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Mat. Sb., 2009, Volume 200, Number 12, Pages 107–120 (Mi msb7597)  

This article is cited in 13 scientific papers (total in 13 papers)

Spectral multiplicities and asymptotic operator properties of actions with invariant measure

V. V. Ryzhikov

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

Abstract: New sets of spectral multiplicities of ergodic automorphisms of a probability space are proposed. Realizations have been obtained, inter alia, for the sets of multiplicities $\{p,q,pq\}$, $\{p,q,r,pq,pr,rq,pqr\}$ and so on. It is also shown that systems with homogeneous spectrum may have factors over which they form a finite extension. Moreover, these systems feature arbitrary polynomial limits, and thus may serve as useful elements in constructions. A so-called minimal calculus of multiplicities is proposed. Some infinite sets of multiplicities occurring in tensor products are calculated, which involve a Gaussian or a Poisson multiplier. Spectral multiplicities are also considered in the class of mixing actions.
Bibliography: 25 titles.

Keywords: measure preserving action, homogeneous spectrum, spectral multiplicity, weak closure of a subaction.

DOI: https://doi.org/10.4213/sm7597

Full text: PDF file (528 kB)
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English version:
Sbornik: Mathematics, 2009, 200:12, 1833–1845

Bibliographic databases:

UDC: 517.987
MSC: 37A25
Received: 25.06.2009 and 11.09.2009

Citation: V. V. Ryzhikov, “Spectral multiplicities and asymptotic operator properties of actions with invariant measure”, Mat. Sb., 200:12 (2009), 107–120; Sb. Math., 200:12 (2009), 1833–1845

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. A. I. Danilenko, V. V. Ryzhikov, “Spectral Multiplicities of Infinite Measure Preserving Transformations”, Funct. Anal. Appl., 44:3 (2010), 161–170  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Danilenko A.I., Solomko A.V., “Ergodic Abelian actions with homogeneous spectrum”, Dynamical numbers—interplay between dynamical systems and number theory, Contemp. Math., 532, 2010, 137–148  crossref  mathscinet  zmath  isi
    3. Danilenko A.I., “On New Spectral Multiplicities for Ergodic Maps”, Studia Math., 197:1 (2010), 57–68  crossref  mathscinet  zmath  isi  elib  scopus
    4. Danilenko A.I., Ryzhikov V.V., “Mixing constructions with infinite invariant measure and spectral multiplicities”, Ergodic Theory Dynam. Systems, 31:3 (2011), 853–873  crossref  mathscinet  zmath  isi  elib  scopus
    5. Tikhonov S.V., “Homogeneous spectrum and mixing transformations”, Dokl. Math., 83:1 (2011), 80–83  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. S. V. Tikhonov, “Mixing transformations with homogeneous spectrum”, Sb. Math., 202:8 (2011), 1231–1252  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. V. V. Ryzhikov, “Spectral multiplicity for powers of weakly mixing automorphisms”, Sb. Math., 203:7 (2012), 1065–1076  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Danilenko A.I., “New spectral multiplicities for mixing transformations”, Ergod. Theory Dyn. Syst., 32:2 (2012), 517–534  crossref  mathscinet  zmath  isi  elib  scopus
    9. Kulaga-Przymus J. Parreau F., “Disjointness properties for Cartesian products of weakly mixing systems”, Colloq. Math., 128:2 (2012), 153–177  crossref  mathscinet  zmath  isi  elib  scopus
    10. Solomko A.V., “New spectral multiplicities for ergodic actions”, Studia Math., 208:3 (2012), 229–247  crossref  mathscinet  zmath  isi  elib  scopus
    11. Danilenko A.I., “A survey on spectral multiplicities of ergodic actions”, Ergod. Th. Dynam. Sys., 33:1 (2013), 81–117  crossref  mathscinet  zmath  isi  elib  scopus
    12. Danilenko A.I., Lemanczyk M., “Spectral Multiplicities for Ergodic Flows”, Discret. Contin. Dyn. Syst., 33:9 (2013), 4271–4289  crossref  mathscinet  zmath  isi  scopus
    13. R. A. Konev, V. V. Ryzhikov, “On the Collection of Spectral Multiplicities $\{2,4,…,2^n\}$ for Totally Ergodic $\mathbb{Z}^2$-Actions”, Math. Notes, 96:3 (2014), 360–368  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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