On the Characterization of Expansion Maps for Self-Affine Tilings |
| |
Authors: | Richard Kenyon Boris Solomyak |
| |
Institution: | 1. Department of Mathematics, Brown University, Providence, RI, 02912, USA 2. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA
|
| |
Abstract: | We consider self-affine tilings in ℝ
n
with expansion matrix φ and address the question which matrices φ can arise this way. In one dimension, λ is an expansion factor of a self-affine tiling if and only if |λ| is a Perron number, by a result of Lind. In two dimensions, when φ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston:
if a complex λ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for
φ to be an expansion matrix for any n, assuming only that φ is diagonalizable over ℂ. We conjecture that this condition on φ is also sufficient for the existence of a self-affine tiling. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|