A. I. Bashtanov, “Generic Mixing Transformations Are Rank $1$”, Mat. Zametki, 93:2 (2013), 163–171; Math. Notes, 93:2 (2013), 209

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Matematicheskie Zametki, 2013, Volume 93, Issue 2, Pages 163–171
DOI: https://doi.org/10.4213/mzm10157 (Mi mzm10157)

This article is cited in 14 scientific papers (total in 14 papers)

Generic Mixing Transformations Are Rank

1

$1$

A. I. Bashtanov

M. V. Lomonosov Moscow State University

Full-text PDF (487 kB) Citations (14)

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DOI: https://doi.org/10.4213/mzm10157

Abstract: In 2007, S. V. Tikhonov introduced a complete metric on the space of mixing transformations. This metric generates a topology called the leash topology. Tikhonov posed the following problem: what conditions should be satisfied by a mixing transformation

T

$T$ for its conjugacy class to be dense in the space of mixing transformations equipped with the leash topology. We show the conjugacy class to be dense for every mixing transformation

T

$T$ . As a corollary, we find that a generic mixing transformation is rank

1

$1$ .

Keywords: mixing transformation, probability space, conjugacy class, Tikhonov metric, leash topology.

Received: 13.06.2012
Revised: 09.10.2012

English version:
Mathematical Notes, 2013, Volume 93, Issue 2, Pages 209–216
DOI: https://doi.org/10.1134/S0001434613010227

Bibliographic databases:

Document Type: Article

UDC: 517.987.5+938.5

Language: Russian

Citation: A. I. Bashtanov, “Generic Mixing Transformations Are Rank

1

$1$ ”, Mat. Zametki, 93:2 (2013), 163–171; Math. Notes, 93:2 (2013), 209–216

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm10157

https://doi.org/10.4213/mzm10157

https://www.mathnet.ru/eng/mzm/v93/i2/p163

This publication is cited in the following 14 articles:

S. V. Tikhonov, “O peremeshivayuschikh deistviyakh lokalno kompaktnykh grupp”, Matem. zametki, 117:4 (2025), 561–574
V. V. Ryzhikov, “Generic correlations and ergodic averages for strongly and mildly mixing automorphisms”, Math. Notes, 116:3 (2024), 521–526
V. V. Ryzhikov, “Generic extensions of ergodic systems”, Sb. Math., 214:10 (2023), 1442–1457
Terrence Adams, Anthony Quas, Encyclopedia of Complexity and Systems Science Series, Ergodic Theory, 2023, 35
Adam Kanigowski, Mariusz Lemańczyk, Encyclopedia of Complexity and Systems Science Series, Ergodic Theory, 2023, 109
Terrence Adams, Anthony Quas, Encyclopedia of Complexity and Systems Science, 2022, 1
Anatoly Stepin, Sergey Tikhonov, Contemporary Mathematics, 772, Topology, Geometry, and Dynamics, 2021, 311
V. V. Ryzhikov, “Measure-preserving rank one transformations”, Trans. Moscow Math. Soc., 81:2 (2020), 229–259
Adam Kanigowski, Mariusz Lemańczyk, Encyclopedia of Complexity and Systems Science, 2020, 1
Y. Gutman, W. Huang, S. Shao, X. D. Ye, “Almost sure convergence of the multiple ergodic average for certain weakly mixing systems”, Acta. Math. Sin.-English Ser., 34:1 (2018), 79–90
V. V. Ryzhikov, “Thouvenot's Isomorphism Problem for Tensor Powers of Ergodic Flows”, Math. Notes, 104:6 (2018), 900–904
Bashtanov A.I., “Conjugacy Classes Are Dense in the Space of Mixing $\mathbb{Z}^d$ -Actions”, Math. Notes, 99:1-2 (2016), 9–23
I. Yaroslavtsev, “On the Asymmetry of the Past and the Future of the Ergodic $\mathbb{Z}$ -Action”, Math. Notes, 95:3 (2014), 438–440
V. V. Ryzhikov, “Ergodic homoclinic groups, Sidon constructions and Poisson suspensions”, Trans. Moscow Math. Soc., 75 (2014), 77–85

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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