  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.
