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It is well known that the moduli space of flat connections on a trivialprincipal bundle MxG, where G is a connected Lie group, is isomorphic to therepresentation variety Hom(\pi_1(M), G)/G. For a tiling T, viewed as a markedcopy of R^d, we define a new kind of bundle called pattern equivariant bundleover T and consider the set of all such bundles. This is a topologicalinvariant of the tiling space induced by T, which we call PREP(T), and we showthat it is isomorphic to the direct limit lim_{f_n} Hom(\pi_1(\Gamma_n), G)/G,where \Gamma_n are the approximants to the tiling space and f_n are mapsbetween them. G can be any group. As an example, we choose G to be thesymmetric group S_3 and we calculate this direct limit for the Period Doublingtiling and its double cover, the Thue-Morse tiling, obtaining differentresults. This is the simplest topological invariant that can distinguish thesetwo examples.

Pattern Equivariant Representation Variety of Tiling Spaces for Any Group G