Escape of mass and entropy for geodesic flows

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 H\"older 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, in particular, to compute the entropy at infinity of the geodesic flow.