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

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

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

Bibliographic databases:

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

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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• http://mi.mathnet.ru/eng/msb3743
• https://doi.org/10.4213/sm3743
• http://mi.mathnet.ru/eng/msb/v198/i4/p135

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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
2. V. V. Ryzhikov, “Spectral multiplicities and asymptotic operator properties of actions with invariant measure”, Sb. Math., 200:12 (2009), 1833–1845
3. Tikhonov S.V., “Homogeneous spectrum and mixing transformations”, Dokl. Math., 83:1 (2011), 80–83
4. S. V. Tikhonov, “Mixing transformations with homogeneous spectrum”, Sb. Math., 202:8 (2011), 1231–1252
5. S. V. Tikhonov, “A Note on Rochlin's Property in the Space of Mixing Transformations”, Math. Notes, 90:6 (2011), 925–926
6. Danilenko A.I., “New spectral multiplicities for mixing transformations”, Ergod. Th. Dynam. Sys., 32:2 (2012), 517–534
7. S. V. Tikhonov, “Genericity of a multiple mixing”, Russian Math. Surveys, 67:4 (2012), 779–780
8. S. V. Tikhonov, “Bernoulli shifts and local density property”, Moscow University Mathematics Bulletin, 67:1 (2012), 29–35
9. Solomko A.V., “New spectral multiplicities for ergodic actions”, Studia Math., 208:3 (2012), 229–247
10. Danilenko A.I., “A survey on spectral multiplicities of ergodic actions”, Ergod. Th. Dynam. Sys., 33:1 (2013), 81–117
11. Tikhonov S.V., “Complete metric on mixing actions of general groups”, J. Dyn. Control Syst., 19:1 (2013), 17–31
12. A. I. Bashtanov, “Generic Mixing Transformations Are Rank $1$”, Math. Notes, 93:2 (2013), 209–216
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
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
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
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
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