This is joint work with W. Geller (IUPUI), N. Jonoska (Univ. South Florida), S. Silberger (Hofstra) and W. Thurston (UCD).
ABSTRACT: A finite collection of finite sets tiles the integers if and only if the integers can be expressed as a disjoint union of translates of members of the collection. We associate with such a tiling a doubly infinite sequence with symbols the sets in the collection. The set of all such sequences is a sofic system, called a tiling system.
For example, if the collection consists of the sets {0} and {0,1}, then the tiling system is the collection of all doubly infinite sequences with symbols R (red, the "color" of {0}) and B (blue, the "color" of {0,1}) such that between any two consecutive appearances of R, there are an even number of B, i.e., the "even system." This sofic system is closely related to the "Golden Mean" shift of finite type. Another such system is {R,B_B}, corresponding to the sets {0} and {0,2}. Many shifts of finite type (one of the basic classes of examples in symbolic dynamics - they, "occur in nature"), e.g., the full 2-shift, cannot be realized (up to topological conjugacy) as tiling systems. However we show that FOR EVERY SHIFT OF FINITE TYPE, SOME POWER CAN BE REALIZED AS A TILING SYSTEM.