Smooth mixing transformations with loosely Bernoulli cartesian product

TL;DR

This paper establishes criteria for zero-entropy systems to be loosely Bernoulli and demonstrates the existence of smooth mixing zero-entropy loosely Bernoulli transformations with certain product properties.

Contribution

It introduces a new criterion for zero-entropy systems to be loosely Bernoulli compatible with mixing and constructs examples with specific product behaviors.

Findings

Zero-entropy systems can be loosely Bernoulli if induced from irrational rotations.

Constructed smooth mixing zero-entropy loosely Bernoulli transformations.

The Cartesian product of these transformations with themselves is also loosely Bernoulli.

Abstract

A zero-entropy system is said to be loosely Bernoulli if it can be induced from an irrational rotation of the circle. We provide a criterion for zero-entropy systems to be loosely Bernoulli that is compatible with mixing. Using these criteria, we show the existence of smooth mixing zero-entropy loosely Bernoulli transformations whose cartesian product with themselves is loosely Bernoulli.