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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| Published in | Acta mathematica Sinica. English series Vol. 28; no. 10; pp. 1937 - 1954 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
01.10.2012
Springer Nature B.V |
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| Abstract | 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. |
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| AbstractList | In this paper, we discuss the rational maps ... with the positive real parameter ... . It is shown that the immediately attracting basin ... of ... is always a Jordan domain if the Julia set of ... is not a Cantor set. Furthermore, ... is a quasidisk if there is no parabolic fixed point on the boundary of ... . It is also shown that if the Julia set of F ... 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 Sierpiski curve is given. (ProQuest: ... denotes formulae and non-USASCII text omitted) 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. In this paper, we discuss the rational maps with the positive real parameter λ . It is shown that the immediately attracting basin B λ of ∞ for F λ is always a Jordan domain if the Julia set of F λ is not a Cantor set. Furthermore, B λ is a quasidisk if there is no parabolic fixed point on the boundary of B λ . It is also shown that if the Julia set of F λ 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 Sierpiński curve is given. In this paper, we discuss the rational maps $F_\lambda (z) = z Delta + \lambda /z Delta ,n \geqslant 2$ with the positive real parameter lambda . It is shown that the immediately attracting basin B sub( ) lambda of infinity for F sub( ) lambda is always a Jordan domain if the Julia set of F sub( ) lambda is not a Cantor set. Furthermore, B sub( ) lambda is a quasidisk if there is no parabolic fixed point on the boundary of B sub( ) lambda It is also shown that if the Julia set of F sub( ) lambda 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 Sierpiski curve is given. |
| Author | Wei Yuan QIU Lan XIE Yong Cheng YIN |
| AuthorAffiliation | School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China |
| Author_xml | – sequence: 1 givenname: Wei Yuan surname: Qiu fullname: Qiu, Wei Yuan email: wyqiu@fudan.edu.cn organization: School of Mathematical Sciences, Fudan University – sequence: 2 givenname: Lan surname: Xie fullname: Xie, Lan organization: School of Mathematical Sciences, Fudan University – sequence: 3 givenname: Yong Cheng surname: Yin fullname: Yin, Yong Cheng organization: School of Mathematical Sciences, Fudan University |
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| DOI | 10.1007/s10114-012-0586-1 |
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| Keywords | Jordan domain local connectivity 37F10 Sierpiński curve Julia set Fatou component |
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| Notes | 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 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
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| References | McMullen (CR1) 1988 Devaney (CR13) 2009; 206 CR19 Steinmetz (CR16) 2006; 10 CR18 Devaney, Look (CR7) 2005; 13 Roesch (CR15) 2006 Devaney (CR10) 2007; 11 Steinmetz (CR17) 2006; 6 McMullen (CR31) 1994 Pommerenke (CR27) 1975 Carleson, Gamelin (CR20) 1993 Milnor (CR21) 2006 Lyubich, Minsky (CR30) 1997; 47 Carleson, Jones, Yoccoz (CR24) 1994; 25 Devaney (CR5) 2004; 5 Devaney (CR6) 2005; 185 Devaney, Pilgrim (CR14) 2009; 202 CR29 Blanchard, Devaney, Garijo (CR3) 2008; 18 CR25 Douady, Hubbard (CR28) 1985; 18 Devaney, Josić, Shapiro (CR4) 2004; 14 Devaney (CR11) 2007; 27 Milnor (CR23) 2000; 261 Blanchard, Devaney, Look (CR2) 2005; 25 Devaney, Look (CR9) 2006; 30 Whyburn (CR22) 1971 Tan, Yin (CR26) 1996; 39 Devaney, Look, Uminsky (CR8) 2005; 54 Devaney, Rocha, Siegmund (CR12) 2007; 154 R. Devaney (586_CR8) 2005; 54 P. Roesch (586_CR15) 2006 J. Milnor (586_CR21) 2006 L. Carleson (586_CR20) 1993 R. Devaney (586_CR5) 2004; 5 586_CR25 C. T. McMullen (586_CR1) 1988 P. Blanchard (586_CR2) 2005; 25 N. Steinmetz (586_CR17) 2006; 6 586_CR19 586_CR18 R. Devaney (586_CR12) 2007; 154 R. Devaney (586_CR7) 2005; 13 N. Steinmetz (586_CR16) 2006; 10 J. Milnor (586_CR23) 2000; 261 R. Devaney (586_CR4) 2004; 14 L. Tan (586_CR26) 1996; 39 P. Blanchard (586_CR3) 2008; 18 M. Lyubich (586_CR30) 1997; 47 L. Carleson (586_CR24) 1994; 25 R. Devaney (586_CR11) 2007; 27 G. T. Whyburn (586_CR22) 1971 A. Douady (586_CR28) 1985; 18 R. Devaney (586_CR14) 2009; 202 R. Devaney (586_CR10) 2007; 11 R. Devaney (586_CR6) 2005; 185 R. Devaney (586_CR9) 2006; 30 Ch. Pommerenke (586_CR27) 1975 586_CR29 C. T. McMullen (586_CR31) 1994 R. Devaney (586_CR13) 2009; 206 |
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| Snippet | 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 ∞... In this paper, we discuss the rational maps with the positive real parameter λ . It is shown that the immediately attracting basin B λ of ∞ for F λ is always a... In this paper, we discuss the rational maps ... with the positive real parameter ... . It is shown that the immediately attracting basin ... of ... is always a... In this paper, we discuss the rational maps $F_\lambda (z) = z Delta + \lambda /z Delta ,n \geqslant 2$ with the positive real parameter lambda . It is shown... |
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| SubjectTerms | 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 |
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| Title | Fatou Components and Julia Sets of Singularly Perturbed Rational Maps with Positive Parameter |
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