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Dynamical Systems and Matching Symmetry in Beta-Expansions

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.

Identiferoai:union.ndltd.org:CALPOLY/oai:digitalcommons.calpoly.edu:theses-4134
Date01 June 2022
CreatorsZieber, Karl
PublisherDigitalCommons@CalPoly
Source SetsCalifornia Polytechnic State University
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceMaster's Theses

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