A Characterization of hyperbolic potentials of rational maps

TL;DR

This paper characterizes hyperbolic potentials of rational maps, showing that for nonuniformly hyperbolic maps, every H"older continuous potential has a unique, exponentially mixing equilibrium state.

Contribution

It provides a characterization of potentials satisfying a pressure inequality via expanding properties of equilibrium states, extending understanding of hyperbolic potentials.

Findings

Unique equilibrium states for all H"older potentials in nonuniformly hyperbolic maps

Equilibrium states are exponentially mixing

Characterization of hyperbolic potentials in terms of expanding properties

Abstract

Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a H\"older continuous potential $\phi: J(f)\rightarrow \R$ and the pressure $P(f,\phi). In the case where $\sup_{J(f)}\phi<P(f,phi)$, the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials $\phi$ for which this property is satisfied for some iterate of $f$, in terms of the expanding properties of the corresponding equilibrium states. A direct consequence of this result is that for a nonuniformly hyperbolic rational map every H\"older continuous potential has a unique equilibrium state and that this measure is exponentially mixing.