Topological Ergodic Shadowing and Chaos on Uniform Spaces

Abstract

This paper firstly proves that every dynamical system defined on a Hausdorff uniform space with topologically ergodic shadowing is topologically mixing, thus topologically chain mixing. Then, the following is proved: (1) every weakly mixing dynamical system defined on a second countable Baire–Hausdorff uniform space is chaotic in the sense of both Li–Yorke and Auslander–Yorke; (2) every point transitive dynamical system defined on a Hausdorff uniform space is either almost equicontinuous or sensitive.