New Techniques for Classifying Williams Solenoids

Abstract

In this paper we compare topological methods and dynamical methods corresponding to recent developments in the classification of inverse limit spaces of one dimensional maps on graphs. We elucidate the role of shifting the periodic base point in applying Williams' theory. We exploit the Fox calculus to define and apply the Bowen-Franks trace---a shift equivalence invariant of free group homomorphisms. We show that augmented cohomology of certain suspensions associated with wrapping rules in a substitution yields augmented dimension groups that have a relatively simple product structure. We complete the classification of a family of examples of generalized solenoids initiated by R.~F. Williams.