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Escape of mass and entropy for geodesic flows

Published online by Cambridge University Press:  28 June 2017

FELIPE RIQUELME
Affiliation:
IMA, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile email [email protected]
ANIBAL VELOZO
Affiliation:
Princeton University, Princeton, NJ 08544-1000, USA email [email protected]

Abstract

In this paper, we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure-theoretical entropy is upper semicontinuous when there is no loss of mass. In the case where mass is lost, 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 determined 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 us, in particular, to compute the entropy of the geodesic flow at infinity.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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