Escape of mass and entropy for geodesic flows

Abstract

In this paper we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure theoretic entropy is upper semicontinuous when there is no loss of mass. In case we are losing mass, the critical exponents of parabolic subgroups of the fundamental group have a significant meaning. More precisely, the failure of upper-semicontinuity of the entropy is determinated by the maximal parabolic critical exponent. We also study the pressure of positive Hölder continuous potentials going to zero through the cusps. We prove that the pressure map $t\mapsto P(tF)$ is differentiable until it undergoes a phase transition, after which it becomes constant. This description allows, in particular, to compute the entropy at infinity of the geodesic flow.

References in corpus (1)

• Phase transitions for geodesic flows and the geometric potential

Cited by in corpus (5)

• Continuity properties of Lyapunov exponents for surface diffeomorphisms
• Escape of entropy for countable Markov shifts
• Pressure metrics for deformation spaces of quasifuchsian groups with parabolics
• Pressure at infinity on countable Markov shifts
• Pressure, Poincaré series and box dimension of the boundary