Classifying dynamical systems by their recurrence properties
type: Item
instance of: scholarly article
title: Classifying dynamical systems by their recurrence properties
zbMATH Open document ID: 1072.37017
DOI: 10.12775/TMNA.2004.018
published in: Topological Methods in Nonlinear Analysis
publication date: 2005-03-30T00:00:00Z
review text: 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.
reviewed by: Krzysztof Ciesielski
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37B05
zbMATH DE Number: 2149526
zbMATH Keywords: recurrence
zbMATH Keywords: transitivity
zbMATH Keywords: mixing
zbMATH Keywords: topological ergodicity
author: Eli Glasner
MaRDI profile type: Publication
full work available at URL: tmna.2004.018
OpenAlex ID: W34711698
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description: scientific article; zbMATH DE number 2149526