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

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.