Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage.
In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when Tβ,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which Tβ,⍺ exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which Tβ,⍺ generates a subshift of finite type.
","abstract_html":"<p>Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage.</p> <p>In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when T<sub>β,⍺</sub> generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which T<sub>β,⍺</sub> exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which T<sub>β,⍺</sub> generates a subshift of finite type.</p>","abstract_has_math":false,"creators":["Zieber, Karl"],"institution":null,"degree_name":"MS in Mathematics","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Erin Pearse","Mathematics","College of Science and Mathematics"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-06-01T07:00:00Z","date_published":"2022-06-01T07:00:00Z","updated_at":"2026-07-24T01:32:42Z","subjects":["Dynamical Systems","Beta Expansions","Matching"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.15368/theses.2022.90"],"render_values":[{"text":"10.15368/theses.2022.90","href":"https://doi.org/10.15368/theses.2022.90","code":true}]}]},"links":{"outbound_url":"https://digitalcommons.calpoly.edu/theses/2454","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Erin Pearse","Mathematics","College of Science and Mathematics"]},{"key":"dc:creator","label":"Author","values":["Zieber, Karl"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2022-08-01T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Dynamical Systems","Beta Expansions","Matching"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.calpoly.edu/theses/2454","10.15368/theses.2022.90"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage.
In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when Tβ,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which Tβ,⍺ exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which Tβ,⍺ generates a subshift of finite type.
"]},{"key":"dc:title","label":"Title","values":["Dynamical Systems and Matching Symmetry in Beta-Expansions"]}]}],"canonical_facts":{"dc:contributor":["Erin Pearse","Mathematics","College of Science and Mathematics"],"dc:creator":["Zieber, Karl"],"dc:date.available":["2022-08-01T07:00:00Z"],"dc:description.abstract":["Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage.
In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when Tβ,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which Tβ,⍺ exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which Tβ,⍺ generates a subshift of finite type.
"],"dc:identifier":["https://digitalcommons.calpoly.edu/theses/2454","10.15368/theses.2022.90"],"dc:subject":["Dynamical Systems","Beta Expansions","Matching"],"dc:title":["Dynamical Systems and Matching Symmetry in Beta-Expansions"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["MS in Mathematics"]},"updated_at":"2026-07-24T01:32:42Z"}