Resonances for Graph Directed Markov Systems

Abstract

In this paper we introduce and study a certain zeta function and its zeros for conformal graph directed Markov systems (GDMS). These zeros are referred to as resonances. We specify a list of geometric, combinatoric and analytic conditions on the GDMS under which this zeta function is indeed well defined and even holomorphic on the whole complex plane. In addition, we prove that there is a half-plane where there are no zeros. Finally, we transfer a result of Guillopé et al. in \cite{GuLiZw} on the zeros of the Selberg zeta function to our setting. More precisely, we give an upper bound for the number of resonances in a strip in terms of the Hausdorff dimension of the limit set of the GDMS. We also briefly discuss relations to other zeta functions, in particular to the Selberg zeta function associated to a Kleinian group of Schottky type and to the geometric zeta function associated to a fractal string. Since the definition of the zeta function introduced in our paper is based on the transfer operator associated to the GDMS, these relations to other zeta functions indicate that our zeta function is a natural generalization of these zeta functions to conformal GDMSs.