Type of Document Dissertation Author Hoggard, John W. Author's Email Address email@example.com 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
- computer algorithms
- polynomial Schwarzian derivative
- Julia sets
Date of Defense 2000-07-31 Availability unrestricted AbstractA 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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