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

Abstract

Let 0 < θ < 1 be an irrational number with continued fraction expansion θ = [a1, a2, a3, . . .], and consider the quadratic polynomial Pθ : z → e 2πiθz + z2. By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if log a n = script O sign (√n) as n →, then the Julia set of Pθ is locally connected and has Lebesgue measure zero. In particular, it follows that for almost every 0 < θ < 1, the quadratic Pθ has a Siegel disk whose boundary is a Jordan curve passing through the critical point of Pθ. By standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.