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

Abstract

In this paper, we study the asymptotic expansions for the zero of the pressure function $s\mapsto P(s\varphi(\varepsilon,\cdot)+\xi(\varepsilon,\cdot))$ for perturbed potentials $\varphi(\varepsilon,\cdot)$ and $\xi(\varepsilon,\cdot)$ defined on the shift space with countable state space. In our main result, we give a sufficient condition for the solution $s=s(\varepsilon)$ of $P(s\varphi(\varepsilon,\cdot)+\xi(\varepsilon,\cdot))=0$ to have the $n$-order asymptotic expansion for the small parameter $\varepsilon$. In addition, we also obtain the case where the order of the expansion of the solution $s=s(\varepsilon)$ is less than the order of the expansion of the perturbed potentials. Our results can be applied to problems concerning asymptotic behaviours of Hausdorff dimensions given by the Bowen formula: conformal graph directed Markov systems and other concrete examples.