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DCDS

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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