Chaos and weak mixing on uniform spaces

Abstract

Let f0,∞={fn}n=0∞ be a sequence of uniformly continuous self-maps on a uniform space X. We prove that under some natural conditions, topological weak mixing implies dense distributional η-chaos in a sequence, and generic η-chaos is equivalent to dense η′-chaos for (X,f0,∞). We give several equivalent characterizations of sensitivity for (X,f0,∞), and as applications, we get the relationships between sensitivity and Li-Yorke sensitivity, and further obtain that topological weak mixing implies Li-Yorke sensitivity for a class of abelian (X,f0,∞). We show that (X×Y,f0,∞×g0,∞) is sensitive (resp. Li-Yorke sensitive and multi-sensitive) if and only if (X,f0,∞) or (Y,g0,∞) is sensitive (resp. Li-Yorke sensitive and multi-sensitive). We also prove that distributional chaos (resp. Li-Yorke chaos, sensitivity and Li-Yorke sensitivity) is equivalent between (X,f0,∞) and its N-th iteration system. Finally, we confirm that distributional chaos in a sequence, Li-Yorke chaos, sensitivity and Li-Yorke sensitivity are all preserved under topological equi-conjugacy.