Entropy in the cusp and phase transitions for geodesic flows

Godofredo Iommi, Felipe Riquelme, Anibal Velozo

Published 2015-11-12Version 1

In this paper we study the geodesic flow for a class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. We develop and study the corresponding thermodynamic formalism. We propose a definition of pressure that satisfies the variational principle and establish conditions under which a potential has a (unique) equilibrium measure. We compute the entropy contribution of the cusps. This allow us to prove certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one side on symbolic methods and Markov partitions and on the other on approximation properties at level of groups.