How regular can the boundary of a quadratic Siegel disk be?

Buff, Xavier; Chéritat, Arnaud

doi:10.1090/S0002-9939-06-08578-9

How regular can the boundary of a quadratic Siegel disk be?
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by Xavier Buff and Arnaud Chéritat PDF

Proc. Amer. Math. Soc. 135 (2007), 1073-1080 Request permission

Abstract:

In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are $C^n$ but not $C^{n+1}$ Jordan curves.

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