Entropy in the cusp and phase transitions for geodesic flows

TL;DR

This paper investigates the entropy behavior and phase transitions in geodesic flows on certain non-compact negatively curved manifolds, combining symbolic, geometric, and thermodynamic methods.

Contribution

It introduces new results on entropy loss, cusp contributions, and phase transition phenomena in the thermodynamic formalism for these flows.

Findings

Entropy contribution of the cusps computed.

Pressure function is real analytic until a phase transition.

Phase transition causes the pressure to become constant.

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 side on symbolic methods and Markov partitions and on the other on geometric techniques and approximation properties at level of groups.