Fatou Components and Julia Sets of Singularly Perturbed Rational Maps with Positive Parameter

In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point...

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Bibliographic Details
Published inActa mathematica Sinica. English series Vol. 28; no. 10; pp. 1937 - 1954
Main Authors Qiu, Wei Yuan, Xie, Lan, Yin, Yong Cheng
Format Journal Article
LanguageEnglish
Published Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.10.2012
Springer Nature B.V
Subjects
Basins
Boundaries
Cantor集
Fatou分支
Julia集
Mathematical analysis
Mathematics
Mathematics and Statistics
Orbits
Studies
Zinc
参数摄动
吸引盆
固定点
地图
抛物线
Jordan domain
local connectivity
37F10
Sierpiński curve
Julia set
Fatou component
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Summary:In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.
Bibliography:Julia set, Fatou component, Jordan domain, local connectivity, Sierpifiski curve
In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.
11-2039/O1
SourceType-Scholarly Journals-1
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ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-012-0586-1