Gibbs measures and factor codes in symbolic dynamics
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: MacDonald, Sophie Querida Tennant
issued: 2023-04-18T15:52:32Z
issued: 2023
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: 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 …
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/84237
note: GIBBS MEASURES AND FACTOR CODES IN SYMBOLIC DYNAMICSbySOPHIE QUERIDA TENNANT MACDONALDA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2023© Sophie Querida Tennant MacDonald, 2023The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the dissertation entitled:Gibbs measures and factor codes in symbolic dynamicssubmitted by Sophie MacDonald in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin MathematicsExamining committee:Brian Marcus, Professor, Mathematics, UBCCo-SupervisorTom Meyerovitch, Professor, Mathematics, Ben Gurion University of the NegevCo-SupervisorAndrew Rechnitzer, Professor, Mathematics, UBCUniversity ExaminerBruce Shepherd, Professor, Computer Science, UBCUniversity ExaminerAdditional Supervisory Committee Members:Lior Silberman, Professor, Mathematics, UBCSupervisory Committee MemberOmer Angel, Professor, Mathematics, UBCSupervisory Committee MemberiiAbstractThis dissertation consists principally of four of the author’s research articles, included as Chapters 2through 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 factorcodes, with Chapter 3 involving both in an essential way.The research in Chapter 2, in the thermodynamic formalism of dynamical systems, shows that twocommonly used definitions of a Gibbs measure coincide for an arbitrary subshift over an arbitrarycountable group, and that two different forms of one of these definitions are equivalent under certainregularity hypotheses. The main innovation is a more careful decomposition of holonomies of the Gibbsrelation than had previously appeared.Chapter 3 is concerned with the Dobru…
has type: Thesis/Dissertation
date issued: 2023-05
is shown at: 10.14288/1.0431092
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Marcus, Brian, 1949-
contributor: Meyerovitch, Tom
type: Text
identifier URI: http://hdl.handle.net/2429/84237