Preprint A531/2007
Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps

Marcelo Viana | Oliveira, Krerley

**Keywords: **
equilibrium state | non-uniform hyperbolicity | Ruelle-Perron-Frobenius operator

We develop a Ruelle-Perron-Frobenius transfer operator approach
to the ergodic theory of a large class of non-uniformly expanding
transformations on compact manifolds. For Holder continuous
potentials not too far from constant, we prove that the
transfer operator has a positive eigenfunction, piecewise Holder
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.