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

TL;DR

This paper explores the spectral properties of self-affine tilings in Euclidean space, establishing a link between algebraic eigenvalues, discrete spectrum, and the Meyer property, under certain algebraic conditions.

Contribution

It demonstrates that the discrete spectrum of the tiling dynamical system is characterized by Pisot families of eigenvalues and relates this to the Meyer property of control points.

Findings

Discrete spectrum exists iff the eigenvalues form a Pisot family.

System is not weakly mixing iff it has a relatively dense discrete spectrum.

Meyer property of control points is equivalent to the spectral condition.

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 $\C$ 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.