On weak mixing, minimality and weak disjointness of all iterates

TL;DR

This paper explores the relationship between weak mixing, minimality, and disjointness in topological dynamical systems, extending known results, constructing counterexamples, and clarifying the roles of these properties.

Contribution

It extends the weakly mixing and strongly transitive system result to non-invertible cases and constructs examples showing limitations of weak mixing and minimality assumptions.

Findings

Weakly mixing and strongly transitive systems are $ riangle$-transitive.

Constructed examples of multi-transitive non-weakly mixing systems.

Constructed examples of weakly mixing non multi-transitive systems.

Abstract

The article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map $f\times f^2 \times...\times f^m$, where $f\colon X\ra X$ is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is $\Delta$-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Colloq. Math. 120 (2010), no. 1, 127--138]. The first one is a multi-transitive non weakly mixing system, and the second one is a weakly mixing non multi-transitive system. The examples are special spacing shifts. The later shows that the assumption of minimality in the Multiple Recurrence Theorem can not be replaced by weak mixing.