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

This article is cited in 5 scientific papers (total in 5 papers)

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
Received: 13.06.2012
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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    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. V. V. Ryzhikov, “Ergodic homoclinic groups, Sidon constructions and Poisson suspensions”, Trans. Moscow Math. Soc., 75 (2014), 77–85  mathnet  crossref  elib
    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  mathnet  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    5. V. V. Ryzhikov, “Thouvenot's Isomorphism Problem for Tensor Powers of Ergodic Flows”, Math. Notes, 104:6 (2018), 900–904  mathnet  crossref  crossref  isi  elib
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