Synchronization points: growth, asymptotics, congruences, and the synchronization zeta function
Alexander Fel'shtyn, Mateusz Slomiany
TL;DR
This work introduces the synchronization zeta function for pairs of self-maps and develops a comprehensive analytic-dynamical framework around synchronization points. It proves a Polya–Carlson-type dichotomy: the synchronization zeta of a synchronously tame pair on suitable spaces is either rational or has a natural boundary, and provides explicit growth-rate formulas in the Abelian dual setting. The paper establishes Gauss congruences under rationality, relates the growth rate to topological entropy via specification, and analyzes asymptotic behavior of the synchronization counts. It also reveals deep connections to Reidemeister theory, showing the synchronization zeta coincides with Reidemeister zeta on duals of certain groups and interpreting Reidemeister torsion as a special zeta value, including in mapping-torus contexts. Together, these results link algebraic, geometric, and dynamical perspectives on coincidence phenomena and offer broad rationality results across Axiom A, pseudo-Anosov, and finite-set dynamics.
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.