Statistical properties of equilibrium states for rational maps

Abstract

Equilibrium states of rational maps for Hölder continuous potentials are not $\phi$-mixing, mainly due to the presence of critical points. Here we prove that for disks the normalized return times of arbitrary orders are, in the limit, Poisson distributed as the radius of the disks go to zero. The return times are normalized by the measure of the disks. We also show that rational maps are weakly Bernoulli with respect to the partition given by Denker and Urbanski.