Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps
Marcelo Viana | Oliveira, Krerley
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.