Synchronization Points and Associated Dynamical Invariants

This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces, toral automorphisms and compact abelian group epimorphisms. Synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered. Furthermore, these systems reveal strong connections between the new invariants and problems of active interest in number theory, relating to heights and greatest common divisors.