Varieties of Mixing

TL;DR

This paper extends the concepts of transitivity and mixing in dynamical systems, introducing new notions and establishing a dichotomy that systems are either mixing or factor onto equicontinuous minimal systems.

Contribution

It introduces strong chain transitivity and vague transitivity, extending mixing concepts, and proves a dichotomy linking these properties to equicontinuous minimal systems.

Findings

Extended notions of transitivity and mixing in dynamical systems.

Established a dichotomy: systems are either mixing or factor onto equicontinuous minimal systems.

Proved that minimal systems are weak mixing if and only if they are vague mixing.

Abstract

We consider extensions of the notion of topological transitivity for a dynamical system $(X,f)$. In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each there is a notion of mixing, defined by transitivity of the product system $(X \times X, f \times f)$. These extend the concept of weak mixing which is associated with topological transitivity. Using the barrier functions of Fathi and Pageault, we obtain for each of these extended notions a dichotomy result that a transitive system of each type either satisfies the corresponding mixing condition or else factors onto an appropriate type of equicontinuous minimal system. The classical dichotomy result for minimal systems follows when it is shown that a minimal system is weak mixing if and only if it is vague mixing.