Loosely Bernoulli Odometer-Based Systems Whose Corresponding Circular Systems Are Not Loosely Bernoulli

TL;DR

This paper demonstrates that the functor linking odometer-based systems to circular systems does not preserve the loosely Bernoulli property, providing examples where this property is lost or gained through the transformation.

Contribution

It shows that the functor $unctor$ does not preserve the loosely Bernoulli property, contrasting with other properties it preserves, through explicit examples.

Findings

Loosely Bernoulli odometer systems can correspond to non-loosely Bernoulli circular systems.

A circular system can be loosely Bernoulli while its odometer counterpart is not.

Examples include systems with zero entropy that differ in the loosely Bernoulli property.

Abstract

M. Foreman and B. Weiss obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal{F}$ mapping odometer-based systems, $\mathcal{OB}$, to circular systems, $\mathcal{CB}$. This functor transfers the classification problem from $\mathcal{OB}$ to $\mathcal{CB}$, and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal{F}$ preserves other dynamical properties. We show that $\mathcal{F}$ does not preserve the loosely Bernoulli property by providing positive and zero entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.