Laminations : a topological approach
Author
Mimbs , Debra L .
Title
Laminations : a topological approach
Description
Given a topological space Z and a function , f : Z → Z , one may examine the sequence of iterates of f , i.e . {f[supercript n](z)}n∈N where N = {0,1,2...} for z ∈ 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 n 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 .
Degree Awarded
Ph.D .
Language
eng
Type
Text
File Type
PDF
File Size
1.82MB
Use
Adobe Reader to view
Physical Description
1 online resource (vi, 64 p.) : ill. (some col.)
Advisor/Chair
Mayer , John
Committee Members
Oversteegen , Lex Aban , Imaculada Blokh , Alexander Liem , Vo Thanh Siegrist , Kyle
Date of Degree
2010
Thesis Note
Thesis (Ph.D.)--University of Alabama at Birmingham, 2010.
Department
Mathematics
School
College of Arts and Sciences
Date
2010
Subject LC
Topological manifolds Julia sets Fractals Iterative methods (Mathematics) Dynamics -- Mathematics
Keywords
Complex dynamics Laminations Julia set Fatou set
Availability
UNRESTRICTED
Publisher
University of Alabama at Birmingham. Graduate School.
Collection
Electronic Theses and Dissertations
Distributor
Mervyn H. Sterne Library (University of Alabama at Birmingham)
Signed Approval Form
Signed Thesis/Dissertation Approval Forms are on file in the UAB Graduate School office. If you need assistance please contact: gradschool@uab.edu or 205.934.8227

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