C*-Algebras and Topological Dynamical Systems on Nets and for Product Systems over Semigoups

Abstract

An asymptotic measure expansiveness is introduced and its relationship with dominated splitting is considered. We show recurrence and multiple recurrence results for topological dynamical systems indexed by an arbitrary directed partial semigroup with respect to acoideal basis suitable for this semigroup, but otherwise arbitrary.Extending the work of Cuntz and Vershik, we develop ageneral notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as irreversible analogues of the dynamical systems considered by Schmidt. We show a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type.