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Minimal self-joinings of infinite mixing actions of rank 1

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Abstract

We prove that measure-preserving actions of rank 1 of the groups ℤn and ℝn on a Lebesgue space with a σ-finite measure have minimal self-joinings.

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References

  1. D. J. Rudolph, “An example of measure-preserving map with minimal self-joinings, and applications,” J. AnalyseMath. 35 (1), 97–122 (1979).

    MathSciNet  MATH  Google Scholar 

  2. J. Aaronson and M. Nadkarni, “L ∞ eigenvalues and L 2 spectra of non-singular transformations,” Proc. London Math. Soc. (3) 55 (3), 538–570 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. V. Ryzhikov and J.-P. Thouvenot, “On the centralizer of an infinite mixing rank-one transformation,” Funktsional. Anal. Prilozhen. 49 (3), 88–91 (2015) [Functional Anal. Appl. 49 (3), 230–233 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  4. A. M. Vershik, I. M. Gel’fand, and M. I. Graev, “Representations of the group of diffeomorphisms,” Uspekhi Mat. Nauk 30 (6 (186)), 3–50 (1975) [Russian Math. Surveys 30 (6), 1–50 (1975)].

    MathSciNet  MATH  Google Scholar 

  5. R. S. Ismagilov, “Unitary representations of group of diffeomorphisms of space Rn, n ≥ 2,” Funktsional. Anal. Prilozhen. 9 (2), 71–72 (1975) [Functional Anal. Appl. 9 (2), 154–155 (1975)].

    MathSciNet  Google Scholar 

  6. I. P. Kornfel’d, Ya. G. Sinai, and S.V. Fomin, Ergodic Theory (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  7. Yu. A. Neretin, “On the correspondence between boson Fock space and the L2 space with respect to Poisson measure,” Mat. Sb. 188 (11), 19–50 (1997) [Sb.Math. 188 (11), 1587–1616 (1997)].

    Article  MathSciNet  Google Scholar 

  8. E. Roy, “Poisson suspensions and infinite ergodic theory,” Ergodic Theory Dynam. Systems 29 (2), 667–683 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. V. Ryzhikov, “Ergodic homoclinic groups, Sidon constructions and Poisson suspensions,” Trudy Moskov. Mat. Obshch. 75 (1), 93–103 (2014) [Trans.MoscowMath. Soc. 75, 77–85 (2014)].

    MATH  Google Scholar 

  10. V. V. Ryzhikov and J.-P. Thouvenot, “Disjointness, divisibility, and quasi-simplicity of measure-preserving actions,” Funktsional. Anal. Prilozhen. 40 (3), 85–89 (2006) [Functional Anal. Appl. 40 (3), 237–240 (2006)].

    Article  MathSciNet  MATH  Google Scholar 

  11. A. B. Katok and A. M. Stepin, “Approximations in ergodic theory,” UspekhiMat. Nauk 22 (5 (137)), 81–106 (1967) [Russian Math. Surveys 22 (5), 77–102 (1967)].

    MathSciNet  MATH  Google Scholar 

  12. A. I. Bashtanov, “Conjugacy classes are dense in the space of mixing Zd-actions,” Math. Notes 99 (1–2), 9–23 (2016).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    I. V. Klimov & V. V. Ryzhikov

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  1. I. V. Klimov
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  2. V. V. Ryzhikov
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Correspondence to I. V. Klimov.

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Original Russian Text © I. V. Klimov, V. V. Ryzhikov, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 6, pp. 851–856.

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Klimov, I.V., Ryzhikov, V.V. Minimal self-joinings of infinite mixing actions of rank 1. Math Notes 102, 787–791 (2017). https://doi.org/10.1134/S0001434617110189

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  • DOI: https://doi.org/10.1134/S0001434617110189

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