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arXiv · 1109.0646

A Characterization of hyperbolic potentials of rational maps

Irene Inoquio-Renteria·Juan Rivera-Letelier
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Abstract

Consider a rational map $f$ of degree at least 2 acting on its Julia set $J(f)$, a H\"older continuous potential $\phi: J(f)\rightarrow \R$ and the pressure $P(f,\phi). In the case where $\sup_{J(f)}\phi<P(f,phi)$, the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials $\phi$ 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.

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Irene Inoquio-Renteria, Juan Rivera-Letelier. 2011-09-03. A Characterization of hyperbolic potentials of rational maps. https://arxiv.org/abs/1109.0646

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