THE MAXIMAL ENTROPY MEASURE DETECTS NON-UNIFORM HYPERBOLICITY

Abstract

We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure.

With the same tools we give an extension of the result of Carleson, Jones and Yoccoz that semi-hyperbolicity characterizes those polynomial maps whose basin of attraction of infinity is a John domain, to rational maps having a completely invariant attracting basin.