Linearly Repetitive Delone Sets

Linearly repetitive Delone sets are the simplest aperiodic repetitive Delone sets of the Euclidean space, e.g. any self similar Delone set is linearly repetitive. We present here some combinatorial, ergodic and mixing properties of their associated dynamical systems. We also give a characterization of such sets via the patch frequencies. Finally, we explain why a linearly repetitive Delone set is the image of a lattice by a bi-Lipschitz map. © 2015 Springer Basel.