Perturbation analysis in thermodynamics using matrix representations of Ruelle transfer operators

TL;DR

This paper investigates how small changes in potentials affect thermodynamic quantities like pressure and entropy in complex dynamical systems, using matrix representations of Ruelle transfer operators to analyze convergence and stability.

Contribution

It introduces a matrix-based framework for analyzing perturbations in thermodynamic formalism of subshifts, providing conditions for convergence of Gibbs measures and entropy.

Findings

Characterizes limit points of thermodynamic quantities under perturbations.

Provides necessary and sufficient conditions for Gibbs measure convergence.

Uses asymptotic eigenvalue expansion techniques for Ruelle transfer operators.

Abstract

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 assumed to be no 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 transfer operators. Consequently, 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 illustrate the relation between potentials and convergence of Gibbs measures by using asymptotic expansion techniques for eigenvalues of Ruelle transfer operators.