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Mat. Sb., 2007, Volume 198, Number 4, Pages 135–158 (Mi msb3743)  

This article is cited in 16 scientific papers (total in 16 papers)

A complete metric in the set of mixing transformations

S. V. Tikhonov

Russian State University of Trade and Economics

Abstract: A metric in the set of mixing measure-preserving transformations is introduced making of it a complete separable metric space. Dense and massive subsets of this space are investigated. A generic mixing transformation is proved to have simple singular spectrum and to be a mixing of arbitrary order; all its powers are disjoint. The convolution powers of the maximal spectral type for such transformations are mutually singular if the ratio of the corresponding exponents is greater than 2. It is shown that the conjugates of a generic mixing transformation are dense, as are also the conjugates of an arbitrary fixed Cartesian product.
Bibliography: 28 titles.

DOI: https://doi.org/10.4213/sm3743

Full text: PDF file (577 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2007, 198:4, 575–596

Bibliographic databases:

UDC: 517.938
MSC: Primary 28D05; Secondary 54E35
Received: 04.10.2006

Citation: S. V. Tikhonov, “A complete metric in the set of mixing transformations”, Mat. Sb., 198:4 (2007), 135–158; Sb. Math., 198:4 (2007), 575–596

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. V. Ryzhikov, “Pairwise $\varepsilon$-Independence of the Sets $T^iA$ for a Mixing Transformation $T$”, Funct. Anal. Appl., 43:2 (2009), 155–157  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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. Tikhonov S.V., “Homogeneous spectrum and mixing transformations”, Dokl. Math., 83:1 (2011), 80–83  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. S. V. Tikhonov, “Mixing transformations with homogeneous spectrum”, Sb. Math., 202:8 (2011), 1231–1252  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. S. V. Tikhonov, “A Note on Rochlin's Property in the Space of Mixing Transformations”, Math. Notes, 90:6 (2011), 925–926  mathnet  crossref  crossref  mathscinet  isi
    6. Danilenko A.I., “New spectral multiplicities for mixing transformations”, Ergod. Th. Dynam. Sys., 32:2 (2012), 517–534  crossref  mathscinet  zmath  isi  elib  scopus
    7. S. V. Tikhonov, “Genericity of a multiple mixing”, Russian Math. Surveys, 67:4 (2012), 779–780  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. S. V. Tikhonov, “Bernoulli shifts and local density property”, Moscow University Mathematics Bulletin, 67:1 (2012), 29–35  mathnet  crossref  mathscinet
    9. Solomko A.V., “New spectral multiplicities for ergodic actions”, Studia Math., 208:3 (2012), 229–247  crossref  mathscinet  zmath  isi  elib  scopus
    10. Danilenko A.I., “A survey on spectral multiplicities of ergodic actions”, Ergod. Th. Dynam. Sys., 33:1 (2013), 81–117  crossref  mathscinet  zmath  isi  elib  scopus
    11. Tikhonov S.V., “Complete metric on mixing actions of general groups”, J. Dyn. Control Syst., 19:1 (2013), 17–31  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. I. Bashtanov, “Generic Mixing Transformations Are Rank $1$”, Math. Notes, 93:2 (2013), 209–216  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. I. Yaroslavtsev, “On the Asymmetry of the Past and the Future of the Ergodic $\mathbb{Z}$-Action”, Math. Notes, 95:3 (2014), 438–440  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. Bashtanov A.I., “Conjugacy Classes Are Dense in the Space of Mixing a"Currency Sign (D) -Actions”, Math. Notes, 99:1-2 (2016), 9–23  mathnet  crossref  mathscinet  zmath  isi  scopus
    15. Adams T.M., Nobel A.B., “Entropy and the uniform mean ergodic theorem for a family of sets”, Trans. Am. Math. Soc., 369:1 (2017), 605–622  crossref  mathscinet  zmath  isi  scopus
    16. Cameron J., Fang J., Mukherjee K., “Mixing and weakly mixing abelian subalgebras of type II1 factors”, J. Funct. Anal., 272:7 (2017), 2697–2725  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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