Rigidity and mapping class group for abstract tiling spaces (Q2884096)

The author proposes the concept of an abstract self-affine tiling action as a way of axiomatizing the essential properties of the translation actions associated to self-affine tilings or Delone sets. Explicitly, an abstract self-affine tiling action is an action of \(\mathbb{R}^d\) on a compact metric space \((X,d)\) satisfying four properties called local freedom, weak expansivity, phase stability and the existence of an expanding linear map. Such objects include the translation actions on the compact spaces associated to aperiodic repetitive tilings and Delone sets in \(\mathbb{R}^d \). An intrinsic characterization of such actions among all \(\mathbb{R}^d\)-actions on a compact space is given. Using this new formalism, the author proves a topological rigidity result for tiling actions of arbitrary rank.NEWLINENEWLINEIn the self-similar case, the author shows a translational rigidity result, proving that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. The concepts of the mapping class group of a tiling space and of a general linear group of a tiling are introduced, and it is shown that these two groups are naturally isomorphic and are discrete. The author provides an explicit example of computation of such groups for the five-fold symmetric Penrose tiling in an appendix.