Alex Clark and Lorenzo Sadun
When Shape Matters: Deformations of Tiling Spaces
(170K, LaTeX with four embedded postscript figures)
ABSTRACT. We investigate the dynamics of tiling dynamical systems and their
deformations. If two tiling systems have identical combinatorics,
then the tiling spaces are homeomorphic, but their dynamical
properties may differ. There is a natural map $\mathcal I$ from the parameter space of possible shapes of tiles to $H^1$ of a model tiling space, with values in $\R^d$. Two tiling spaces that have the same image
under $\mathcal I$ are mutually locally derivable (MLD). When the
difference of the images is ``asymptotically negligible'', then the
tiling dynamics are topologically conjugate, but generally not MLD. For
substitution tilings, we give a simple test for a cohomology class to
be asymptotically negligible, and show that infinitesimal deformations
of shape result in topologically conjugate dynamics only when the
change in the image of $\mathcal I$ is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be
topologically weakly mixing.