Deformations of substitution tiling spaces
by
Lorenzo Sadun
University of Texas at Austin

R.F. Williams and I proved that any tiling space (of finite local complexity w.r.t. translations) is homeomorphic to a Cantor set bundle over a torus. The homeomorphism involves deforming the shapes and sizes of the tiles, without changing the combinatorial structure of the tilings. The question is how such deformations affect the translational dynamics of the tiling space.

Alex Clark and I previously solved this problem for substitution tilings in one dimension. In this talk, I extend these results to substitution tilings in arbitrary dimensions. The space of deformations is naturally associated with the first Cech cohomology of the tiling space with values in R^d. This cohomology can be decomposed into eigenspaces of the substitution operator. The effect of each eigenspace depends only on the magnitude of the eigenvalue: The Perron-Frobenius eigenspace corresponds to linear transformations of R^d, other eigenspaces with eigenvalue of magnitude 1 or greater fundamentally change the dynamics, while eigenspaces with eigenvalue less than one have no effect.

Date received: February 25, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajz-61.