UBC Theses and Dissertations
Gibbs measures and factor codes in symbolic dynamics MacDonald, Sophie Querida Tennant
This dissertation consists principally of four of the author’s research articles, included as Chapters 2 through 5, all within or related to the area of symbolic dynamics, especially to shifts of finite type (SFTs) and sofic shifts. More specifically, each of the articles involves either Gibbs measures or factor codes, with Chapter 3 involving both in an essential way. The research in Chapter 2, in the thermodynamic formalism of dynamical systems, shows that two commonly used definitions of a Gibbs measure coincide for an arbitrary subshift over an arbitrary countable group, and that two different forms of one of these definitions are equivalent under certain regularity hypotheses. The main innovation is a more careful decomposition of holonomies of the Gibbs relation than had previously appeared. Chapter 3 is concerned with the Dobrushin and Lanford-Ruelle theorems, which relate Gibbs and equilibrium measures. This chapter establishes these theorems for irreducible sofic shifts, by lifting to an irreducible SFT and using cyclic structure to reduce to the classical mixing SFT case. The result was already known, by work of Haydn-Ruelle and Baladi, but the proof presented in this chapter is more self-contained. In particular, the argument makes extensive use of the properties of doubly transitive points, and relates them in a novel way to the Gibbs relation studied in Chapter 2. The work in Chapter 4 resolves two new cases of a problem in automata theory that originated in symbolic dynamics. The methods combine the notion of stability, introduced by Culik-Karhumaki-Kari and used by Trahtman in the solution to the road colouring problem, with the framework of graph homomorphisms introduced by Ashley-Marcus-Tuncel. The chapter also includes several new algorithms relevant to the problem. Chapter 5 solves a symbolic dynamics problem related to zero-error coding. The main result, which generalizes Krieger’s embedding theorem, characterizes the subshifts of a given mixing SFT on which a given sliding block code is injective. The proof follows the overall strategy of Krieger’s proof, but with significant technical innovations required to make the embedding compatible with the code.
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