All ETDs from UAB

Advisory Committee Chair

John Mayer

Advisory Committee Members

Inmaculada Aban

Alexander Blokh

Vo Thanh Liem

Kyle Siegrist

Document Type

Dissertation

Date of Award

2010

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

Given a topological space Z and a function, f : Z --> Z, one may examine the sequence of iterates of f, i.e. {fn(z)} where n is in the set N = {0, 1, 2, ...} for z in the space Z, called the orbit of z. Then one may classify the points of Z based upon their behavior under iterates of f. Specifically, let P : C --> C, where C denotes the complex plane, be a polynomial of degree at least two. We denote by F(P) the Fatou set, which is the maximal open set on which the iterates of P form a normal family in the sense of Montel. Further, we let J (P ) = C - F (P ) denote the Julia set, the set on which the dynamics is chaotic. It is well known that J(P) is a nonempty, perfect, compact set, which is either connected or has uncountably many components. We consider the case where J(P) is connected. As J(P) is where complex, chaotic behavior occurs, and the dynamics on F(P) are well understood, we are interested in studying the behavior of P on J(P). Typically, Julia sets exhibit very complex behavior. Thus, we desire to simplify our study of Julia sets by using a less complicated model. One method of modeling Julia sets is to use a lamination, which is a closed collection of chords in the unit disc, D, any two of which intersect at most at an endpoint on the boundary of D. By requiring that the lamination be sibling d-invariant, one achieves a space whose dynamics are easier to study than a Julia set, while the dynamics on the two spaces are related.

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