{"id":{"repo_id":"houston","oai_identifier":"oai:uh-ir.tdl.org:10657/19574"},"canonical_url":"https://search.dev.ndltd.org/etd/houston/oai:uh-ir.tdl.org:10657/19574","repository":{"repo_id":"houston","name":"University of Houston","base_url":"https://uh-ir.tdl.org/server/oai/request"},"display":{"title":"Equilibrium Measures for Symbolic Spaces Using Dimension Theory","abstract":"The theory of thermodynamic formalism is well developed for subshifts of finite type. However, less is known about equilibrium measures for more general shift spaces beyond existence and, in some cases, uniqueness. Rather than constructing equilibrium measures as a limit of a sequence of measures or via a fixed point theorem, equilibrium measures can be constructed using dimension theoretic techniques. This approach can be thought of as a generalization of Hausdorff measure, and we outline a number of criteria that guarantee that this construction produces an equilibrium state. For two-sided shifts, we will show that subshifts of finite type satisfy these conditions. We will also show that equilibrium measures can be constructed for shifts with synchronizing words under additional assumptions. Another notable application of this approach is a construction of the measure of maximal entropy for finite horizon dispersing billiards. For one-sided subshifts of finite type, we construct the eigenmeasure and eigenfunction that appear in the Ruelle-Peron-Frobenius theorem using this dimension theoretic machinery.","abstract_html":"The theory of thermodynamic formalism is well developed for subshifts of finite type. However, less is known about equilibrium measures for more general shift spaces beyond existence and, in some cases, uniqueness. Rather than constructing equilibrium measures as a limit of a sequence of measures or via a fixed point theorem, equilibrium measures can be constructed using dimension theoretic techniques. This approach can be thought of as a generalization of Hausdorff measure, and we outline a number of criteria that guarantee that this construction produces an equilibrium state. For two-sided shifts, we will show that subshifts of finite type satisfy these conditions. We will also show that equilibrium measures can be constructed for shifts with synchronizing words under additional assumptions. Another notable application of this approach is a construction of the measure of maximal entropy for finite horizon dispersing billiards. For one-sided subshifts of finite type, we construct the eigenmeasure and eigenfunction that appear in the Ruelle-Peron-Frobenius theorem using this dimension theoretic machinery.","abstract_has_math":false,"creators":["Day, Jason 1995-"],"institution":"University of Houston","degree_name":"Doctor of Philosophy","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Climenhaga, Vaughn"],"committee_chairs":[],"committee_members":["Torok, Andrew","Nicol, Matthew","Gunaratne , Gemunu"],"year":2025,"date_issued":"2025-05","date_published":"2025-05","updated_at":"2026-07-24T02:32:27Z","subjects":["Mathematics"],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10657/19574","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Climenhaga, Vaughn"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Torok, Andrew","Nicol, Matthew","Gunaratne , Gemunu"]},{"key":"dc:creator","label":"Author","values":["Day, Jason 1995-"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-06-23T19:05:49Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-05"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Houston"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["English"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10657/19574"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The theory of thermodynamic formalism is well developed for subshifts of finite type. However, less is known about equilibrium measures for more general shift spaces beyond existence and, in some cases, uniqueness. Rather than constructing equilibrium measures as a limit of a sequence of measures or via a fixed point theorem, equilibrium measures can be constructed using dimension theoretic techniques. This approach can be thought of as a generalization of Hausdorff measure, and we outline a number of criteria that guarantee that this construction produces an equilibrium state. For two-sided shifts, we will show that subshifts of finite type satisfy these conditions. We will also show that equilibrium measures can be constructed for shifts with synchronizing words under additional assumptions. Another notable application of this approach is a construction of the measure of maximal entropy for finite horizon dispersing billiards. For one-sided subshifts of finite type, we construct the eigenmeasure and eigenfunction that appear in the Ruelle-Peron-Frobenius theorem using this dimension theoretic machinery."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Equilibrium Measures for Symbolic Spaces Using Dimension Theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Climenhaga, Vaughn"],"dc:contributor.committeemember":["Torok, Andrew","Nicol, Matthew","Gunaratne , Gemunu"],"dc:creator":["Day, Jason 1995-"],"dc:date.accessioned":["2025-06-23T19:05:49Z"],"dc:date.issued":["2025-05"],"dc:description.abstract":["The theory of thermodynamic formalism is well developed for subshifts of finite type. However, less is known about equilibrium measures for more general shift spaces beyond existence and, in some cases, uniqueness. Rather than constructing equilibrium measures as a limit of a sequence of measures or via a fixed point theorem, equilibrium measures can be constructed using dimension theoretic techniques. This approach can be thought of as a generalization of Hausdorff measure, and we outline a number of criteria that guarantee that this construction produces an equilibrium state. For two-sided shifts, we will show that subshifts of finite type satisfy these conditions. We will also show that equilibrium measures can be constructed for shifts with synchronizing words under additional assumptions. Another notable application of this approach is a construction of the measure of maximal entropy for finite horizon dispersing billiards. For one-sided subshifts of finite type, we construct the eigenmeasure and eigenfunction that appear in the Ruelle-Peron-Frobenius theorem using this dimension theoretic machinery."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/10657/19574"],"dc:language.iso":["English"],"dc:subject":["Mathematics"],"dc:title":["Equilibrium Measures for Symbolic Spaces Using Dimension Theory"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["University of Houston"]},"updated_at":"2026-07-24T02:32:27Z"}