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.
Categories: math.DS