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 Mat. Zametki, 2000, Volume 68, Issue 5, Pages 643–647 (Mi mz985)

On the Spectrum of Cartesian Powers of Classical Automorphisms

O. N. Ageev

N. E. Bauman Moscow State Technical University

Abstract: We prove the following statement: the set of all essential spectral multiplicities of $T^{(n)}=T\times…\times T$($n$ times) is $\{n,n(n-1),…,n!\}$ on $\{\operatorname{const}\}^\perp$ for Chacon transformations $T$, or, equivalently, the operator $T^{(n)}$ has a simple spectrum on the subspace $C_{\operatorname{sim}}$ of all functions that are invariant with respect to permutations of the coordinates. As an immediate consequence of this fact, we have the disjointness of all convolution powers of the spectral measure for Chacon transformations. If $n=2$, then $T\times T$ has a homogeneous spectrum of multiplicity 2 on $\{\operatorname{const}\}^\perp$, i.e., this is a solution of Rokhlin"s problem for Chacon transformations. Similar statements are considered for other classical automorphisms.

DOI: https://doi.org/10.4213/mzm985

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English version:
Mathematical Notes, 2000, 68:5, 547–551

Bibliographic databases:

UDC: 517.9

Citation: O. N. Ageev, “On the Spectrum of Cartesian Powers of Classical Automorphisms”, Mat. Zametki, 68:5 (2000), 643–647; Math. Notes, 68:5 (2000), 547–551

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz/v68/i5/p643

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This publication is cited in the following articles:
1. D. V. Anosov, “Spectral Multiplicity in Ergodic Theory”, Proc. Steklov Inst. Math., 290, suppl. 1 (2015), 1–44
2. Ageev, ON, “On asymmetry of the future and the past for limit self-joinings”, Proceedings of the American Mathematical Society, 131:7 (2003), 2053
3. Danilenko, AI, “Explicit solution of Rokhlin's problem on homogeneous spectrum and applications”, Ergodic Theory and Dynamical Systems, 26 (2006), 1467
4. Ageev, O, “Mixing with staircase multiplicity functions”, Ergodic Theory and Dynamical Systems, 28 (2008), 1687
5. Danilenko A.I., “(C, F)-Actions in Ergodic Theory”, In Memory of Alexander Reznikov, Progress in Mathematics, 265, eds. Kapranov M., Kolyada S., Manin Y., Moree P., Potyagailo L., Birkhauser Verlag Ag, 2008, 325–351
6. Roy, E, “Poisson suspensions and infinite ergodic theory”, Ergodic Theory and Dynamical Systems, 29 (2009), 667
7. Katok A., Lemanczyk M., “Some new cases of realization of spectral multiplicity function for ergodic transformations”, Fundamenta Mathematicae, 206 (2009), 185–215
8. Lemanczyk M., Parreau F., Roy E., “Joining Primeness and Disjointness From Infinitely Divisible Systems”, Proc. Amer. Math. Soc., 139:1 (2011), 185–199
9. Kulaga-Przymus J., Parreau F., “Disjointness Properties for Cartesian Products of Weakly Mixing Systems”, Colloq. Math., 128:2 (2012), 153–177
10. Danilenko A.I., “A Survey on Spectral Multiplicities of Ergodic Actions”, Ergod. Theory Dyn. Syst., 33:Part 1 (2013), 81–117
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