Christopher J. Bishop, Alexandre Eremenko, Kirill Lazebnik
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引用次数: 0
Abstract
A rational lemniscate is a level set of |r| where \(r: \widehat {\mathbb{C}}\rightarrow \widehat {\mathbb{C}}\) is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert’s lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge’s theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.
期刊介绍:
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