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 Hölder continuous potential $ϕ: J(f)\rightarrow \R$ and the pressure $P(f,ϕ). In the case where$\sup_{J(f)}ϕ<P(f,phi)$, the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials$ϕ$for which this property is satisfied for some iterate of$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 Hölder continuous potential has a unique equilibrium state and that this measure is exponentially mixing.

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