A characterization of hyperbolic potentials of rational maps

作者：Inoquio Renteria Irene^*; Rivera Letelier Juan

来源：Bulletin of the Brazilian Mathematical Society, 2012, 43(1): 99-127.

DOI：10.1007/s00574-012-0007-1

• Ergodic theory
• thermodynamic formalism
• complex dynamical systems
• non-uniformly hyperbolic systems

Consider a rational map f of degree at least 2 acting on its Julia set J(f), a Holder continuous potential phi : J(f) -%26gt; R and the pressure P(f, phi). In the case where %26lt;br%26gt;sup phi %26lt; P(f, phi), %26lt;br%26gt;J(f) %26lt;br%26gt;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 off, 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 Holder continuous potential has a unique equilibrium state and that this measure is exponentially mixing.

• 出版日期2012-3