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Ergodicity of Rokhlin cocycles

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Abstract

We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such groups. Finally, we examine the structure of self-joinings of extensions.

We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss on the class of automorphisms disjoint from all weakly mixing transformations.

Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker dynamical systems.

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References

  1. J. Aaronson,The eigenvalues of non-singular transformations, Israel J. Math.45 (1983), 297–312.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Aaronson,An Introduction to Infinite Ergodic Theory, Math. Surveys Monographs50, Amer. Math. Soc., Providence, RI, 1997.

    MATH  Google Scholar 

  3. J. Aaronson, T. Hamachi and K. Schmidt,Associated actions and uniqueness of cocycles, inAlgorithms, Fractals and Dynamics (Y. Takahashi, ed.), Plenum Press, New York, 1995, pp. 1–26.

    Chapter  Google Scholar 

  4. J. Aaronson, M. Lemańczyk, C. Mauduit and H. Nakada,Koksma inequality and group extensions of Kronecker transformations, inAlgorithms, Fractals and Dynamics (Y. Takahashi, ed.), Plenum Press, New York, 1995, pp. 27–50.

    Chapter  Google Scholar 

  5. J. Aaronson and M. Nadkarni,L ∞ eigenvalues and L 2 spectra of non-singular transformations, Proc. London Math. Soc.55 (1987), 538–570.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Aaronson, M. Lemańczyk and D. Volný,A cut salad of cocycles, Fund. Math.157 (1998), 99–119.

    MathSciNet  MATH  Google Scholar 

  7. L. M. Abramov and V. A. Rokhlin,The entropy of a skew product of measure preserving transformations, Amer. Math. Soc. Transl.48 (1965), 225–265.

    Google Scholar 

  8. H. Anzai,Ergodic skew product transformation on the torus, Osaka J. Math.3 (1951), 83–99.

    MathSciNet  MATH  Google Scholar 

  9. N. Bourbaki,Éléments de mathématiques, Livre III: Topologie générale, Hermann, Paris, 1958.

    Google Scholar 

  10. I. P. Cornfeld, S. V. Fomin and Y. G. Sinai,Ergodic Theory, Springer, New York, 1982.

    Book  MATH  Google Scholar 

  11. C. Foiaş and S. Stratila,Ensembles de Kronecker dans la théorie ergodique, C.R. Acad. Sci. Paris267, 20A (1967), 166-168.

    Google Scholar 

  12. H. Furstenberg,Strict ergodicity and transformations of the torus, Amer. J. Math.,83 (1961), 573–601.

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Furstenberg,Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, Math. Systems Theory1 (1967), 1–49.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.

    MATH  Google Scholar 

  15. H. Furstenberg and B. Weiss,The finite multipliers of infinite ergodic transformations, Lecture Notes in Math.688 (1978), 127–132.

    Article  MathSciNet  Google Scholar 

  16. E. Glasner and B. Weiss,Processes disjoint from weak mixing, Trans. Amer. Math. Soc.316 (1989), 689–703.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Guivarc’h,Propriétés ergodiques, en mesure infinie, de certains systemes dynamiques fibrés, Ergodic Theory Dynam. Systems9 (1989), 433–453.

    MathSciNet  MATH  Google Scholar 

  18. T. Hamachi and M. Osikawa,Ergodic groups of automorphisms and Krieger’s theorems, Seminar on Math. Sci., Keio Univ.3 (1991).

  19. E. Hewitt and K. A. Ross,Abstract Harmonic Analysis II, Springer-Verlag, New York, Heidelberg, Berlin, 1970.

    MATH  Google Scholar 

  20. F. den Hollander and J. E. Steiff,Mixing properties of the generalized T, T −1 -process, J. Analyse Math.72 (1997), 165–202.

    Article  MATH  Google Scholar 

  21. B. Host, J.-F. Méla and F. Parreau,Non-singular transformations and spectral analysis of measures, Bull. Soc. Math. France119 (1991), 33–90.

    MathSciNet  MATH  Google Scholar 

  22. A. del Junco and M. Lemańczyk,Simple automorphisms are disjoint from Gaussian automorphisms, Studia Math.133 (1999), 249–256.

