Perturbed infinite-state Markov systems with holes and its application

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.