Entropy in the cusp and phase transitions for geodesic flows

Abstract

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain 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 hand on symbolic methods and Markov partitions, and on the other on geometric techniques and approximation properties at the level of groups. Keywords. KeyWords Plus:COUNTABLE MARKOV SHIFTS

NEGATIVELY CURVED MANIFOLDS

SUSPENSION FLOWS

THERMODYNAMIC FORMALISM

EQUILIBRIUM MEASURES

SYMBOLIC DYNAMICS

KLEINIAN-GROUPS

GIBBS MEASURES

MAPS

PARTITIONS