  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 ${\mathbb
R}^d$. 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.
