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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Julia sets converging to the unit disk


Authors: Robert L. Devaney and Antonio Garijo
Journal: Proc. Amer. Math. Soc. 136 (2008), 981-988
MSC (2000): Primary 37F10, 37F40
Published electronically: November 23, 2007
MathSciNet review: 2361872
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the family of rational maps $ F_\lambda(z) = z^n + \lambda/z^d$, where $ n,d \geq 2$ and $ \lambda$ is small. If $ \lambda$ is equal to 0, the limiting map is $ F_0(z)=z^n$ and the Julia set is the unit circle. We investigate the behavior of the Julia sets of $ F_\lambda$ when $ \lambda$ tends to 0, obtaining two very different cases depending on $ n$ and $ d$. The first case occurs when $ n=d=2$; here the Julia sets of $ F_\lambda$ converge as sets to the closed unit disk. In the second case, when one of $ n$ or $ d$ is larger than $ 2$, there is always an annulus of some fixed size in the complement of the Julia set, no matter how small $ \vert\lambda\vert$ is.


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Additional Information

Robert L. Devaney
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email: bob@bu.edu

Antonio Garijo
Affiliation: Dep. d’Eng. Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans, 26, 43007 Tarragona, Spain

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09084-3
PII: S 0002-9939(07)09084-3
Received by editor(s): November 29, 2006
Published electronically: November 23, 2007
Additional Notes: The second author was supported by MTM2005-02139/Consolider (including a FEDER contribution) and CIRIT 2005 SGR01028.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.