Dynamics and Geometry in Negatively Curved Manifolds

This is a topic course in dynamical systems in which we will study many dynamical properties of the geodesic flow in negative curvature. We will first review all the geometric background needed to study the dynamical properties. In terms of dynamics, we will study ergodicity and mixing for two standard invariant measures for the geodesic flow: the Bowen-Margulis measure and the Liouville measure.

By the end of this course, we aim to prove the following two rigidity results:

1)Measure rigidity theorem by A. Katok: In the case of a surface with negative curvature, if the Bowen-Magulis measure coincides with the Liouville measure then the metric has constant negative curvature.

2)Marked length spectrum rigidity theorem by J.P. Otal: Two negatively curved metric on a surface with the same marked length spectrum are isometric.