Escape of mass and entropy for geodesic flows

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 Hölder continuous potentials going to zero through the cusps. We prove that the pressure map t Þ Ñ P ptF q 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.

Regularity of entropy, geodesic currents and entropy at infinity

“…See also [EKP15,ELMV12] for related works in finite volume rank one homogeneous dynamics. It follows from [RV19] in the geometrically finite case and [Vel17] more generally that this entropy at infinity coincides with our definition.…”

“…On the one hand, SPR manifolds are a very general and interesting class of manifolds, much larger than the well known and well studied class of finite volume, or even geometrically finite hyperbolic manifolds, as illustrated by Theorem 1.7. It may be an optimal class to get such result in the sense that we guess that phase transitions for the entropy can happen when the manifold is not SPR (see [ST]), analogous to those obtained by Riquelme-Velozo [RV19] for the pressure when varying a potential on geometrically finite manifolds.…”

“…We introduce a notion of entropy at infinity (see Section 7), which measures the highest possible complexity of the (topological) dynamics outside a compact set in the manifold. Note that another definition of entropy at infinity appears in [BBG14, RV19,Vel17], which is somehow the maximal entropy of a sequence of invariant probability measures diverging to infinity. See also [EKP15,ELMV12] for related works in finite volume rank one homogeneous dynamics.…”

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

“…In turn, Schapira and Tapie were motivated, in part, by work on strongly positive recurrent potentials for countable Markov shifts due to Gurevich-Savchenko [23], Sarig [56,57], Ruette [51], and Boyle-Buzzi-Gómez [8]. Other relevant precursors to our results include the work Iommi-Riquelme-Velozo [24], Riquelme-Velozo [49], and Velozo [65].…”

Pressure, Poincaré series and box dimension of the boundary

“…One of the main goals of this paper is to prove that for geometrically finite manifolds this is indeed the case. It is important to mention that under the geometrically finite assumption we have that h 8 pΓq " sup P δ P , where the supremum runs over the parabolic subgroups of Γ (see [RV,Theorem 1.3]). In other words, the entropy at infinity is determined by the critical exponent of the parabolic subgroups of Γ.…”