On quadratic Siegel disks with a class of unbounded type rotation numbers

In this paper we explore a class of quadratic polynomials having Siegel disks with unbounded type rotation numbers. We prove that any boundary point of Siegel disks of these polynomials is a Lebesgue density point of their filled-in Julia sets, which generalizes the corresponding result of McMullen for bounded type rotation numbers. As an application, this result can help us construct more quadratic Julia sets with positive area. Moreover, we also explore the canonical candidate model for quasiconformal surgery of quadratic polynomials with Siegel disks. We prove that for any irrational rotation number, any boundary point of ``Siegel disk'' of the canonical candidate model is a Lebesgue density point of its ``filled-in Julia set'', in particular the critical point $1$ is a measurable deep point of the ``filled-in Julia set''.