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Title:A global Boettcher's theorem
Author(s):Kline, Bradford J.
Doctoral Committee Chair(s):Miles, Joseph B.
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
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 $\varphi.$
Issue Date:1995
Type:Text
Language:English
URI:http://hdl.handle.net/2142/21910
Rights Information:Copyright 1995 Kline, Bradford J.
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9543631
OCLC Identifier:(UMI)AAI9543631


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