Synchronizing Dynamical Systems: Shift Spaces and $K$-Theory

Abstract

Building on our previous work, we give a thorough presentation of the techniques developed for synchronizing dynamical systems in the special case of synchronizing shift spaces. Following work of Thomsen, we give a construction of the homoclinic, the heteroclinic, and synchronizing heteroclinic $C^\ast$-algebras along with the synchronizing ideal of a shift space in terms of Bratteli diagrams. The algebras introduced in our previous work (the synchronizing ideal, and synchronizing heteroclinic algebra) are discussed in detail. In the sofic shift case, these algebras are shown to be related to the $C^\ast$-algebras of its minimal left and minimal right presentations. Several specific examples are discussed to demonstrate these techniques. For the even shift we give a complete computation of all the associated invariants. We discuss these algebras for a sofic shift that is not of almost finite type and for a number of strictly non-sofic synchronizing shifts. In particular we discuss the rank of the $K$-theory of the homoclinic algebra of a shift space and its synchronizing ideal and its implications. We also give a construction for producing from any minimal shift a synchronizing shift whose set of non-synchronizing points is exactly the original minimal shift.