Advisory Committee Chair
John C Mayer
Advisory Committee Members
Mark Beasley
Nikita Selinger
Bulent Tosun
Dongsheng Wu
Document Type
Dissertation
Date of Award
2023
Degree Name by School
Doctor of Philosophy (PhD) College of Arts and Sciences
Abstract
In this thesis, we first discuss the concept of laminations which was initially introduced by Thurston [1] as a way to understand quadratic polynomials and their parameter space. Laminations are a combinatorial way to model Julia sets. We go into detail explaining the introductory definitions in hopes it makes the following chapters easier to follow. In the next chapter, we go into detail discussing unicritical laminations and laminations with a single critical moment (SCM). We show there is a one - to - one correspondence between globally unicritical laminations and single critical moment laminations. The goal in this correspondence is to get a relationship between the Julia sets in parameter space. We then extend this result in chapter three. We consider the notion of “locally” unicritical laminations to find a similar one - to - one correspondence. Fixed points and fixed point portraits are an important part of this correspondence as they are what forces the “local” part. We give a count for the number of global fixed point portraits. Finally, we show a method for realizing laminations to Julia sets through mapping the orbits of critical points. We choose to find the Julia set where the critical points are periodic. This is the center of a bulb in parameter space. Any parameters in the same bulb will produce laminations topologically conjugate to the lamination we find. We parameterize our complex polynomial by the critical points and use a system of equations to solve for the coefficients. We also discuss some possibilities of reducing the number of solutions we find.
Recommended Citation
Burdette, Brittany, "The Global and Local Correspondence Between Unicritical and Maximally Critical Laminations and the Realization of Laminations to Julia Sets" (2023). All ETDs from UAB. 445.
https://digitalcommons.library.uab.edu/etd-collection/445