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

This article is cited in 2 scientific papers (total in 2 papers)

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

Full text: PDF file (498 kB)
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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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\paper Spectral multiplicity for powers of weakly mixing automorphisms
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\vol 203
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\pages 149--160
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\transl
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\pages 1065--1076
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866281941}


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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. V. V. Ryzhikov, “Bounded ergodic constructions, disjointness, and weak limits of powers”, Trans. Moscow Math. Soc., 74 (2013), 165–171  mathnet  crossref  mathscinet  zmath  elib
    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  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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