Although we have seen many aspects of symbolic dynamics, there are still many more that we have not mentioned. This final chapter serves as a guide to the reader for some of the more advanced topics. Our treatment of each topic only sketches some of its most important features, and we have not included some important topics. For each topic we have tried to give sufficient references to research papers so that the reader may learn more. In many places we refer to papers for precise proofs and sometimes even for precise definitions. The survey paper of Boyle [Boy5] contains descriptions of some additional topics.

More on Shifts of Finite Type and Sofic Shifts

THE CORE MATRIX

Any shift of finite type X can be recoded to an edge shift XG, and we can associate the matrix AG to X. This matrix is not unique, but any two such matrices are shift equivalent, and in particular they must have the same Jordan form away from zero. This gives us a way of associating to X a particular Jordan form, or, equivalently, a particular similarity class of matrices. By Theorem 7.4.6, this similarity class is an invariant of conjugacy, and, by Proposition 12.2.3, it gives a constraint on finite-to-one factors between irreducible shifts of finite type.