Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem

Abstract

For the unitary dual mapping of an automorphism of a torsion-free, finite rank nilpotent group, we prove the Pólya–Carlson dichotomy between rationality and the natural boundary for the analytic behavior of its Artin–Mazur dynamical zeta function. We also establish Gauss congruences for the Reidemeister numbers of the iterations of endomorphisms of groups in this class. Our method is the twisted Burnside–Frobenius theorem proven in the paper for automorphisms of this class of groups, and a calculation of the Reidemeister numbers via a product formula and profinite completions.