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2. Geodesic Flows on Compact Spaces


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      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 //////)


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          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?


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              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?


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                  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.