Rigidity and sensitivity on uniform spaces

Abstract

In this paper, we introduce the concepts of weak uniformity, uniform rigidity and multi-sensitivity for uniform (not necessarily compact or metric) spaces and obtain some equivalent characterizations of uniform rigidity. In particular, we prove that a dynamical system (X,f) defined on a Hausdorff uniform space is uniformly rigid if and only if (X,fn) is uniformly rigid for some/all n∈N if and only if its hyperspatial dynamical system is uniformly rigid or weakly rigid. Besides, we show that every non-minimal point transitive dynamical system defined on a Hausdorff uniform space with dense Banach almost periodic points is sensitive and obtain the equivalence of the multi-sensitivity between original dynamical system and its hyperspatial dynamical system for Hausdorff uniform spaces.