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Smooth, mixing transformations with loosely Bernoulli Cartesian square

Part of: Ergodic theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  04 June 2021

FRANK TRUJILLO Open the ORCID record for FRANK TRUJILLO [Opens in a new window]
FRANK TRUJILLO*
Affiliation:
CNRS, IMJ-PRG, Institut de Mathématiques de Jussieu, UMR7586 Bâtiment Sophie Germain, 75205Paris Cedex 13, France
*
e-mail: frank.trujillo@imj-prg.fr
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Abstract

A zero-entropy system is said to be loosely Bernoulli if it can be induced from an irrational rotation of the circle. We provide a criterion for zero-entropy systems to be loosely Bernoulli that is compatible with mixing. Using this criterion, we show the existence of smooth mixing zero-entropy loosely Bernoulli transformations whose Cartesian square is loosely Bernoulli.

Keywords

classical ergodic theoryclassification in ergodic theorysmooth dynamics

MSC classification

Primary: 37A05: Measure-preserving transformations 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37C05: Smooth mappings and diffeomorphisms 37C10: Vector fields, flows, ordinary differential equations
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. 1252 - 1283
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Abramov, L. M.. On the entropy of a flow. Dokl. Akad. Nauk SSSR 128 (1959), 873–875.Google Scholar
Dye, H. A.. On groups of measure preserving transformations. I. Amer. J. Math. 81(1) (1959), 119–159.CrossRefGoogle Scholar
Fayad, B.. Rank one and mixing differentiable flows. Invent. Math. 160(2) (2005), 305–340.CrossRefGoogle Scholar
Fayad, B.. Smooth mixing flows with purely singular spectra. Duke Math. J. 132(2) (2006), 371–391.CrossRefGoogle Scholar
Fayad, B. and Qu, Y.. Continuous spectrum for a class of smooth mixing Schrödinger operators. Ergod. Th. & Dynam. Sys. 39(2) (2019), 357–369.CrossRefGoogle Scholar
Fayad, B. R.. Analytic mixing reparametrizations of irrational flows. Ergod. Th. & Dynam. Sys. 22(2) (2002), 437–468.CrossRefGoogle Scholar
Feldman, J.. NewK-automorphisms and a problem of Kakutani. Israel J. Math. 24(1) (1976), 16–38.CrossRefGoogle Scholar
Ferenczi, S.. Systèmes localement de rang un. Ann. Inst. Henri Poincaré Probab. Stat. 20(1) (1984), 35–51.Google Scholar
Ferenczi, S.. Systems of finite rank. Colloq. Math. 73 (1997), 35–65.CrossRefGoogle Scholar
Gerber, M.. A zero-entropy mixing transformation whose product with itself is loosely Bernoulli. Israel J. Math. 38(1) (1981), 1–22.CrossRefGoogle Scholar
Gerber, M. and Kunde, P.. A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli. J. Anal. Math. 141 (2020), 521–583.CrossRefGoogle Scholar
Halmos, P. R.. Approximation theories for measure preserving transformations. Trans. Amer. Math. Soc. 55(1) (1944), 1–18.CrossRefGoogle Scholar
Halmos, P. R.. In general a measure preserving transformation is mixing. Ann. of Math. 45(4) (1944), 786–792.CrossRefGoogle Scholar
Janvresse, É. and de la Rue, T.. The Pascal adic transformation is loosely Bernoulli. Ann. Inst. Henri Poincaré Probab. Stat. 40(2) (2004), 133–139.CrossRefGoogle Scholar
Kakutani, S. Induced measure preserving transformations. Proc. Imperial Acad. 19(10) (1943), 635–641.Google Scholar
Kanigowski, A. and De La Rue, T.. Product of two staircase rank one transformations that is not loosely Bernoulli. Preprint, 2019, arXiv:1812.08027.Google Scholar
Kanigowski, A. and Wei, D.. Product of two Kochergin flows with different exponents is not standard. Studia Math. 244(3) (2019), 265–283.CrossRefGoogle Scholar
Katok, A.. Combinatorial Constructions in Ergodic Theory and Dynamics (University Lecture Series, 30). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Katok, A. B.. Entropy and approximations of dynamical systems by periodic transformations. Funct. Anal. Appl. 1(1) (1967), 66–74.CrossRefGoogle Scholar
Katok, A. B.. Time change, monotone equivalence, and standard dynamical systems. Dokl. Akad. Nauk SSSR 223(4) (1975), 789–792.Google Scholar
Katok, A. B. and Stepin, A. M.. Approximation of ergodic dynamic systems by periodic transformations. Dokl. Akad. Nauk SSSR 171(6) (1966), 1268–1271.Google Scholar
Katok, A. B. and Stepin, A. M.. Approximations in ergodic theory. Uspekhi Mat. Nauk 22(5(137)) (1967), 81–106.Google Scholar
Katok, A. B. and Stepin, A. M.. Metric properties of homeomorphisms that preserve measure. Uspekhi Mat. Nauk 25(2(152)) (1970), 193–220.Google Scholar
Ornstein, D. S., Rudolph, D. J. and Weiss, B.. Equivalence of Measure Preserving Transformations (Memoirs of the American Mathematical Society, 37). American Mathematical Society, Providence, RI, 1982.CrossRefGoogle Scholar
Ratner, M.. Horocycle flows are loosely Bernoulli. Israel J. Math. 31(2) (1978), 122–132.CrossRefGoogle Scholar
Ratner, M.. The Cartesian square of the horocycle flow is not loosely Bernoulli. Israel J. Math. 34(1) (1979), 72–96.CrossRefGoogle Scholar
Rohlin, V.. A ‘general’ measure-preserving transformation is not mixing. Dokl. Akad. Nauk SSSR 60 (1948), 349–351.Google Scholar
Stepin, A. M.. The spectrum and approximation of metric automorphisms by periodic transformations. Funktsional. Anal. i Prilozhen. 1(2) (1967), 77–80.Google Scholar
Swanson, L.. Loosely Bernoulli Cartesian products. Proc. Amer. Math. Soc. 73(1) (1979), 73–78.CrossRefGoogle Scholar
Yoccoz, J.-C.. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Astérisque 231 (1995), 89–242.Google Scholar