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.