Rigidity and mapping class group for abstract tiling spaces

Abstract

AbstractWe study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝdon a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝd. In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.