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Proprietes generiques de processus croises

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Abstract

LetS φ be the skew product transformation(x, g)↦(Sx, gφ(x)) defined on Ω×G, where Ω is a compact metric space,G a compact metric group with its Haar measureh. IfS is a μ-continuous transformation where μ is a Borel measure on Ω, ergodic with respect toS, we study the setE 0 of μ-continuous applications φ:Ω→G such that μ⩀h is ergodic (with respect toS φ). For example,E 0 is residual in the group of μ-continuous applications from Ω toG with the uniform convergence topology. We also study the weakly mixing case. Some arithmetic applications are given.

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Bibliographie

  1. R. L. Alder and P. C. Shields,Skew products of Bernoulli shifts with rotations, Israel J. Math.12 (1972), 215–222.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. L. Adler and P. C. Shields,Skew products of Bernoulli shifts with rotations II, Israel J. Math.19 (1974), 228–236.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. J. P. Conze,Equirépartition et ergodicité de transformations cylindriques, publications des séminaires, Université de Rennes, 1976.

  5. J. P. Conze et M. Keane,Ergodicité d'un flot cylindrique, préprint.

  6. Corput, J. G., Van der,Diophantische Ungleichungen I. Zur Gleichverteilung modulo Eins, Acta Math.56 (1931), 373–456.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. P. R. Halmos and H. Samelson,On monothetic groups, Proc. Nat. Acad. Sci. U.S.A.28 (1942), 254–258.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. R. Halmos,Lectures on Ergodic Theory, Chelsea Publishing Company, New York, 1956.

    MATH  Google Scholar 

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

  12. M. Keane et G. Rauzy,Stricte ergodicité des échanges d'intervalles, à paraître.

  13. L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Pure and Applied Math., Wiley-Interscience Publication, 1974.

  14. P. Liardet,Répartition et ergodicité, Séminaire D.P.P. Paris, 19o année, no 10, 1977–78, 12p.

  15. G. Rauzy,Sur une suite liée à la discrépance de la suite (nα) n∈N, Univers. d'Aix-Marseille II, 5p. (non publié).

  16. G. Rauzy,Répartition modulo 1, SMF-Asterisque41–42 (1977), 81–101.

    MathSciNet  MATH  Google Scholar 

  17. M. Stewart,Irregularities of uniform distribution, à paraître.

  18. W. Veech,Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl Theorem mod 2, Trans. Amer. Math. Soc.140 (1969), 1–33.

    MathSciNet  MATH  Google Scholar 

  19. W. Veech,Well distributed sequences of integers, Trans. Amer. Math. Soc.161 (1971), 63–70.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Veech,Some questions of uniform distribution, Ann. of Math. (2)94 (1971), 125–138.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Veech,Finite group extensions of irrational rotations, Israel J. Math.21 (1975), 240–259.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire associé au C.N.R.S.no 225 U.E.R. de Mathématiques, Université de Provence, 3 place Victor-Hugo, 13331, Marseille Cedex 3, France

    Pierre Liardet

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  1. Pierre Liardet
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Liardet, P. Proprietes generiques de processus croises. Israel J. Math. 39, 303–325 (1981). https://doi.org/10.1007/BF02761676

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  • DOI: https://doi.org/10.1007/BF02761676

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