On the Julia set of a typical quadratic polynomial with a Siegel disk

Abstract

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.