Dynamical Systems Seminar

The geodesic flow on a compact locally CAT(-1) metric space, first studied by Gromov, is a far-reaching generalization of the geodesic flow on a closed negative curvature Riemannian manifold. While one expects these flows to exhibit similar behaviour to the classical case, the lack of smooth structure has been a major obstacle to extending many of the finer aspects of the dynamical theory to this setting. Our new approach to this problem is to show that such geodesic flows are Smale flows.

A Smale flow is a topological flow equipped with a continuous bracket operation which is an abstraction of the local product structure from uniform hyperbolicity. In 1987, Pollicott showed that a version of Bowens construction of symbolic dynamics for Axiom A flows can be extended to this setting. By symbolic dynamics, we mean there exists a suspension flow over a shift of finite type which describes the original dynamics. By taking additional care in the construction, we are able to verify that the roof function can be taken to be Lipschitz in our setting. This is achieved by using carefully chosen geometric rectangles as the building blocks for the construction.

With this additional ingredient, the symbolic dynamics machine switches on and ergodic-theoretic results which are true for Axiom A flows are extended to this setting. For example, we obtain that the Bowen-Margulis measure for the geodesic flow is Bernoulli and satisfies the Central Limit Theorem. This is joint work with Dave Constantine and Jean-Francois Lafont.