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

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

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English version:
Sbornik: Mathematics, 2009, 200:12, 1833–1845

Bibliographic databases:

UDC: 517.987
MSC: 37A25

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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• https://doi.org/10.4213/sm7597
• http://mi.mathnet.ru/eng/msb/v200/i12/p107

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

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
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
3. Danilenko A.I., “On New Spectral Multiplicities for Ergodic Maps”, Studia Math., 197:1 (2010), 57–68
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
5. Tikhonov S.V., “Homogeneous spectrum and mixing transformations”, Dokl. Math., 83:1 (2011), 80–83
6. S. V. Tikhonov, “Mixing transformations with homogeneous spectrum”, Sb. Math., 202:8 (2011), 1231–1252
7. V. V. Ryzhikov, “Spectral multiplicity for powers of weakly mixing automorphisms”, Sb. Math., 203:7 (2012), 1065–1076
8. Danilenko A.I., “New spectral multiplicities for mixing transformations”, Ergod. Theory Dyn. Syst., 32:2 (2012), 517–534
9. Kulaga-Przymus J. Parreau F., “Disjointness properties for Cartesian products of weakly mixing systems”, Colloq. Math., 128:2 (2012), 153–177
10. Solomko A.V., “New spectral multiplicities for ergodic actions”, Studia Math., 208:3 (2012), 229–247
11. Danilenko A.I., “A survey on spectral multiplicities of ergodic actions”, Ergod. Th. Dynam. Sys., 33:1 (2013), 81–117
12. Danilenko A.I., Lemanczyk M., “Spectral Multiplicities for Ergodic Flows”, Discret. Contin. Dyn. Syst., 33:9 (2013), 4271–4289
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
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