Positive topological entropy and $\Delta$-weakly mixing sets

TL;DR

This paper introduces $\Delta$-weakly mixing sets in dynamical systems, showing their abundance in systems with positive topological entropy and establishing their existence under certain ergodic measures, thus generalizing known results.

Contribution

It defines $\Delta$-weakly mixing sets, proves their residuality in systems with positive entropy, and demonstrates their existence under specific ergodic measures, extending previous theories.

Findings

$\Delta$-weakly mixing sets are residual in systems with positive entropy

Existence of $\Delta$-weakly mixing sets in systems with non-measurable distal ergodic measures

Generalization of several known results and resolution of open questions

Abstract

The notion of $\Delta$-weakly mixing set is introduced, which shares similar properties of weakly mixing sets. It is shown that if a dynamical system has positive topological entropy, then the collection of $\Delta$-weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace. The existence of $\Delta$-weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained. Moreover, Our results generalize several well known results and also answer several open questions.