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Joint ergodicity and mixing

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Abstract

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ1σ2,...,σ, ofG, the sequence ((1/N)Σ N−1 n=0 σ n1 f 1·σ n2 f 2· ··· · σ ns f sconverges inL 2(G) for everyf 1,f 2,…,f s∈L ∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π s i=1 ∫ G f 1 d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.

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References

  1. D. Berend and V. Bergelson,Jointly ergodic measure-preserving transformations, Isr. J. Math.49 (1984), 307–314.

    MATH  MathSciNet  Google Scholar 

  2. J. W. S. Cassels,An embedding theorem for fields, Bull. Austral. Math. Soc.14 (1976), 193–198.

    MATH  MathSciNet  Google Scholar 

  3. J. P. Conze and E. Lesigne,Théorèmes ergodiques pour des mesures diagonales, Bull. Soc. Math. France112 (1984), 143–175.

    MATH  MathSciNet  Google Scholar 

  4. I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai,Ergodic Theory, Springer-Verlag, New York, 1982.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. H. Furstenberg, Y. Katznelson and D. Ornstein,The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc.7 (1982), 527–552.

    MathSciNet  Google Scholar 

  8. H. Furstenberg and B. Weiss,Topological dynamics and combinatorial number theory, J. Analyse Math.34 (1978), 61–85.

    MATH  MathSciNet  Google Scholar 

  9. H. Furstenberg and B. Weiss, personal communication.

  10. E. Hewitt and K. A. Ross,Abstract Harmonic Analysis, Vol. I, Springer-Verlag, Berlin, 1963.

    MATH  Google Scholar 

  11. L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York, 1974.

    MATH  Google Scholar 

  12. E. Lesigne,Resolution d'une equation fonctionelle, Bull. Soc. Math. France112 (1984), 177–196.

    MATH  MathSciNet  Google Scholar 

  13. K. Petersen,Ergodic Theory, Cambridge University Press, Cambridge, 1983.

    MATH  Google Scholar 

  14. A. J. van der Poorten,Hermite interpolation and p-adic exponential polynomials, J. Austral. Math. Soc.22 (Ser. A) (1976), 12–26.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    Daniel Berend

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  1. Daniel Berend
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Berend, D. Joint ergodicity and mixing. J. Anal. Math. 45, 255–284 (1985). https://doi.org/10.1007/BF02792552

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  • DOI: https://doi.org/10.1007/BF02792552

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