In the 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 al-
gebraic 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 ex-
plain 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.
Key words and phrases. Dynamical zeta functions, Reidemeister torsion,
Nielsen and Reidemeister numbers, fixed point classes and lifting classes,
Lefschetz zeta function, Nielsen zeta function, Reidemeister zeta function,
functional equation, Dold congruences, topological entropy , Pontryagin du-
ality, space of irreducible unitary representations, Rochlin invariant, topology
of an attraction domain.
AMS classification: Primary 58F20; Secondary 55M20, 57Q10.