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Let $0\lt \theta <1$ be an irrational number with continued fraction expansion $\theta=[a_1, a_2, a_3, \ldots]$, and consider the quadratic polynomial $P_\theta : z \mapsto e^{2\pi i \theta} z +z^2$. By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if \[\log a_n = {\mathcal{O}} (\sqrt{n}) \quad \mbox{as}\quad n \to \infty,\] then the Julia set of $P_\theta$ is locally connected and has Lebesgue measure zero. In particular, it follows that for almost every $0\lt \theta < 1$, the quadratic $P_\theta$ has a Siegel disk whose boundary is a Jordan curve passing through the critical point of $P_\theta$. By standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.
@article{10_4007_annals_2004_159_1,
author = {C. L. Petersen and S. Zakeri},
title = {On the {Julia} set of a typical quadratic polynomial with a {Siegel} disk},
journal = {Annals of mathematics},
pages = {1--52},
publisher = {mathdoc},
volume = {159},
number = {1},
year = {2004},
doi = {10.4007/annals.2004.159.1},
mrnumber = {2051390},
zbl = {1069.37038},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.159.1/}
}
TY - JOUR AU - C. L. Petersen AU - S. Zakeri TI - On the Julia set of a typical quadratic polynomial with a Siegel disk JO - Annals of mathematics PY - 2004 SP - 1 EP - 52 VL - 159 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.159.1/ DO - 10.4007/annals.2004.159.1 LA - en ID - 10_4007_annals_2004_159_1 ER -
%0 Journal Article %A C. L. Petersen %A S. Zakeri %T On the Julia set of a typical quadratic polynomial with a Siegel disk %J Annals of mathematics %D 2004 %P 1-52 %V 159 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.159.1/ %R 10.4007/annals.2004.159.1 %G en %F 10_4007_annals_2004_159_1
C. L. Petersen; S. Zakeri. On the Julia set of a typical quadratic polynomial with a Siegel disk. Annals of mathematics, Tome 159 (2004) no. 1, pp. 1-52. doi: 10.4007/annals.2004.159.1
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