On local aspects of topological weak mixing, sequence entropy and chaos

Abstract

AbstractIn this paper we show that for every$n\geq 2$there are minimal systems with perfect weakly mixing sets of order$n$and all weakly mixing sets of order$n+ 1$trivial. We present some relations between weakly mixing sets and topological sequence entropy; in particular, we prove that invertible minimal systems with non-trivial weakly mixing sets of order three always have positive topological sequence entropy. We also study relations between weak mixing of sets and other well-established notions from qualitative theory of dynamical systems like (regional) proximality, chaos and equicontinuity in a broad sense.