Perturbation analysis in thermodynamics using matrix representations of Ruelle operators and its application to graph IFS

摘要

We study perturbations of topological pressures, Gibbs measures and measure-theoretic entropies of these measures concerning perturbed potentials defined on topologically transitive subshift of finite type. The subshift with respect to non-perturbed system is not assumed to be topologically transitive. Therefore, the subshift of the perturbed systems and the subshift of the unperturbed system are different. We reduce this situation to a perturbation problem of certain irreducible nonnegative matrices generated by Ruelle operators. Moreover, under suitable conditions of potentials, we characterize the limit points of those thermodynamics and give a necessary and sufficient condition for convergence of Gibbs measures and the measure-theoretic entropy of this measure when the subshift of the non-perturbed system has 2 or 3 transitive components with the maximal pressure. Finally we apply our results to a problem of collapse of perturbed graph iterated function systems.