Topology of substitutions
by
Marcy Barge
Montana State University
Coauthors: Bev Diamond (College of Charleston), Charles Holton (U. C. Berkeley)

A substitution is a map s from a finite set A (called the alphabet) into the set of finite, nonempty, words with letters in A. Iteration of s produces the language L(s) of allowable finite words - those that appear as factors of words obtained by iterating s repeatedly on some letter - and a collection W(s) of allowable biinfinite words - those whose finite factors all lie in L(s). Under mild assumptions (primitivity and non-periodicity of s), W(s) is a Cantor set and the shift homeomorphism on W(s) has very nice properties: it's minimal and uniquely ergodic. There is also a natural map s' on W(s) induced by s. This map is injective but not surjective, and the latter defect has been a serious inconvenience to researchers in substitutive dynamics. Through the process of suspension (with respect to the shift), W(s) can be embedded in a continuum T(s) (this is the tiling space associated with s) and the map s' extended to a homeomorphism s^'' of T(s) (s^''s is called inflation and substitution). We will show how consideration of this suspension, and the ``desubstitution'' made possible by the invertibility of s^'', leads to the resolution of two rather different seeming problems: the branching problem in the language L(s); and the topological classifcation of hyperbolic one-dimensional attractors.

Date received: February 6, 2001