A characterization of hyperbolic potentials of rational maps

Abstract

Consider a rational map f of degree at least 2 acting on its Julia set J(f), a Hölder continuous potential φ: J(f) → ℝ and the pressure P(f,φ). In the case where $$\mathop {\sup }\limits_{J(f)} \phi < P(f,\phi ),$$ the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials 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 non-uniformly hyperbolic rational map every Hölder continuous potential has a unique equilibrium state and that this measure is exponentially mixing.