Pisot family self-affine tilings, discrete spectrum, and the Meyer property

Jeong-Yup Lee; Boris Solomyak

doi:10.3934/dcds.2012.32.935

Article Contents

2012, Volume 32, Issue 3: 935-959. Doi: 10.3934/dcds.2012.32.935

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Pisot family self-affine tilings, discrete spectrum, and the Meyer property

Jeong-Yup Lee^1, and
Boris Solomyak^2,

1.
Dept. of Math. Edu., Kwandong University, 522 Naegok-dong, Gangneung, Gangwon 210-701
2.
Box 354350, Department of Mathematics, University of Washington, Seattle WA 98195

Received: August 31, 2010

Revised: February 28, 2011

Published: February 2012

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Abstract

We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $\phi$ for the tiling. Assuming that $\phi$ is diagonalizable over $\mathbb{C}$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of $\phi$ is a "Pisot family." Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.

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Mathematics Subject Classification: Primary: 37B50; Secondary: 52C23.

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