Eli Glasner (Tel-Aviv University)

Classifying dynamical systems by their recurrence properties

Abstract. In his seminal paper of 1967 on disjointness in topological dynamics and ergodic theory H. Furstenberg has shown that a dynamical system (X,T) is weakly mixing iff the collection ${\mathcal{F}}=\{N(U,V):U,V\subset X\ {\text{are non-empty open subsets}}\}$ is a filter base (here N(U,V) is the set of ``times" n such that $T^nU\cap V\ne\emptyset$). In recent years this fact served as a basis for a broad and detailed classification of topologically transitive dynamical systems by their recurrence properties. I will describe some aspects of this new and exciting theory and its connections with combinatorics, harmonic analysis and the theory of topological groups. Works by Glasner & Weiss (1993), Blanchard, Host & Maass (2000), Weiss (2000), Akin & Glasner (2001) and Huang & Ye (2002) will be reviewed.