A Characterization of hyperbolic potentials of rational maps

Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a H"older continuous potential $\varphi :J(f)\to \mathbb{R}$ and the pressure $P(f,\varphi )\mathrm{.}Inthecasewhere$\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 $\varphi$ 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.