All ETDs from UAB

Advisory Committee Chair

Lex Oversteegen

Advisory Committee Members

Alexander Blokh

John K Johnstone

Jia Li

Vo T Liem

John C Mayer

Document Type

Dissertation

Date of Award

2013

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

In this dissertation we study the Julia sets of cubic polynomials using both analytic and topological methods. The main topological method that we employ is the use of invariant laminations to model polynomial Julia Sets. Our particular focus is on polynomials from the boundary of the degree three Principal Hyperbolic Domain. Such polynomials have simple yet nontrivial dynamics. Furthemore, they satisfy certain conditions on their periodic cut points. We call the set of polynomials which satisfy these conditions the Main Cubioid and characterize laminations that correspond to such polynomials. Small perturbations of a quadratic polynomial with an indifferent fixed point yield polynomials from the degree 2 principal hyperbolic domain. This is not true for cubic polynomials. We study such cubic polynomials and show that they admit quadratic-like dynamics except in highly pathological cases. We define an extended closure of the degree three principal hyperbolic domain and show that if a polynomial with a nonrepelling fixed point is not from this extended closure then it admits quadratic-like dynamics. Finally we conjecture that both the Main Cubioid and the extended closure are equal to the closure of the degree three principle hyperbolic domain. We prove some relationships between the three sets that support the conjecture.

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