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Quasi-factors and relative entropy for infinite-measure-preserving transformations

Israel Journal of Mathematics volume 185, pages 43–60 (2011)Cite this article

Abstract

We extend the definition of quasi-factors for infinite-measure-preserving transformations. The existence of a system with zero Krengel entropy and a quasi-factor with positive entropy is obtained. On the other hand, relative zero-entropy for conservative systems implies relative zero-entropy of any quasi-factor with respect to its natural projection onto the factor. This extends (and is based upon) results of Glasner, Thouvenot and Weiss [6, 7]. Following and extending Glasner and Weiss [8], we also prove that any conservative measure-preserving system with positive entropy in the sense of Danilenko and Rudolph [3] admits any probability-preserving system with positive entropy as a factor. Some applications and connections with Poisson-suspensions are presented.

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References

  1. J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  2. J. Aaronson and M. Keane, Isomorphism of random walks, Israel Journal of Mathematics 87 (1994), 37–63.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Danilenko and D. Rudolph, Conditional entropy theory in infinite measure and a question of Krengel, Israel Journal of Mathematics 172 (2009), 93–117.

    Article  MathSciNet  MATH  Google Scholar 

  4. É. Janvresse, T. Meyerovitch, E. Roy and Thierry de la Rue, Poisson suspensions and entropy for infinite transformations, Transactions of the American Mathematical Society 362 (2010), 3069–3094.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Glasner, Quasifactors in ergodic theory, Israel Journal of Mathematics 45 (1983), 198–208.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Glasner, J. -P. Thouvenot and B. Weiss, Entropy theory without a past, Ergodic Theory and Dynamical Systems 20 (2000), 1355–1370.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems, Journal of the American Mathematical Society 8 (1995), 665–686.

    MathSciNet  MATH  Google Scholar 

  8. E. Glasner and B. Weiss, Quasifactors of ergodic systems with positive entropy, Israel Journal of Mathematics 134 (2003), 363–380.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. C. Kieffer, A simple development of the Thouvenot relative isomorphism theory, Annals of Probability 12 (1984), 204–211.

    Article  MathSciNet  MATH  Google Scholar 

  10. U. Krengel, Entropy of conservative transformations, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 7 (1967), 161–181.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Ornstein and B. Weiss, Any flow is an orbit factor of any flow, Ergodic Theory and Dynamical Systems 4 (1984), 105–116.

    Article  MathSciNet  MATH  Google Scholar 

  12. W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, New York-Amsterdam, 1969.

    MATH  Google Scholar 

  13. E. Roy, Measures de poisson, infinie divisibilité et propriétés ergodiques, Ph. D. thesis, Université Paris, 2005.

  14. E. Roy, Poisson suspensions and infinite ergodic theory, Ergodic Theory and Dynamical Systems 29 (2009), 667–683.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Smorodinsky and J.-P. Thouvenot, Bernoulli factors that span a transformation, Israel Journal of Mathematics 32 (1979), 39–43.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.-P. Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli, Israel Journal of Mathematics 21 (1975), 177–207.

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Tom Meyerovitch

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  1. Tom Meyerovitch
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Correspondence to Tom Meyerovitch.

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Meyerovitch, T. Quasi-factors and relative entropy for infinite-measure-preserving transformations. Isr. J. Math. 185, 43–60 (2011). https://doi.org/10.1007/s11856-011-0100-y

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  • DOI: https://doi.org/10.1007/s11856-011-0100-y

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