Phase transitions for geodesic flows and the geometric potential

Anibal Velozo

Published 2017-04-09Version 1

In this paper we discuss the phenomenon of phase transitions for the geodesic flow on some geometrically finite negatively curved manifolds. We define a class of potentials going slowly to zero through the cusps of $X$ for which, modulo taking coverings, the pressure map $t\mapsto P(tF)$ exhibits a phase transition. By careful choice of the metric at the cusp we can show that the geometric potential (or unstable jacobian) $F^{su}$ belongs to this class of potentials (modulo an additive constant). This results in particular apply for the geodesic flow on a $M$-puncture sphere for every $M\ge 3$.