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https://hdl.handle.net/2142/21910
Description
Title
A global Boettcher's theorem
Author(s)
Kline, Bradford J.
Issue Date
1995
Doctoral Committee Chair(s)
Miles, Joseph B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We prove a global result for rational functions that is analogous to a local theorem of L. E. Boettcher (1904). Under the hypotheses that f is a complex rational function with a superattractive fixed point $\alpha$ of order $p \ge 2$ and that $\alpha$ is the only critical point of f in the immediate basin of attraction of $\alpha,$ we prove that there exists a conformal map $w = \varphi(z)$ of the entire immediate basin of attraction of $\alpha$ onto the unit disk such that $(\varphi \circ f \circ \varphi\sp{-1})(w) = w\sp{p}.$
In our proof, the conjugating function $\varphi$ appears as the unique fixed point of a certain contraction operator on a complete metric space of one-to-one analytic functions. We make extensive use of the topological concept of a branched covering space in defining the contraction operator and, hence, in obtaining the global existence of $\varphi.$
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