Loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli

Abstract Foreman and Weiss [Measure preserving diffeomorphisms of the torus are unclassifiable. Preprint , 2020, arXiv:1705.04414] obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal {F}$ (see [Foreman Weiss, From odometers circular systems: global structure theorem. J. Mod. Dyn. 15 (2019), 345–423]) mapping odometer-based systems, {OB}$ {CB}$ . This transfers classification problem from it preserves weakly mixing extensions, compact factor maps, rank-one property, certain types isomorphisms. Thus is natural ask whether other dynamical properties. We show that does not preserve loosely Bernoulli property providing positive zero-entropy examples systems whose corresponding Bernoulli. also construct system has zero entropy