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Conformal Geometry and Dynamics

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Uniformly quasiregular mappings of Lattès type

Author: Volker Mayer
Journal: Conform. Geom. Dyn. 1 (1997), 104-111
MSC (1991): Primary 30C65; Secondary 58Fxx
Published electronically: December 16, 1997
MathSciNet review: 1482944
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Abstract: Using an analogy of the Lattès’ construction of chaotic rational functions, we show that there are uniformly quasiregular mappings of the $n$-sphere $\overline {{\Bbb R}}^n$ whose Julia set is the whole sphere. Moreover there are analogues of power mappings, uniformly quasiregular mappings whose Julia set is ${\Bbb S}^{n-1}$ and its complement in ${\Bbb S}^{n}$ consists of two superattracting basins. In the chaotic case we study the invariant conformal structures and show that Lattès type rational mappings are either rigid or form a 1-parameter family of quasiconformal deformations.

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Volker Mayer
Affiliation: U.R.A. 751, UFR de Mathématiques Pures et Appliquées, Université de Lille I, 59655 Villeneuve d’Ascq, Cedex, France
MR Author ID: 333982

Received by editor(s): March 7, 1997
Received by editor(s) in revised form: September 22, 1997
Published electronically: December 16, 1997
Article copyright: © Copyright 1997 American Mathematical Society