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

Published online by Cambridge University Press:  01 April 2008

KRERLEY OLIVEIRA  and
MARCELO VIANA
KRERLEY OLIVEIRA
Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, 57072-090 Maceio, Brazil (email: [email protected])
MARCELO VIANA
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110 - Jardim Botanico, 22460-320 Rio de Janeiro, Brazil (email: [email protected])
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 501 - 533
Copyright
Copyright © Cambridge University Press 2008

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