Julia sets converging to the unit disk
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- by Robert L. Devaney and Antonio Garijo PDF
- Proc. Amer. Math. Soc. 136 (2008), 981-988 Request permission
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 $|\lambda |$ is.References
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Additional Information
- Robert L. Devaney
- Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
- MR Author ID: 57240
- Email: bob@bu.edu
- Antonio Garijo
- Affiliation: Dep. d’Eng. Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans, 26, 43007 Tarragona, Spain
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 981-988
- MSC (2000): Primary 37F10, 37F40
- DOI: https://doi.org/10.1090/S0002-9939-07-09084-3
- MathSciNet review: 2361872