Ergodic theory of the geodesic flow and entropy at infinity

Abstract

In this thesis we study the geodesic flow on a non-compact pinched negatively curved manifold. We prove the upper semi-continuity of the entropy map and relate the escape of mass phenomenon with the topological entropy at infinity of the geodesic flow. We also study the thermodynamic formalism of the geodesic flow. We obtain a complete description of the pressure map of potentials that vanish at infinity, and construct Hölder potentials that exhibit phase transitions. We remark that phase transitions for regular potentials is a feature that can only occur in the non-compact situation. We introduce the family of strongly positive recurrent potentials and prove some important properties of such potentials. We also obtain large deviation bounds for the geodesic flow on geometrically finite manifolds and very strongly positive recurrent potentials.