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


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.


C. L. Petersen

IMFUFA, Roskilde University, Roskilde, Denmark

S. Zakeri

Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794, United States