Outer Billiards, Arithmetic Graphs and Multigrid Flows

I considered the first construction in my book, Outer Billiards on Kites [S2]. The arithmetic graphs served as the main tool for understanding the dynamics of outer billiards on kites. See §2 for definitions. The second construction, which is quite general, is based on patterns of oriented lines in the Euclidean plane. See §3 for definitions. The concrete instance of the multigrid construction that is related to outer billiards on kites is closely connected to Sturmian sequences . I am grateful to John Smillie for his beautiful explanation of the renormalization theory for Sturmian sequences. I am also grateful to Smillie (and also Mark Sapir) for suggesting to me that my arithmetic graphs might be related to Sturmian sequences. This connection is far from being worked out. I call my main result the Coarse Isomorphism Theorem. I have not yet tried to prove the Coarse Isomorphism Theorem, though I have some idea