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Summary: SMOOTH SIEGEL DISKS VIA SEMICONTINUITY: A REMARK
ON A PROOF OF BUFF AND CHERITAT
ARTUR AVILA
Abstract. Recently, Xavier Buff and Arnaud Cheritat have provided an el-
egant proof of the existence of quadratic Siegel disks with smooth boundary.
In this short note, we show how results of Yoccoz and Risler can be used to
conclude the same result. Our proof is a small modification of the argument
given by Buff and Cheritat.
1. Introduction
Recently, in [BC1], Xavier Buff and Arnaud Cheritat gave a new proof of the
following unpublished result of Perez-Marco: there exists a quadratic map with a
Siegel disk whose boundary is a smooth (C
) Jordan curve. Their proof involves
both techniques of renormalization of [Y] and estimates for parabolic explosion.
Our aim in this note is to show that the same result follows easily from renor-
malization theory via two known results (of Yoccoz and Risler) by some general
abstract reasoning (which is really just a small modification of [BC1]).
We would like to note that the method of parabolic explosion (coupled with
renormalization) allows much greater control of the dynamics. In particular, in
[BC1] it is also possible to conclude that the Siegel disks are accumulated by small
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