Escape of mass and entropy for geodesic flows

TL;DR

This paper investigates the ergodic properties of geodesic flows on negatively curved manifolds, focusing on entropy behavior, mass loss effects, and phase transitions in pressure functions.

Contribution

It establishes upper semicontinuity of measure-theoretic entropy without mass loss and links entropy failure to parabolic critical exponents, also analyzing pressure phase transitions.

Findings

Entropy is upper semicontinuous without mass loss.

Failure of entropy semicontinuity relates to maximal parabolic critical exponents.

Pressure function exhibits a phase transition, becoming constant after.

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 H\"older 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.