Synchronization points: growth, asymptotics, congruences, and the synchronization zeta function

TL;DR

This paper introduces the synchronization zeta function for pairs of self-maps, explores its properties, growth rates, and asymptotics, and connects these concepts with entropy and torsion in topological dynamics.

Contribution

It provides a new framework for analyzing synchronization points via the zeta function and derives explicit formulas and congruences, extending understanding of dynamical systems on Abelian groups.

Findings

Explicit formula for synchronization growth rate

Gauss congruences for synchronization points

Asymptotic behavior under rational zeta function

Abstract

In this paper, we introduce the synchronization zeta function associated with a pair of self-maps of a topological space and investigate its properties. We also define the growth rate of synchronization points and derive an explicit formula in the setting of endomorphisms of compact, connected Abelian groups. In addition, we establish Gauss congruences and describe the asymptotic behavior for the sequence of numbers of synchronization points, under the assumption that the synchronization zeta function is rational. Further, we discuss connections with topological entropy and Reidemeister torsion.