On the characterization of expansion maps for self-affine tilings

Details On the characterization of expansion maps for self-affine tilings On the characterization of expansion maps for self-affine tilings

2008-01-13

arxiv.org/abs/0801.1993v2

Discrete Comput. Geom. (2010) 43: 577-593

Mathematics

Metric Geometry

Revised version. A typo corrected (after publication!) in the definition of the set $\Omega$ at the bottom of p.13

Scientific paper

We consider self-affine tilings in $\R^n$ with expansion matrix $\phi$ and address the question which matrices $\phi$ can arise this way. In one dimension, $\lambda$ is an expansion factor of a self-affine tiling if and only if $|\lambda|$ is a Perron number, by a result of Lind. In two dimensions, when $\phi$ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex $\lambda$ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for $\phi$ to be an expansion matrix for any $n$, assuming only that $\phi$ is diagonalizable over the complex numbers. We conjecture that this condition on $\phi$ is also sufficient for the existence of a self-affine tiling.

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