Asymptotic solution of Bowen equation for perturbed potentials on shift spaces with countable states

TL;DR

This paper analyzes the asymptotic behavior of solutions to the Bowen equation for perturbed potentials on countable shift spaces, providing conditions for their expansion and applications to Hausdorff dimension problems.

Contribution

It establishes sufficient conditions for the asymptotic expansion of the Bowen equation's solution under perturbations on countable shift spaces, extending previous results.

Findings

Derived n-order asymptotic expansions for the solution s(ε).

Identified cases where the solution's expansion order is less than that of the potentials.

Applied results to Hausdorff dimension problems in dynamical systems.

Abstract

We study the asymptotic solution of the equation of the pressure function $s\mapsto P(s\varphi(\epsilon,\cdot)+\psi(\epsilon,\cdot))$ for perturbed potentials $\varphi(\epsilon,\cdot)$ and $\psi(\epsilon,\cdot)$ defined on the shift space with countable state space. In our main result, we give a sufficient condition for the solution $s=s(\epsilon)$ of $P(s\varphi(\epsilon,\cdot)+\psi(\epsilon,\cdot))=0$ to have the $n$-order asymptotic expansion for the small parameter $\epsilon$. In addition, we also obtain the case where the order of the expansion of the solution $s=s(\epsilon)$ is less than the order of the expansion of the perturbed potentials. Our results can be applied to problems concerning asymptotic behaviors of Hausdorff dimensions obtained from Bowen formula: conformal graph directed Markov systems, an infinite graph directed systems with contractive infinitesimal similitudes mappings, and other concrete examples.