Title page for ETD etd-08142000-10410017


Type of Document Dissertation
Author Hoggard, John W.
Author's Email Address hoggard@math.vt.edu
URN etd-08142000-10410017
Title Accuracy of Computer Generated Approximations to Julia Sets
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Rossi, John F. Committee Chair
Haskell, Peter E. Committee Member
Linnell, Peter A. Committee Member
Olin, Robert F. Committee Member
Wheeler, Robert L. Committee Member
Keywords
  • tangent
  • meromorphic
  • computer algorithms
  • polynomial Schwarzian derivative
  • Julia sets
Date of Defense 2000-07-31
Availability unrestricted
Abstract
A Julia set for a complex function $f$ is the set of all points in the

complex plane where the iterates of $f$ do not form a normal family.

A picture of the Julia set for a function can be generated with a

computer by coloring pixels (which we consider to be small squares)

based on the behavior of the point at the center of each pixel. We

consider the accuracy of computer generated pictures of Julia sets.

Such a picture is said to be accurate if each colored pixel actually

contains some point in the Julia set. We extend previous work to show

that the pictures generated by an algorithm for the family $lambda

e^z$ are accurate, for appropriate choices of parameters in the

algorithm. We observe that the Julia set for meromorphic functions

with polynomial Schwarzian derivative is the closure of those points

which go to infinity under iteration, and use this as a basis for an

algorithm to generate pictures for such functions. A pixel in our

algorithm will be colored if the center point becomes larger than some

specified bound upon iteration. We show that using our algorithm, the

pictures of Julia sets generated for the family $lambda an(z)$ for

positive real $lambda$ are also accurate. We conclude with a

cautionary example of a Julia set whose picture will be inaccurate for

some apparently reasonable choices of parameters, demonstrating that

some care must be exercised in using such algorithms. In general,

more information about the nature of the function may be needed.

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