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 $\varphi :J(f)\to \mathbb{R}$ and the pressure $P(f,\varphi ).Inthecasewhere$\sup_{J(f)}ϕ<P(f,phi)$,theuniquenessandstochasticpropertiesofthecorrespondingequilibriumstateshavebeenextensivelystudied.Inthispaperwecharacterizethosepotentials$ϕ$forwhichthispropertyissatisfiedforsomeiterateof$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.

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)

References in corpus (2)

• Nice inducing schemes and the thermodynamics of rational maps
• Measures with positive Lyapunov exponent and conformal measures in rational dynamics

Cited by in corpus (1)

• High-order phase transitions in the quadratic family