Pattern-equivariant cohomology theory was developed by Ian Putnam
and Johannes Kellendonk in 2003, for tilings whose tiles appear in
fixed orientations. In this dissertation, we generalize this theory
in two ways: first, we define this cohomology to apply to tiling
spaces, rather than individual tilings. Second, we allow tilings
with tiles appearing in multiple orientations - possibly infinitely
many. Along the way, we prove an approximation theorem, which has
use beyond pattern-equivariant cohomology. This theorem states that
a function which is a topological conjugacy can be approximated
arbitrarily closely by a function which preserves the local
structure of a tiling space. The approximation theorem is limited
to translationally finite tilings, and we conjecture that it is not
true in the infinite case.
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