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March 2012, 32(3): 935-959. doi: 10.3934/dcds.2012.32.935
Pisot family self-affine tilings, discrete spectrum, and the Meyer property
Jeong-Yup Lee 1, and Boris Solomyak 2,
Dept. of Math. Edu., Kwandong University, 522 Naegok-dong, Gangneung, Gangwon 210-701, South Korea
Box 354350, Department of Mathematics, University of Washington, Seattle WA 98195, United States
Received September 2010 Revised March 2011 Published October 2011
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.
Keywords: no weakly mixing, Meyer sets., Pisot family, discrete spectrum, self-affine tilings.
Mathematics Subject Classification: Primary: 37B50; Secondary: 52C2.
Citation: Jeong-Yup Lee, Boris Solomyak. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 935-959. doi: 10.3934/dcds.2012.32.935
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