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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2012, Volume 13, Issue 2, Pages 237–272 (Mi adm76)

RESEARCH ARTICLE

The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters

R. I. Grigorchuka, Ya. S. Krylyukb

a Department of Mathematics, Mailstop 3368 Texas A&M University College Station, TX 77843-3368, USA
b Mathematics Department, De Anza College, 21250 Stevens Creek Blvd, Cupertino, CA 95014, USA

Abstract: It is shown that the KNS-spectral measure of the typical Schreier graph of the action of $3$-generated $2$-group of intermediate growth constructed by the first author in 1980 on the boundary of binary rooted tree coincides with the Kesten’s spectral measure, and coincides (up to affine transformation of $\mathbb R$) with the density of states of the corresponding diatomic linear chain.
Jacoby matrix associated with Markov operator of simple random walk on these graphs is computed. It shown shown that KNS and Kesten's spectral measures of the Schreier graph based on the orbit of the point $1^{\infty}$ are different but have the same support and are absolutely continuous with respect to the Lebesgue measure.

Keywords: group of intermediate growth, diatomic linear chain, random walk, spectral measure, Schreier graph, discrete Laplacian.

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MSC: 20F, 20P, 37A, 60J, 82D
Accepted:03.04.2012
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Citation: R. I. Grigorchuk, Ya. S. Krylyuk, “The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters”, Algebra Discrete Math., 13:2 (2012), 237–272

Citation in format AMSBIB
\Bibitem{GriKry12} \by R.~I.~Grigorchuk, Ya.~S.~Krylyuk \paper The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters \jour Algebra Discrete Math. \yr 2012 \vol 13 \issue 2 \pages 237--272 \mathnet{http://mi.mathnet.ru/adm76} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3027509} \zmath{https://zbmath.org/?q=an:06120573} 

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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. Dudko, R. Grigorchuk, “On spectra of Koopman, groupoid and quasi-regular representations”, J. Mod. Dyn., 11 (2017), 99–123
2. A. Dudko, R. Grigorchuk, “On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. A. Katok, Y. Pesin, F. Hertz, Amer. Math. Soc., 2017, 51–66
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