Topology of an attraction domain, dynamical zeta functions and Reidemeister torsion

Abstract

In Section 1 of this paper we consider a flow with a circular chain-recurrent set and describe, with the help of the Reidemeister torsion, the connection between the topology of the attraction domain of an attractor and the dynamics of the flow on the attractor. We show in Theorem 3 that the Reidemeister torsion of a level surface of a Lyapunov function and of the attraction domain of an attractor is calculated as a special value of the twisted Lefschetz zeta function build via closed orbits in the attractor. In Section 2 we continue the study of analytical properties of the Nielsen zeta function. The Nielsen zeta function Nf (z) has a positive radius of convergence which has a sharp estimate in terms of the topological entropy of the map f [15]. In Theorem 5 of Section 2 we propose another proof of the positivity of the radius and give an exact algebraic lower estimate for the radius using the Reidemeister trace formula for the generalized Lefschetz numbers.