Abstract

Roughly, a Delone set is the set of vertices of a tiling using pieces that "do not degenerate" in form. Formally, it is a uniformly separated and coarsely dense subset of the plane.
A natural question raised by Gromov and Furstenberg was answered in the negative by Burago-Kleiner and McMullen: there exist Delone sets that are not bi-Lipschitz equivalent to the standard lattice. In this talk, we will show that such sets can be even made "repetitive", which means that they are the vertices of a quasi-periodic tiling. Nevertheless, we will see that this cannot be the case for "Isfahan like tilings" (as the Penrose one): for all of these, there are even bi-Lipschitz homeomorphisms of the plane sending the Delone set into the standard lattice.



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