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 Mat. Zametki, 2013, Volume 93, Issue 2, Pages 163–171 (Mi mz10157)

Generic Mixing Transformations Are Rank $1$

A. I. Bashtanov

M. V. Lomonosov Moscow State University

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 $T$ 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 $T$. As a corollary, we find that a generic mixing transformation is rank $1$.

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

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

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English version:
Mathematical Notes, 2013, 93:2, 209–216

Bibliographic databases:

UDC: 517.987.5+938.5
Revised: 09.10.2012

Citation: A. I. Bashtanov, “Generic Mixing Transformations Are Rank $1$”, Mat. Zametki, 93:2 (2013), 163–171; Math. Notes, 93:2 (2013), 209–216

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10157
• https://doi.org/10.4213/mzm10157
• http://mi.mathnet.ru/eng/mz/v93/i2/p163

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. I. Yaroslavtsev, “On the Asymmetry of the Past and the Future of the Ergodic $\mathbb{Z}$-Action”, Math. Notes, 95:3 (2014), 438–440
2. V. V. Ryzhikov, “Ergodic homoclinic groups, Sidon constructions and Poisson suspensions”, Trans. Moscow Math. Soc., 75 (2014), 77–85
3. Bashtanov A.I., “Conjugacy Classes Are Dense in the Space of Mixing $\mathbb{Z}^d$-Actions”, Math. Notes, 99:1-2 (2016), 9–23
4. 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
5. V. V. Ryzhikov, “Thouvenot's Isomorphism Problem for Tensor Powers of Ergodic Flows”, Math. Notes, 104:6 (2018), 900–904
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