In this paper, we study new dynamical zeta functions connected with Nielsen fixed point theory. The study of dynamical zeta functions is part of the theory of dynamical systems, but it is also intimately related to algebraic geometry, number theory, topology and statistical mechanics. The paper consists of four parts. Part I presents a brief account of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with analog of Dold congruences for the Reidemeister and Nielsen numbers. In Part IV, we explain how dynamical zeta functions give rise to the Reidemeister torsion, a very important topological invariant which has useful applications in knots theory, quantum field theory and dynamical systems.
Introduction Nielsen fixed point theory The Reidemeister zeta function The Nielsen zeta function Reidemeister and Nielsen zeta functions modulo normal subgroup, minimal dynamical zeta functions Congruences for Reidemeister and Nielsen numbers The Reidemeister torsion.