Dynamical zeta functions of Reidemeister type

Abstract

In this paper we study dynamical representation theory zeta functions counting numbers of fixed irreducible representations for iterations of group endomorphism. The rationality and functional equation for these zeta functions are proven for several classes of groups. We prove Pólya-Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta functions for a large class of automorphisms of infinitely generated Abelian groups. We also establish the connection between the Reidemeister zeta function and dynamical representation theory zeta functions under restriction of endomorphism to a subgroup.