Blaschke products and rational functions with Siegel disks
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Let m be a positive integer. We show that for any given real number α∈ [0,1] and complex number μ with |μ| ≤ 1 which satisfy e^(2πiα)μ^m ≠ 1 there exists a Blaschke product B of degree 2m+1 which has a fixed point of multiplier μ^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 α. Moreover if the given real number α 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 μ^m at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.
Abstract
Let m be a positive integer. We show that for any given real number α∈ [0,1] and complex number μ with |μ| ≤ 1 which satisfy e^(2πiα)μ^m ≠ 1 there exists a Blaschke product B of degree 2m+1 which has a fixed point of multiplier μ^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 α. Moreover if the given real number α 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 μ^m at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.
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