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

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  01 October 2021

MARLIES GERBER  and
PHILIPP KUNDE
MARLIES GERBER*
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN47405, USA
PHILIPP KUNDE
Affiliation:
Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146Hamburg, Germany (e-mail: philipp.kunde@math.uni-hamburg.de)
*
e-mail: gerber@indiana.edu
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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 and Weiss, From odometers to circular systems: a global structure theorem. J. Mod. Dyn.15 (2019), 345–423]) 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.

Keywords

loosely BernoulliKakutani equivalenceodometer

MSC classification

Primary: 37A05: Measure-preserving transformations
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 37B10: Symbolic dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 917 - 973
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Dedication: We dedicate this paper to the memory of Anatole Katok. It utilizes two major contributions of Katok, namely the introduction of standard automorphisms (also called zero-entropy loosely Bernoulli) and his work together with D.V. Anosov on the construction of smooth ergodic diffeomorphisms on the disk.

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