BLASCHKE PRODUCTS AND RATIONAL FUNCTIONS WITH SIEGEL DISKS

Abstract

Let m be a positive integer. We show that for any given real number ∈ (0; 1) and complex number with | | ≤ 1 which satisfy e2i m 1, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier m at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number . Moreover if the given real number is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier m at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point. point of f and λ = f ' (z0) is called the multiplier of z0 if z0 ∈ C. The multiplier of z0 = ∞ is defined as the multiplier of the origin for ψ ◦ f ◦ ψ −1 , where ψ(z) = 1/z. The fixed point z0 is attracting, repelling or indifferent if its multiplier λ satisfies that |λ| 1 or |λ| = 1 respectively. Attracting fixed points belong to the Fatou set and repelling fixed points belong to the Julia set. In the case that z0 is indifferent, the classification is more complicated. The fixed point z0 is parabolic, a Siegel point or a Cremer point if its multiplier is a root of unity, z0 ∈ F (f ) or z0 ∈ J(f ) respectively. Parabolic fixed points belong to the Julia set. The Fatou component containing a Siegel point is called a Siegel disk centered at z0. Non-repelling fixed points capture at least one critical point of f , which is a solution of the equation f ' (z) = 0. In this paper, we investigate rational functions with Siegel disks. Let f :