Abstract tiling actions, expansiveness and local structure

Abstract

A significant amount of literature is devoted to the study of the dynamical properties of the translation actions associated to self-affine tilings or Delone sets. A natural step is to axiomatize these essential properties among all Rd-actions on compact metric spaces. We propose a set of dynamical axioms of such an action which yields a topological conjugacy between the Rd-action and the translation action associated to a self-affine repetitive aperiodic tiling. In particular, we show that these axioms admit an expanding metric on the local cross-section of the phase space, which implies the existence of a local cross-section that is a Cantor set. We also investigate an interesting example of a tiling space that contains non-FLC tilings, which exhibits an unusually complicated local structure.