2019/12/29 by Marlies Gerber, Gerber, Marlies, Philipp Kunde +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS
paper · pdf · doi:10.48550/arxiv.1912.12684
53 pages, 3 figures
arxiv created 2019/12/29 · arxiv updated 2020/01/01
M. Foreman and B. Weiss obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor F mapping odometer-based systems, OB, to circular systems, CB. This functor transfers the classification problem from OB to 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 F preserves other dynamical properties. We show that 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.