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Smooth Siegel disks without number theory

Published online by Cambridge University Press:  01 March 2008

LUKAS GEYER
Affiliation:
Montana State University, Department of Mathematics, P.O. Box 172400, Bozeman, MT 59717–2400, U.S.A. e-mail: geyer@math.montana.edu
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Abstract

X. Buff and A. Chéritat proved that there are quadratic polynomials having Siegel disks with smooth boundaries. Based on a simplification of Avila, we give yet another simplification of their proof. The main tool used is a harmonic function introduced by Yoccoz whose boundary values are the sizes of the Siegel disks. The proof also applies to some other families of polynomials, entire and meromorphic functions.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

[1]Avila, A.Smooth Siegel disks via semicontinuity: a remark on a proof by Buff and Chéritat. arXiv: math.DS/0305272.Google Scholar
[2]Avila, A., Buff, X. and Chéritat, A.. Siegel disks with smooth boundaries. Acta Math. 193, no. 1 (2004), 130.CrossRefGoogle Scholar
[3]Buff, X. and Chéritat, A.. Quadratic Siegel disks with smooth boundaries. Preprint 242, Université Toulouse.Google Scholar
[4]Buff, X. and Chéritat, A.. How regular can the boundary of a quadratic Siegel disk be? Proc. Amer. Math. Soc. 135 (2007), 10731080.CrossRefGoogle Scholar
[5]Carleson, L. and Gamelin, T. W.. Complex dynamics (Springer-Verlag, 1993).CrossRefGoogle Scholar
[6]Pérez–Marco, R.Siegel disks with smooth boundaries. Preprint, 1997.Google Scholar
[7]Rippon, P.On the boundaries of certain Siegel discs. C. R. Acad. Sci., Ser. I 319, no. 8 (1994), 821826.Google Scholar
[8]Yoccoz, J.–C.. Théorème de Siegel, nombres de Brjuno et polynomes quadratiques. Asterisque 231 (1995), 388.Google Scholar

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