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 Mat. Sb. (N.S.), 1988, Volume 136(178), Number 3(7), Pages 307–319 (Mi msb1743)

Dynamical systems with an even-mulriplicity Lebesgue component in the spectrum

O. N. Ageev

Abstract: A general construction of ergodic transformations with Lebesgue component of finite multiplicity is proposed. All known examples with this property can be encompassed within the proposed construction. The spectral and combinatorial properties of the transformations are studied. It is shown that the construction permits one to obtain a continuum of spectrally nonisomorphic transformations with even-multiplicity Lebesgue component. As a rule, the transformations have a continuous spectrum. It is proved that continuum many metrically nonisomorphic transformations having the same spectrum are contained in the proposed class. Proof of all the results uses a combinatorial and approximation technique.
Figures: 4.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:2, 305–317

Bibliographic databases:

UDC: 517.987.5
MSC: 28D05

Citation: O. N. Ageev, “Dynamical systems with an even-mulriplicity Lebesgue component in the spectrum”, Mat. Sb. (N.S.), 136(178):3(7) (1988), 307–319; Math. USSR-Sb., 64:2 (1989), 305–317

Citation in format AMSBIB
\Bibitem{Age88} \by O.~N.~Ageev \paper Dynamical systems with an even-mulriplicity Lebesgue component in the spectrum \jour Mat. Sb. (N.S.) \yr 1988 \vol 136(178) \issue 3(7) \pages 307--319 \mathnet{http://mi.mathnet.ru/msb1743} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=959483} \zmath{https://zbmath.org/?q=an:0695.28009} \transl \jour Math. USSR-Sb. \yr 1989 \vol 64 \issue 2 \pages 305--317 \crossref{https://doi.org/10.1070/SM1989v064n02ABEH003309} 

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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. Ferenczi S., Kwiatkowski J., “Rank and Spectral Multiplicity”, Studia Math., 102:2 (1992), 121–144
2. Goodson G., Kwiatkowski J., Lemanczyk M., Liardet P., “On the Multiplicity Function of Ergodic Group Extensions of Rotations”, Studia Math., 102:2 (1992), 157–174
3. O. N. Ageev, “Mixing in the components and rearrangements of $T_{\alpha,\beta}$”, Russian Math. Surveys, 49:2 (1994), 141–142
4. Kwiatkowski J., Lemanczyk M., “On the Multiplicity Function of Ergodic Group Extensions .2.”, Studia Math., 116:3 (1995), 207–215
5. Fraczek K., “Cyclic Space Isomorphism of Unitary Operators”, Studia Math., 124:3 (1997), 259–267
6. Filipowicz I., “Product Zeta(D)-Actions on a Lebesgue Space and their Applications”, Studia Math., 122:3 (1997), 289–298
7. O. N. Ageev, “The spectral multiplicity function and geometric representations of interval exchange transformations”, Sb. Math., 190:1 (1999), 1–28
8. Oleg Ageev, “The homogeneous spectrum problem in ergodic theory”, Invent math, 160:2 (2005), 417
9. O. N. Ageev, “Nonsingular α-rigid maps”, J Dyn Control Syst, 2009
10. E. H. Abdalaoui, “On the spectrum of α-rigid maps”, J Dyn Control Syst, 2009
11. E. H. El Abdalaoui, M. Lemańczyk, “Approximate transitivity property and Lebesgue spectrum”, Mh Math, 2010
12. E. H. Abdalaoui, M. Lemańczyk, “Approximately transitive dynamical systems and simple spectrum”, Arch. Math, 2011
13. ALEXANDRE I. DANILENKO, “A survey on spectral multiplicities of ergodic actions”, Ergod. Th. Dynam. Sys, 2011, 1
14. V. V. Ryzhikov, “Spectral multiplicity for powers of weakly mixing automorphisms”, Sb. Math., 203:7 (2012), 1065–1076
15. E. H. EL ABDALAOUI, M. G. NADKARNI, “A non-singular transformation whose spectrum has Lebesgue component of multiplicity one”, Ergod. Th. Dynam. Sys, 2014, 1
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