Abstract

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.

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