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

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.