Symbolic codings for geodesic flows in negative curvature
Time and place
12:30 PM on Thursday, November 16th, 2017; NAC 6/111
Dave Constantine (Wesleyan University)
Abstract
The geodesic flow on a compact manifold provides a dynamical tool with which to study the geometry of the manifold. In negative curvature, the properties of this flow are particularly nice -- it has dense orbits but also many closed orbits, is ergodic and mixing for Liouville measure, and so on. However, proving more delicate properties of this flow via its geometric description can be difficult. Symbolic codings for the flow are enormously helpful in this respect, removing the dynamics from the world of Riemannian metrics, tangent vectors and Jacobi fields and recasting them in the language of shift spaces where things are far more concrete and detailed computations are possible. This idea goes back to work of Hedlund and finds its best expression in the work of Bowen, who produced Markov codings for geodesic flows in negative curvature.
In this talk I'll give an overview of the symbolic codings approach to this problem, and try to indicate its usefulness. Then I'll present some recent work with Lafont and Thompson on Markov codings for geodesic flows on CAT(-1) metric spaces. This work extends all of the good dynamical properties of the Riemannian negatively-curved setting to non-smooth metric settings and shows how a very simple but strong metric geometry property lies behind the dynamical regularity that geodesic flows exhibit.
