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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Douady–Earle extension ： ウィキペディア英語版 Douady–Earle extension In mathematics, the Douady–Earle extension, named after Adrien Douady and Clifford Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed unit disk, such that the extension is a diffeomorphism of the open disk. The extension is analytic on the open disk. The extension has an important equivariance property: if the homeomorphism is composed on either side with a Möbius transformation preserving the unit circle the extension is also obtained by composition with the same Möbius transformation. If the homeomorphism is quasisymmetric, the diffeomorphism is quasiconformal. An extension for quasisymmetric homeomorphisms had previously been given by Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia. Equivariant extensions have important applications in Teichmüller theory, for example they lead to a quick proof of the contractibility of the Teichmüller space of a Fuchsian group. ==Definition== By the Radó–Kneser–Choquet theorem, the Poisson integral ''F''''f'' of a homeomorphism ''f'' of the circle defines a harmonic diffeomorphism of the unit disk extending ''f''. If ''f'' is quasisymmetric, the extension is not necessarily quasiconformal, i.e. the complex dilatation :$\backslash displaystyle,\}$ does not necessarily satisfy :$\backslash displaystyle$ However ''F'' can be used to define another analytic extension ''H''''f'' of ''f''−1 which does satisfy this condition. It follows that :$\backslash displaystyle$ is the required extension. For |''a''| < 1 define the Möbius transformation :$\backslash displaystyle.\}$ It preserves the unit circle and unit disk sending ''a'' to 0. If ''g'' is any Möbius transformation preserving the unit circle and disk, then :$\backslash displaystyle$ For |''a''| < 1 define :$\backslash displaystyle$ to be the unique ''w'' with |''w''| < 1 and :$\backslash displaystyle$ For |''a''| =1 set :$\backslash displaystyle$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Douady–Earle extension」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.