A Characterization of hyperbolic potentials of rational maps

2011/09/03 by Irene Inoquio-Renteria, Juan Rivera-Letelier, Inoquio-Renteria, Irene +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1109.0646

A throughout revision of the first version, incorporating a new author, a more precise title, a new section devoted to hyperbolic potentials of a general topological dynamical system and a shortened version of the main technical result (Key Lemma)

arxiv created 2011/09/03 · arxiv updated 2011/09/06

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 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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