Ergodic Theory and Dynamical Systems

Research Article

Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps

KRERLEY OLIVEIRAa1 and MARCELO VIANAa2

a1 Instituto de Matemática, Universidade Federal de Alagoas, 57072-090 Maceio, Brazil (email: krerley@gmail.com)

a2 Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110 - Jardim Botanico, 22460-320 Rio de Janeiro, Brazil (email: viana@impa.br)

Abstract

We develop a Ruelle–Perron–Fröbenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Hölder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, which is piecewise Hölder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here.

(Received July 09 2007)

(Revised November 27 2007)

Footnotes

Dedicated to the memory of William Parry