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Mat. Sb., 2004, Volume 195, Number 12, Pages 95–108 (Mi msb867)  

This article is cited in 17 scientific papers (total in 17 papers)

Non-unique inclusion in a flow and vast centralizer of a generic measure-preserving transformation

A. M. Stepin, A. M. Eremenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The problem of the inclusion in a flow is considered for a measure-preserving transformation. It is shown that if a transformation $T$ has a simple spectrum, then the set of flows including $T$ – provided that it is not empty – consists either of a unique element or of infinitely many spectrally non-equivalent flows.
It is proved that, generically, inclusions in a flow are maximally non-unique in the following sense: the centralizer of a generic transformation contains a subgroup isomorphic to an infinite-dimensional torus. The corresponding proof is based on the so-called dynamical alternative, a topological analogue of Fubini's theorem, a fundamental fact from descriptive set theory about the almost openness of analytic sets, and Dougherty's lemma describing conditions ensuring that the image of a separable metric space is a second-category set.


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English version:
Sbornik: Mathematics, 2004, 195:12, 1795–1808

Bibliographic databases:

UDC: 517.987.5+938.5
MSC: 37A05, 37A10
Received: 17.06.2004

Citation: A. M. Stepin, A. M. Eremenko, “Non-unique inclusion in a flow and vast centralizer of a generic measure-preserving transformation”, Mat. Sb., 195:12 (2004), 95–108; Sb. Math., 195:12 (2004), 1795–1808

Citation in format AMSBIB
\by A.~M.~Stepin, A.~M.~Eremenko
\paper Non-unique inclusion in a~flow and vast centralizer of a~generic measure-preserving transformation
\jour Mat. Sb.
\yr 2004
\vol 195
\issue 12
\pages 95--108
\jour Sb. Math.
\yr 2004
\vol 195
\issue 12
\pages 1795--1808

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    This publication is cited in the following articles:
    1. V. V. Ryzhikov, A. E. Troitskaya, “Tensor roots of isomorphisms and weak limits of transformations”, Math. Notes, 80:4 (2006), 563–566  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. V. Ryzhikov, “Factors, rank, and embedding of a generic $\mathbb Z^n$-action in an $\mathbb R^n$-flow”, Russian Math. Surveys, 61:4 (2006), 786–787  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. S. V. Tikhonov, “Embedding lattice actions in flows with multidimensional time”, Sb. Math., 197:1 (2006), 95–126  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. S. V. Tikhonov, “A complete metric in the set of mixing transformations”, Sb. Math., 198:4 (2007), 575–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. 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
    6. Frączek K., Lemańczyk M., “On the self-similarity problem for ergodic flows”, Proc. Lond. Math. Soc. (3), 99:3 (2009), 658–696  crossref  mathscinet  zmath  isi  elib
    7. Eisner T., “Embedding operators into strongly continuous semigroups”, Arch. Math. (Basel), 92:5 (2009), 451–460  crossref  mathscinet  zmath  isi
    8. Danilenko A.I., Ryzhikov V.V., “On self-similarities of ergodic flows”, Proc. London Math. Soc., 104:3 (2012), 431–454  crossref  mathscinet  zmath  isi  elib
    9. S. V. Tikhonov, “Bernoulli shifts and local density property”, Moscow University Mathematics Bulletin, 67:1 (2012), 29–35  mathnet  crossref  mathscinet
    10. A. I. Bashtanov, “Generic Mixing Transformations Are Rank $1$”, Math. Notes, 93:2 (2013), 209–216  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. Melleray J., Tsankov T., “Generic Representations of Abelian Groups and Extreme Amenability”, Isr. J. Math., 198:1 (2013), 129–167  crossref  mathscinet  zmath  isi
    12. S. Solecki, “Closed subgroups generated by generic measure automorphisms”, Ergod. Th. Dynam. Sys., 34:3 (2014), 1011–1017  crossref  mathscinet  zmath  isi
    13. J. Melleray, “Extensions of generic measure-preserving actions”, Ann. Inst. Fourier, 64:2 (2014), 607–623  crossref  mathscinet  zmath  isi
    14. Hill A., “Centralizers of Rank-One Homeomorphisms”, Ergod. Theory Dyn. Syst., 34:2 (2014), 543–556  crossref  mathscinet  zmath  isi
    15. Kulaga-Przymus J., “on Embeddability of Automorphisms Into Measurable Flows From the Point of View of Self-Joining Properties”, Fundam. Math., 230:1 (2015), 15–76  crossref  mathscinet  zmath  isi  elib
    16. Sergey M. Saulin, Dmitry V. Treschev, “On the Inclusion of a Map Into a Flow”, Regul. Chaotic Dyn., 21:5 (2016), 538–547  mathnet  crossref  mathscinet  zmath  elib
    17. Gao S., Hill A., “Topological isomorphism for rank-1 systems”, J. Anal. Math., 128 (2016), 1–49  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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