for this workshop

## Equilibrium states for dynamical systems arising from geometry

at the

American Institute of Mathematics, San Jose, California

organized by

Keith Burns, Vaughn Climenhaga, Todd Fisher, and Dan Thompson

This workshop, sponsored by AIM and the NSF, will address questions of existence and uniqueness of equilibrium states, and their statistical properties, for dynamical systems arising from geometry, particularly geodesic flows.

Geodesic flows are an important class of systems, whose study mirrors the historical development of the theory of dynamical systems; many major theoretical results were obtained first for geodesic flows, before being generalized to broader classes of systems. The study of geodesic flow for a closed manifold of negative curvature is by now classical. Directions of contemporary study include weakening the hypotheses on the curvature (for example, to non-positive curvature or the assumption of no focal points), weakening the compactness assumption to allow for cusps, and weakening the assumption that the underlying space is a Riemannian manifold (we could consider e.g. a locally CAT(-1) or CAT(0) metric space). Other cases of particular interest are the geodesic flow for the Teichmuller space of quadratic differentials, and certain classes of billiard flow.

The workshop will focus on developing the dynamical theory, particularly thermodynamic formalism, in the various settings described above, building on recent breakthroughs in this area in the last few years.

The main topics for the workshop are:

- Geodesic flows on non-compact negatively curved manifolds
- Geodesic flow beyond non-positive curvature for closed manifolds
- Non-Riemannian geodesic flow and billiard flows
- Teichmuller geodesic flow

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email *workshops@aimath.org*