Bounded type Siegel disks of a family of sine families

Abstract

We prove that for any positive integer n, there is a polynomial P(z) with degree n such that the entire function G(z)=P(sin⁡(z)) has a bounded type Siegel disk bounded by a quasi-circle in the plane which passes through at least one critical point of G. In addition, let 0<θ<1 be an irrational number of bounded type. We prove that for any integer k≥0, the boundary of the Siegel disk of G(z)=∫0sin⁡(z)e2πiθ(1−s2)kds centered at the origin is a quasi-circle passing through exactly two critical points π/2 and −π/2 with multiplicity k+1, respectively.