Abstract
Let 0<θ<1 be an irrational number with continued fraction expansion θ=[a1,a2,a3,…], and consider the quadratic polynomial Pθ:z↦e2πiθz+z2. By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if logan=O(√n)asn→∞, 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.