Cocycles on Tiling Spaces

The space of tilings by rectangles form an interesting collection of multidimensional subshifts. Here, we study cocycles on such tiling spaces and provide an algorithmically decidable necessary and sufficient condition for the tiling space associated with a collection of rectangles to admit a nontrivial cocycle. Along the way, we also prove that the tiling spaces corresponding to a class of rectangular tile sets satisfy certain mixing properties.