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M. Foreman and B. Weiss obtained an anti-classification result for smoothergodic diffeomorphisms, up to measure isomorphism, by using a functor$\mathcal{F}$ mapping odometer-based systems, $\mathcal{OB}$, to circularsystems, $\mathcal{CB}$. 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 ofisomorphisms. Thus it is natural to ask whether $\mathcal{F}$ preserves otherdynamical properties. We show that $\mathcal{F}$ does not preserve the looselyBernoulli property by providing positive and zero entropy examples of looselyBernoulli odometer-based systems whose corresponding circular systems are notloosely Bernoulli. We also construct a loosely Bernoulli circular system whosecorresponding odometer-based system has zero entropy and is not looselyBernoulli.

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