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

Abstract

Let 0 << 1 be an irrational number with continued fraction expan- sion =( a1 ;a 2 ;a 3 ;::: ), and consider the quadratic polynomial P : z 7! e 2i z +z 2 . By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if log an = O( p n )a sn !1 ; then the Julia set of P is locally-connected and has Lebesgue measure zero. In particular, it follows that for almost every 0 << 1, the quadratic P has a Siegel disk whose boundary is a Jordan curve passing through the critical point of P.B y standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.