The Nielsen zeta function (Q1082690)

The author gives a sharp estimate for the radius of convergence of the Nielsen zeta-function \(N_ f\) in terms of the topological entropy of the continuous map \(f: X\to X\) where X is a compact polyhedron. It follows from this estimate that the Nielsen zeta-function has a positive radius of convergence. Furthermore it is proved that for an orientation-preserving homeomorphism of a compact surface the Nielsen zeta-function is either a rational or the radical of a rational function. For some surfaces the Nielsen zeta function is explicitly calculated. (The paper is a modification of the authors' paper [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 143, 156-161 (1985; Zbl 0582.58032)].