Substitution Tilings and Separated Nets with Similarities to the Integer Lattice
Metric Geometry
2009-01-18 v2 Dynamical Systems
Abstract
We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.
Keywords
Cite
@article{arxiv.0810.5225,
title = {Substitution Tilings and Separated Nets with Similarities to the Integer Lattice},
author = {Yaar Solomon},
journal= {arXiv preprint arXiv:0810.5225},
year = {2009}
}