Classifying dynamical systems by their recurrence properties
In the paper [Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)], \textit{H. Furstenberg} started a systematic study of transitive dynamical systems. This work served as a basis for a broad classification of disrete dynamical systems by their recurrence properties. The reviewed paper is devoted to the theory invented by Furstenberg and its connections with combinatorics, harmonic analysis and the theory of topological groups. It is a survey paper, summarizing many results on this subject, several of them obtained by the author. The author gives some comments and writes about important ideas of the theory. Only some proofs are presented, nevertheless, the references are given. The list of references contains 27 positions. Some open problems are also stated. The paper is organised as follows. The following subjects are considered: Furstenberg's theorem; the standard families; \(\mathcal F\) transitivity; disjointness and weak disjointness; the complexity function and scattering; the Weiss-Akin-Glasner theorem; almost equicontinuity, monothetic groups and a fixed-point property; scattering but not weakly mixing systems; topological mild mixing; monothetic Polish groups admitting nontrivial weakly mixing actions; various degrees of scattering; and the standard classes of transitive dynamical systems.
scientific article; zbMATH DE number 2149526