Characterization of hyperbolic potentials

TL;DR

This paper characterizes H"older continuous potentials for rational maps where the uniqueness and stochastic properties of equilibrium states hold for some iterate of the map, extending previous understanding of these dynamical systems.

Contribution

It provides a characterization of potentials ensuring equilibrium state properties for some iterate of a rational map, a novel extension in dynamical systems theory.

Findings

Identifies conditions on potentials for equilibrium state properties

Extends previous results to iterates of rational maps

Provides a new framework for analyzing stochastic properties

Abstract

For a rational map $f$ and a H\"older continuous potential $\phi$ satisfying $$ \sup\phi< P(f,\phi)$$ the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. For a given rational map $f$, in this paper we characterize those H\"older continuous potentials $\phi$ for which this property is satisfied for some iterate of $f$.