Equicontinuity and Sensitivity of Group Actions

Abstract

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.

Keywords

Hausdorff sensitivity / Hausdorff equicontinuity / Topological equicontinuity / Even continuity

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