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 $s\mapsto P(s\phi (\epsilon ,\cdot )+\xi (\epsilon ,\cdot ))$ for perturbed potentials $\phi (\epsilon ,\cdot )$ and $\xi (\epsilon ,\cdot )$ 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 $P(s\phi (\epsilon ,\cdot )+\xi (\epsilon ,\cdot ))=0$ 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 behaviours of Hausdorff dimensions given by the Bowen formula: conformal graph directed Markov systems and other concrete examples.