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Invariant measures for Cartesian powers of Chacon infinite transformation

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Abstract

We describe all boundedly finite measures which are invariant by Cartesian powers of an infinite measure preserving version of Chacon transformation. All such ergodic measures are products of so-called diagonal measures, which are measures generalizing in some way the measures supported on a graph. Unlike what happens in the finite-measure case, this class of diagonal measures is not reduced to measures supported on a graph arising from powers of the transformation: it also contains some weird invariant measures, whose marginals are singular with respect to the measure invariant by the transformation. We derive from these results that the infinite Chacon transformation has trivial centralizer, and has no nontrivial factor.

At the end of the paper, we prove a result of independent interest, providing sufficient conditions for an infinite measure preserving dynamical system defined on a Cartesian product to decompose into a direct product of two dynamical systems.

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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.

    Google Scholar 

  2. T. Adams, N. Friedman and C. E. Silva, Rank-one weak mixing for nonsingular transformations, Israel J. Math. 102 (1997), 269–281.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Chang and D. Pollard, Conditioning as disintegration, Statist. Neerlandica 51 (1997), 287–317.

    Article  MathSciNet  MATH  Google Scholar 

  4. I. Dai, X. Garcia, T. Pădurariu and C. E. Silva, On rationally ergodic and rationally weakly mixing rank-one transformations, Ergodic Theory Dynam. Systems 35 (2015), 1141–1164.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. del Junco and M. Keane, On generic points in the Cartesian square of Chacón’s transformation, Ergodic Theory Dynam. Systems 5 (1985), 59–69.

    MathSciNet  MATH  Google Scholar 

  6. A. del Junco, M. Rahe and L. Swanson, Chacon’s automorphism has minimal selfjoinings, J. Analyse Math. 37 (1980), 276–284.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. del Junco, A simple measure-preserving transformation with trivial centralizer, Pacific J. Math. 79 (1978), 357–362.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. A. Friedman, Introduction to ergodic theory, Van Nostrand Reinhold Co., New York- Toronto, Ont.-London, 1970, Van Nostrand Reinhold Mathematical Studies, No. 29.

    MATH  Google Scholar 

  9. P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.

    Book  MATH  Google Scholar 

  10. É. Janvresse, E. Roy and T. de la Rue, Poisson suspensions and SuShis, hal-01205239, 2015, to appear in Annales Scientifiques de l’École Normale Supérieure.

  11. A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 649–743.

    Google Scholar 

  12. D. Maharam, Incompressible transformations, Fund. Math. 56 (1964), 35–50.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. J. Rudolph and C. E. Silva, Minimal self-joinings for nonsingular transformations, Ergodic Theory Dynam. Systems 9 (1989), 759–800.

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, F76801, Saint Étienne du Rouvray, France

    Élise Janvresse & Thierry de la Rue

  2. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Institut Galilée, 99 avenue Jean-Baptiste Clément–, F93430, Villetaneuse, France

    Emmanuel Roy

Authors
  1. Élise Janvresse
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  2. Emmanuel Roy
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  3. Thierry de la Rue
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Correspondence to Élise Janvresse.

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Janvresse, É., Roy, E. & de la Rue, T. Invariant measures for Cartesian powers of Chacon infinite transformation. Isr. J. Math. 224, 1–37 (2018). https://doi.org/10.1007/s11856-018-1634-z

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  • DOI: https://doi.org/10.1007/s11856-018-1634-z

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