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Advisory Committee Chair

Nikita Selinger

Advisory Committee Members

Lex Oversteegen

Alexander Blokh

Da Yan

Bulent Tosun

Wu Dongsheng

Document Type

Dissertation

Date of Award

2021

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

To study the \emph{parameter space of all polynomials of degree $d$} with connected Julia sets, Thurston proposed studying the space of all \emph{$\sigma_d$-invariant laminations}, where $\sigma_d:\mathbb S\to\mathbb S$ is the degree $d$ covering map of the unit circle defined by $\sigma_d(z)=z^d$. A lamination is a family of chords in the unit disk satisfying the following property. No two chords in a lamination intersect inside the disk. Thurston built a topological model for the space of quadratic polynomials $f(z) = z^2 +\lambda$ using a parametrization of the space of quadratic invariant laminations. He completed this approach for the space of quadratic polynomials but the case of higher degree has remained elusive. Our goal is to gain a better understanding of the space of cubic polynomials $f(z) = z^3 +b z^2+ \lambda z$. We have studied a particular slice of the space of cubic polynomials. We call polynomials of the form $f(z) = z^3 +\lambda^2 z$ cubic symmetric polynomials. In the same spirit as Thurston's work, we will parametrize space of cubic symmetric laminations which will provide a model for the space of cubic symmetric polynomials.

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