Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures

Leandro Cioletti, Artur O. Lopes

Published 2016-08-12Version 1

In this work we study the Ruelle Operator associated to continuous potentials on general compact spaces. We present a new criterion for uniqueness of eigenprobabilities on general compact setting. This is obtained by showing that set of eigenprobabilities for any continuous potential, coincides with the set of the DLR-Gibbs measures for a suitable quasilocal specification. In particular, we proved that the phase transition in the DLR sense is equivalent to existence of more than one eingenprobability for the dual of the Ruelle operator. We also consider bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions with respect to eigenprobabilities. We give very general sufficient conditions for the existence of integrable and almost everywhere positive eigenfunctions associated to the maximal eigenvalue. An approximation theory is presented for sequences of Ruelle operators and several continuity results are obtained, among them limits theorem for the pressure functionals. Techniques of super and sub solutions to the eigenvalue problem is developed and employed to construct (semi-explicitly) eigenfunctions to a very large class of continuous potentials having low regularity.