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Mat. Sb., 2007, Volume 198, Number 5, Pages 137–159 (Mi msb3729)  

This article is cited in 23 scientific papers (total in 23 papers)

Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions

V. V. Ryzhikov

M. V. Lomonosov Moscow State University

Abstract: A class of automorphisms of the Lebesgue space such that their symmetric powers have simple spectrum is considered. In the framework of rank-one constructions mixing automorphisms with this property are constructed. The paper also contains results on weak limits, the local rank, and the spectral multiplicity of powers of automorphisms. Spectral properties of the stochastic Chacon automorphism are discussed.
Bibliography: 23 titles.


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English version:
Sbornik: Mathematics, 2007, 198:5, 733–754

Bibliographic databases:

UDC: 517.9
MSC: Primary 37A30; Secondary 47A35, 28D05
Received: 03.10.2006 and 13.11.2006

Citation: V. V. Ryzhikov, “Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions”, Mat. Sb., 198:5 (2007), 137–159; Sb. Math., 198:5 (2007), 733–754

Citation in format AMSBIB
\by V.~V.~Ryzhikov
\paper Weak limits of powers, simple spectrum of
symmetric products, and rank-one mixing constructions
\jour Mat. Sb.
\yr 2007
\vol 198
\issue 5
\pages 137--159
\jour Sb. Math.
\yr 2007
\vol 198
\issue 5
\pages 733--754

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    This publication is cited in the following articles:
    1. Ageev O., “Mixing with staircase multiplicity functions”, Ergodic Theory Dynam. Systems, 28:6 (2008), 1687–1700  crossref  mathscinet  zmath  isi  elib  scopus
    2. V. V. Ryzhikov, “Spectral multiplicities and asymptotic operator properties of actions with invariant measure”, Sb. Math., 200:12 (2009), 1833–1845  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Katok A., Lemańczyk M., “Some new cases of realization of spectral multiplicity function for ergodic transformations”, Fund. Math., 206 (2009), 185–215  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. I. Danilenko, V. V. Ryzhikov, “Spectral Multiplicities of Infinite Measure Preserving Transformations”, Funct. Anal. Appl., 44:3 (2010), 161–170  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Danilenko A.I., “On new spectral multiplicities for ergodic maps”, Studia Math., 197:1 (2010), 57–68  crossref  mathscinet  zmath  isi  elib  scopus
    6. Danilenko A.I., Ryzhikov V.V., “Mixing constructions with infinite invariant measure and spectral multiplicities”, Ergodic Theory Dynam. Systems, 31:3 (2011), 853–873  crossref  mathscinet  zmath  isi  elib  scopus
    7. Tikhonov S.V., “Homogeneous spectrum and mixing transformations”, Dokl. Math., 83:1 (2011), 80–83  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. Lemańczyk M., Parreau F., Roy E., “Joining primeness and disjointness from infinitely divisible systems”, Proc. Amer. Math. Soc., 139:1 (2011), 185–199  crossref  mathscinet  zmath  isi  elib  scopus
    9. Danilenko A.I., Park K.K., “Rank-one flows of transformations with infinite ergodic index”, Proc. Amer. Math. Soc., 139:1 (2011), 201–207  crossref  mathscinet  zmath  isi  elib  scopus
    10. S. V. Tikhonov, “Mixing transformations with homogeneous spectrum”, Sb. Math., 202:8 (2011), 1231–1252  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Danilenko A.I., “New spectral multiplicities for mixing transformations”, Ergod. Th. Dynam. Sys., 32 (2012), 517–534  crossref  mathscinet  elib  scopus
    12. Danilenko A.I., Ryzhikov V.V., “On self-similarities of ergodic flows”, Proc. London Math. Soc., 104:3 (2012), 431–454  crossref  mathscinet  zmath  isi  elib  scopus
    13. V. V. Ryzhikov, “Spectral multiplicity for powers of weakly mixing automorphisms”, Sb. Math., 203:7 (2012), 1065–1076  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. V. V. Ryzhikov, “Simple spectrum of the tensor product of powers of a mixing automorphism”, Trans. Moscow Math. Soc., 73 (2012), 183–191  mathnet  crossref  mathscinet  zmath  elib
    15. Kulaga-Przymus J., Parreau F., “Disjointness properties for Cartesian products of weakly mixing systems”, Colloq. Math., 128:2 (2012), 153–177  crossref  mathscinet  zmath  isi  elib  scopus
    16. Danilenko A.I., “A survey on spectral multiplicities of ergodic actions”, Ergodic Theory Dynam. Systems, 33:1 (2013), 81–117  crossref  zmath  isi  elib  scopus
    17. V. V. Ryzhikov, “The Local Rank of an Ergodic Symmetric Power $T^{\odot n}$ Does Not Exceed $n! n^{-n}$”, Funct. Anal. Appl., 47:1 (2013), 76–79  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    18. E.H.oucein El Abdalaoui, Mariusz Lemańczyk, Thierry de la Rue, “On spectral disjointness of powers for rank-one transformations and Möbius orthogonality”, Journal of Functional Analysis, 2013  crossref  isi  scopus
    19. S. V. Tikhonov, “Approximation of Mixing Transformations”, Math. Notes, 95:2 (2014), 255–269  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    20. R. A. Konev, V. V. Ryzhikov, “On the Collection of Spectral Multiplicities $\{2,4,…,2^n\}$ for Totally Ergodic $\mathbb{Z}^2$-Actions”, Math. Notes, 96:3 (2014), 360–368  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    21. V. V. Ryzhikov, “On the Asymmetry of Multiple Asymptotic Properties of Ergodic Actions”, Math. Notes, 96:3 (2014), 416–422  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    22. V. V. Ryzhikov, “Ergodic homoclinic groups, Sidon constructions and Poisson suspensions”, Trans. Moscow Math. Soc., 75 (2014), 77–85  mathnet  crossref  elib
    23. V. V. Ryzhikov, A. E. Troitskaya, “Peremeshivayuschie potoki s odnorodnym spektrom kratnosti 2”, Fundament. i prikl. matem., 21:5 (2016), 191–197  mathnet
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