Rigidity and sensitivity on uniform spaces

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, f(n)) is uniformly rigid for some/all n is an element of 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. (C) 2018 Elsevier B.V. All rights reserved.