The symbolic dynamics of tiling the integers
by
Ethan M. Coven
Wesleyan University and MSRI
Coauthors: William Geller, Sylvia Silberger, William Thurston

We show that, up to powers of the shift, every mixing shift of finite type can be realized as a tiling system. Specifically, if X is a mixing shift of finite type, then there is a tiling system T and a positive integer m such that T is the union of m subsets T_i which are cyclically permuted by the shift S, and S^m on X is topologically conjugate to S^m on each T_i. An almost immediate corollary is that the sets of topological entropies of tiling systems and of shifts of finite type are the same.

Definitions: a collection of subsets of the integers tiles the integers iff the integers can be expressed as a disjoint union of translates of members of that collection. Considering only finite collections of finite subsets of the integers such that no two members are translates of each other, each tiling of the integers by members of {P} determines a doubly infinite sequence x on alphabet {P} by x_i = P iff i is in P. The set of sequences determined by tilings of the integers by members of a fixed collection is a sofic system, called a tiling system.

Date received: February 23, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabe-07.