Dynamical Systems and Matching Symmetry in Beta-Expansions

Author

Karl Zieber, California Polytechnic State University, San Luis ObispoFollow

DOI: https://doi.org/10.15368/theses.2022.90
Available at: https://digitalcommons.calpoly.edu/theses/2454

Date of Award

6-2022

Degree Name

MS in Mathematics

Department/Program

Mathematics

College

College of Science and Mathematics

Advisor

Erin Pearse

Advisor Department

Mathematics

Advisor College

College of Science and Mathematics

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 T_β,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which T_β,⍺ 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_β,⍺ generates a subshift of finite type.