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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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