
## 2. Geodesic Flows on Compact Spaces

1. #### Problem 2.1.

[Ilya Gekhtman and Vaughn Climenhaga] Let $\tilde{M}$ be contractible negatively curved Riemannian manifold, and $\Gamma \subset \operatorname{Isom}(\tilde{M})$ be a discrete group of isometries which acts cocompactly.

1. Consider a nearest-neighbor random walk on $\Gamma$, which produces an exit measure on the boundary $\partial\Gamma$. Is this measure a Gibbs state?
2. If $\tilde{M}$ is nonpositively curved, what can be said about the measures induced on the boundary by Brownian motion? Is it a Gibbs state? What else can be said about it? (In the negative curvature case, this is known by //////)
• #### Problem 2.2.

[Jayadev Athreya] Let $\varphi_t$ be geodesic flow on the unit tangent bundle of a compact surface of constant negative curvature. Let $S_n$ denote the number of simple closed geodesics with length $\le n$. Find precise growth rates for $S_n$. More generally, what can be said about variable negative curvature?
• #### Problem 2.3.

[Dan Thompson] Glue together two hyperbolic manifolds along some geodesic segment, and consider the corresponding $\operatorname{CAT}(-1)$ geodesic flow. Does the Bowen-Margulis measure satisfy exponential decay of correlations?
• #### Problem 2.4.

[Jayadev Athreya] Consider the geodesic flow on a translation surface. The Lebesgue measure is a 0-entropy measure, and the measure of maximal entropy sits on saddle connections. Are there other interesting invariant measures with intermediate entropy?

Cite this as: AimPL: Equilibrium states for dynamical systems arising from geometry, available at http://aimpl.org/equibdynsysgeom.