TY - JOUR

T1 - Pisot family self-affine tilings, discrete spectrum, and the Meyer property

AU - Lee, No matches for: J.-Y.

AU - Solomyak, B.

PY - 2010

Y1 - 2010

N2 - 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 ϕ for the tiling. Assuming that ϕ is diagonalizable over $\C$ 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 ϕ is a "Pisot family". Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.

AB - 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 ϕ for the tiling. Assuming that ϕ is diagonalizable over $\C$ 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 ϕ is a "Pisot family". Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.

UR - http://arxiv.org/abs/1002.0039

M3 - Article

SN - 1078-0947

VL - 32

SP - 935

EP - 959

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

ER -