Mixing sets for non-mixing transformations

TL;DR

This paper studies lightly mixing sets in measure-preserving transformations, establishing new hierarchies and providing stronger mixing realizations than previous results, especially for mildly mixing transformations.

Contribution

It introduces a hierarchy for lightly mixing and related properties, and demonstrates that mildly mixing transformations have dense algebras where they are lightly mixing.

Findings

Mildly mixing transformations have dense algebras of lightly mixing sets.

Hierarchy established for lightly mixing, sweeping out, and uniform sweeping out properties.

Stronger mixing realizations achieved than previous extensions of Jewett-Krieger Theorem.

Abstract

For different classes of measure preserving transformations, we investigate collections of sets that exhibit the property of lightly mixing. Lightly mixing is a stronger property than topological mixing, and requires that a lim inf is positive. In particular, we give a straightforward proof that any mildly mixing transformation T has a dense algebra C such that T is lightly mixing on C. Also, we provide a hierarchy for the properties of lightly mixing, sweeping out and uniform sweeping out for dense collections, and dense algebras of sets. As a result, stronger mixing realizations are given for several types of transformations than those given by previous extensions of the Jewett-Krieger Theorem.