On the characterization of expansion maps for self-affine tilings

TL;DR

This paper investigates which matrices can serve as expansion maps for self-affine tilings in Euclidean space, extending known one- and two-dimensional results to higher dimensions under certain conditions.

Contribution

It establishes a necessary condition for diagonalizable matrices to be expansion maps of self-affine tilings in any dimension, proposing a conjecture on sufficiency.

Findings

Necessary condition for diagonalizable matrices to be expansion maps

Extension of Perron number conditions to higher dimensions

Conjecture on sufficiency of the condition for existence of tilings

Abstract

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.