Tiling spaces are covering spaces over irrational tori

Abstract

We study tiling spaces in the diffeological context. We prove some basic diffeological properties for tiling spaces and analyze two different fiber bundle structures of tiling spaces over irrational tori. We use the diffeological classification of irrational tori which captures their arithmetical escence in order to inherit the diffeological equivalence in the context of one-dimensional tiling spaces.