    MathSciNet  MATH  Google Scholar 

  23. A. del Junco and D. Rudolph,On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems7 (1987), 531–557.

    MathSciNet  MATH  Google Scholar 

  24. S. Kalikow,T, T −1 transformation is not loosely Bernoulli, Ann. of Math.115, (1982), 393–409.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Kallman,Certain quotient spaces are countably separated, III, J. Funct. Anal.22 (1976), 225–241.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. A. Kirillow,Dynamical systems, factors and representations of groups, Uspekhi Mat. Nauk22 (1967), 63–75.

    Google Scholar 

  27. M. Lemańczyk, E. Lesigne and D. Skrenty,Multiplicative Gaussian cocycles, Aequationes Math.61 (2001), 162–178.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Lemańczyk and F. Parreau,On the disjointness problem for Gaussian automorphisms, Proc. Amer. Math. Soc.127 (1999), 2073–2081.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Lemańczyk, F. Parreau and J.-P. Thouvenot,Gaussian automorphism whose ergodic selfjoinings are Gaussian, Fund. Math.164 (2000), 253–293.

    MathSciNet  MATH  Google Scholar 

  30. C. Moore and K. Schmidt,Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson, Proc. London Math. Soc.40 (1980), 443–475.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Newton,On canonical factors of ergodic dynamical systems J. London Math. Soc.19 (1979), 129–136.

    Article  MathSciNet  MATH  Google Scholar 

  32. W. Parry,Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge, 1981.

    MATH  Google Scholar 

  33. M. Queffélec,Substitution Dynamical Systems. Spectral Analysis, Lecture Notes in Math.1294 (1987).

  34. E. A. Robinson,A general condition for lifting theorems, Trans. Amer. Math. Soc.,330 (1992), 725–755.

    Article  MathSciNet  MATH  Google Scholar 

  35. W. A. Rokhlin,On the fundamental ideas of measure theory, Uspekhi Mat. Nauk, Amer. Math. Soc. Transl.10, (1962), 1–54.

    Google Scholar 

  36. D. Rudolph,Classifying isometric extensions of a Bernoulli shift, Israel J. Math.34 (1978), 36–60.

    MathSciNet  MATH  Google Scholar 

  37. D. Rudolph,k-fold mixing lifts to weakly mixing isometric extensions, Ergodic Theory Dynam. Systems5 (1985), 445–447.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Rudolph,Z n and R n cocycle extensions and complementary algebras, Ergodic Theory Dynam. Systems6 (1986), 583–599.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Rudolph,Fundamentals of Measurable Dynamics—Ergodic Theory on Lebesgue Spaces, Clarendon Press, Oxford, 1990.

    MATH  Google Scholar 

  40. K. Schmidt,Cocycles of Ergodic Transformation Groups, Lecture Notes in Mathematics, Vol. 1, MacMillan Co. of India, 1977.

  41. K. Schmidt,Spectra of ergodic group actions, Israel J. Math.41 (1982), 151–153.

    Article  MathSciNet  MATH  Google Scholar 

  42. K. Schmidt and P. Walters,Mildly mixing actions of locally compact groups, Proc. London Math. Soc. (3)45 (1982), 506–518.

    Article  MathSciNet  MATH  Google Scholar 

  43. F. Spitzer,Principles of Random Walks, Van Nostrand, Princeton, 1964.

    Google Scholar 

  44. A. Tempelman,Ergodic Theorems for Group Actions, Kluwer Academic Publ., Dordrecht, 1992.

    Google Scholar 

  45. R. Zimmer,Extensions of ergodic group actions, Illinois J. Math.20 (1976), 373–409.

    MathSciNet  MATH  Google Scholar 

  46. R. Zimmer,Ergodic actions with generalized discrete spectrum, Illinois J. Math.20 (1976), 555–588.

    MathSciNet  MATH  Google Scholar 

  47. R. Zimmer,Cocycles and the structure of ergodic group actions, Israel J. Math.26 (1977), 214–220.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicholas Copernicus University, UL. Chopina 12/18, 87-100, Toruń, Poland

    Mariusz Lemańczyk

  2. Laboratoire de Mathématiques et Physique Théorique, Université Françoise Rabelais, Parc de Grandmont, 37200, Tours, France

    Emmanuel Lesigne

Authors
  1. Mariusz Lemańczyk
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  2. Emmanuel Lesigne
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Additional information

Research partially supported by KBN grant 2 P03A 002 14 (1998).

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Lemańczyk, M., Lesigne, E. Ergodicity of Rokhlin cocycles. J. Anal. Math. 85, 43–86 (2001). https://doi.org/10.1007/BF02788075

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  • DOI: https://doi.org/10.1007/BF02788075

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