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Xie, Shaoting; Yin, Jiandong

Equicontinuity and sensitivity of group actions. (English) Zbl 07762268

Chin. Ann. Math., Ser. B 44, No. 4, 501-516 (2023).
Summary: Let \((X, G)\) be a dynamical system \((G\)-system for short), that is, \(X\) is a topological space and \(G\) is an infinite topological group continuously acting on \(X\). In the paper, the authors introduce the concepts of Hausdorff sensitivity, Hausdorff equicontinuity and topological equicontinuity for \(G\)-systems and prove that a minimal \(G\)-system \((X, G)\) is either topologically equicontinuous or Hausdorff sensitive under the assumption that \(X\) is a \(T_3\)-space and they provide a classification of transitive dynamical systems in terms of equicontinuity pairs. In particular, under the condition that \(X\) is a Hausdorff uniform space, they give a dichotomy theorem between Hausdorff sensitivity and Hausdorff equicontinuity for \(G\)-systems admitting one transitive point.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B02 Dynamics in general topological spaces
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
22F05 General theory of group and pseudogroup actions
54C35 Function spaces in general topology
54C05 Continuous maps

Keywords:

Hausdorff sensitivity; Hausdorff equicontinuity; topological equicontinuity; even continuity
PDFBibTeX XMLCite
\textit{S. Xie} and \textit{J. Yin}, Chin. Ann. Math., Ser. B 44, No. 4, 501--516 (2023; Zbl 07762268)
Full Text: DOI

References:

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