Institute for Mathematical Sciences

Preprint ims00-06

C. L. Petersen and S. Zakeri
On the Julia Set of a Typical Quadratic Polynomial with a Siegel disk.

Abstract: Let $0< \theta <1$ be an irrational number with continued fraction expansion $\theta=[a_1, a_2, a_3, \ldots]$, and consider the quadratic polynomial $\pt : z \mapsto e^{2\pi i \theta} z +z^2$. By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if $$\log a_n = {\cal O} (\sqrt{n})\ \operatorname{as}\ n \to \infty ,$$ then the Julia set of $\pt$ is locally-connected and has Lebesgue measure zero. In particular, it follows that for almost every $0< \theta < 1$, the quadratic $\pt$ has a Siegel disk whose boundary is a Jordan curve passing through the critical point of $\pt$. By standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.
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