## 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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