1. Noncompact Phase Spaces

Problem 1.1.
[Keith Burns] What can be said about equilibrium states for the geometric potential associated to geodesic flows on finite volume, complete surfaces? More generally, manifolds? 
Problem 1.2.
[Ilya Gekhtman] Is the harmonic measure from Brownian motion in the finitevolume noncompact setting an equilibrium state? 
Problem 1.3.
[Ilya Gekhtman] Let $M$ be a finite volume, complete, negatively curved manifold. Is the Liouville measure an equilibrium state? The case when the curvature is uniformly bounded below is known, so the case of curvature is unbounded is the remaining case. 
Problem 1.4.
[Jayadev Athreya] What can one say about extreme value theory for (nonuniformly) hyperbolic flows on noncompact spaces? 
Problem 1.6.
[Dan Thompson] Give a complete picture of existence and uniqueness for HoĢlder potentials on the space of geodesics for the geodesic flow on noncompact $\operatorname{CAT}(1)$ spaces. In particular, consider potentials which depend not only the basepoint of a geodesic, but the geodesic itself. 
Problem 1.7.
[Jerome Buzzi and Todd Fisher] For geodesic flows on noncompact metric spaces, can the variational principle have an extremal measure and metric? That is, if $\mathcal M_f$ is the set of $f$invariant probability measures on $X$, $\mathcal D_X$ is the space of distances on $X$ inducing the same topology, $h_d(f) = \displaystyle \sup_{K \subset X} h_d(f,K)$ where the supremum is taken over the topological entropy of compact subsets $K$ with respect to the metric $d$, and:
\begin{eqnarray*} h_{\operatorname{Borel}}(f) & = & \sup_{\mu \in \mathcal M_f} h_\mu(f) \\ h_{\operatorname{top}}(f) & = & \inf_{d \in \mathcal D_X} h_{d}(f) \end{eqnarray*}
The Borel and topological entropies were shown to coincide in [MR1348316] for locally compact spaces. Under what conditions can one find a metric $d$ and measure $\mu$ which realize the common value?
Cite this as: AimPL: Equilibrium states for dynamical systems arising from geometry, available at http://aimpl.org/equibdynsysgeom.