Perturbed infinite-state Markov systems with holes and its application

TL;DR

This paper studies how Gibbs measures for infinite-state Markov systems with holes behave under small perturbations, extending previous finite-state results and applying spectral analysis techniques.

Contribution

It develops a spectral gap approach for infinite-state systems and analyzes measure convergence and splitting behavior under perturbations.

Findings

Convergence of Gibbs measures as perturbation parameter tends to zero.

Representation of limiting measures in perturbed infinite-state systems.

Splitting behavior of Gibbs measures in systems with holes.

Abstract

We consider a perturbed system $(X,\varphi(\epsilon,\cdot))$, where $X$ is a topological Markov shift with a countably infinite state space, and $\varphi(\epsilon,\cdot)$ is a real-valued potential on X depending on a small parameter $\epsilon\in (0,1)$. We assume that for each $\epsilon>0$, the system has a unique transitive component and a unique Gibbs measure (or more generally, a Ruelle-Perron-Frobenius (RPF) measure) $\mu_{\epsilon}$, while the unperturbed system possesses multiple transitive components and Gibbs measures on these components. We investigate the convergence of the measure $\mu_{\epsilon}$ as $\epsilon\to 0$ and the representation of the limiting measure, if it exists. In previous work [T. 2020], we considered the finite state case. Our approach relies on a development of the Schur-Frobenius factorization theorem, which we apply to demonstrate a spectral gap property for Perron complements of Ruelle operators in the infinite-state case. As an application, we examine perturbed piecewise expanding Markov maps with holes, defined over countably infinite partitions. We investigate the splitting behavior of the associated Gibbs measure under perturbation.