The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences

TL;DR

This paper investigates the growth, asymptotic behavior, and zeta functions of Reidemeister and Nielsen numbers in dynamical systems, establishing new congruences and rationality results for these sequences in specific algebraic and topological contexts.

Contribution

It proves Gauss congruences and rationality of Nielsen coincidence zeta functions, and demonstrates the existence of growth rates for sequences of Nielsen numbers in certain dynamical systems.

Findings

Gauss congruences for Reidemeister coincidence numbers

Rationality of Nielsen coincidence zeta functions

Existence of growth rates for Nielsen number sequences

Abstract

In the present paper, taking a dynamical point on view, we study the growth rate and asymptotic behavior of the sequences of the Reidemeister numbers and the sequences of the Reidemeister and the Nielsen coincidence numbers. We also prove the Gauss congruences for the sequence $\{R(\varphi^n,\psi^n)\}$ of the Reidemeister coincidence numbers of the tame pair $(\varphi,\psi)$ of endomorphisms of a torsion-free nilpotent group~$G$ of finite Pr\"ufer rank. Furthermore, we prove the rationality of the Nielsen coincidence zeta function, the Gauss congruences for the sequence $\{N(f^n, g^n)\}$ of the Nielsen coincidence numbers and show that the growth rate exists for the sequence \{$N(f^n, g^n)\}$ of tame pair of maps $(f,g)$ of a compact nilmanifold to itself.