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 Mat. Sb., 2012, Volume 203, Number 7, Pages 149–160 (Mi msb7895)

Spectral multiplicity for powers of weakly mixing automorphisms

V. V. Ryzhikov

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

Abstract: We study the behaviour of the maximal spectral multiplicity $\mathfrak m(R^n)$ for the powers of a weakly mixing automorphism $R$. For some particular infinite sets $A$ we show that there exists a weakly mixing rank-one automorphism $R$ such that $\mathfrak m(R^n)=n$ and $\mathfrak m(R^{n+1})=1$ for all positive integers $n\in A$. Moreover, the cardinality $\operatorname{cardm}(R^n)$ of the set of spectral multiplicities for the power $R^n$ is shown to satisfy the conditions $\operatorname{cardm}(R^{n+1})=1$ and $\operatorname{cardm}(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: all powers $R^{n}$ have homogeneous spectra and the set of limit points of the sequence $\{\mathfrak m(R^n)/n:n\in \mathbb N \}$ is infinite.
Bibliography: 17 titles.

Keywords: weakly mixing transformation, homogeneous spectrum, maximal spectral multiplicity.

DOI: https://doi.org/10.4213/sm7895

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English version:
Sbornik: Mathematics, 2012, 203:7, 1065–1076

Bibliographic databases:

UDC: 517.987
MSC: Primary 37A30; Secondary 47A35, 28D05
Received: 03.06.2011 and 04.02.2012

Citation: V. V. Ryzhikov, “Spectral multiplicity for powers of weakly mixing automorphisms”, Mat. Sb., 203:7 (2012), 149–160; Sb. Math., 203:7 (2012), 1065–1076

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7895
• https://doi.org/10.4213/sm7895
• http://mi.mathnet.ru/eng/msb/v203/i7/p149

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This publication is cited in the following articles:
1. V. V. Ryzhikov, “Bounded ergodic constructions, disjointness, and weak limits of powers”, Trans. Moscow Math. Soc., 74 (2013), 165–171
2. El Abdalaoui, El Houcein, M. Lemańczyk M., Th. de la Rue, “On spectral disjointness of powers for rank-one transformations and Möbius orthogonality”, J. Funct. Anal., 266:1 (2014), 284–317
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