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Stability property of Möbius mappings

Author: T. Iwaniec
Journal: Proc. Amer. Math. Soc. 100 (1987), 61-69
MSC: Primary 30C60
MathSciNet review: 883402
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Abstract: Let $ F$ be an arbitrary class of continuous mappings acting and ranging on domains in $ {{\mathbf{R}}^n}$ which is invariant under similarity transformations of $ {{\mathbf{R}}^n}$ and the restriction of a map to any subdomain. The class Möb of Möbius mappings acting in $ {{\mathbf{R}}^n}$ is of particular interest. Assume that the class $ F$ is "$ c$-uniformly close" to Möb. Then we show that any map in $ F$ is either constant or a local quasiconformal homeomorphism. As a corollary we obtain a distinctly elementary proof of the Local Injectivity Theorem for quasiregular mappings.

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