the first case the condition in Definition 3.2 is clearly satisfied; in the later case we can use inequality (2) to conclude the claim.

Following [CDP, Theorem 1.4, Chapter 3] a $(\lambda,L)$-local-quasi-geodesic is a $\lambda^{\prime}$-quasi-geodesic, where $\lambda^{\prime}\geq 1$ depends only on the constants $\lambda,L,\varrho$. Here $\varrho>0$ denotes a positive constant for which $\widetilde{\mathcal{M}}$ is a $\varrho$-hyperbolic space in the sense of Gromov. In [CDP, Theorem 3.1, Chapter 3] it is proved the stability of quasi-geodesics having infinite length. More precisely, they proved that a $\lambda$-quasi-geodesic is contained in a $D$-neighbourhood of a geodesic line, where $D$ depends only on $\lambda$ and $\varrho$. Combining these results and the discussion above we conclude the following result:

The following proposition summarizes relevant properties of piecewise geodesic curves having interior angles bounded from below away from 0. This proposition will be useful at the end of Section 4.

In this section we will prove that the Hausdorff dimension of the diverging on average radial limit set is equal to $\delta_{\Gamma}^{\infty}.$ We separate the proof into two parts: we first prove that $\delta_{\Gamma}^{\infty}$ is an upper bound for the Hausdorff dimension and then we prove it is a lower bound.

Fix a reference point $o\in\widetilde{\mathcal{M}}$. Let $(K_{n})_{n\in{\mathbb{N}}}$ be an increasing sequence of compact pathwise connected sets with piecewise $C^{1}$ boundaries such that $\mathcal{M}=\bigcup_{n}K_{n}$. Let $\widetilde{K}_{n}$ be a nice pre-image of $K_{n}$ containing $o$. We assume that $\widetilde{K}_{n}\subset\widetilde{K}_{n+1}$, for every $n\in{\mathbb{N}}$, and that $\delta_{\Gamma}^{\infty}=\lim_{n\to\infty}\delta_{K_{n}^{c}}$. For $\xi\in\partial_{\infty}\widetilde{\mathcal{M}}$, let $v_{\xi}\in T^{1}\widetilde{\mathcal{M}}$ be the unit vector based at $o$ pointing towards $\xi$. Denote by $\pi:T^{1}\widetilde{\mathcal{M}}\to\widetilde{\mathcal{M}}$ the canonical projection. Observe that