Journal of the
Korean Mathematical Society JKMS

Let $m$ be a positive integer. We show that for any given real number $\alpha \in [0,1]$ and complex number $\mu$ with $|\mu| \le 1$ which satisfy $e^{2\pi i \alpha} \mu^m \neq 1$, there exists a Blaschke product $B$ of degree $2m+1$ which has a fixed point of multiplier $\mu^m$ 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 $\alpha$. Moreover if the given real number $\alpha$ 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 $\mu^m$ at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.

Keywords: Blaschke product, Siegel disk

MSC numbers: Primary 37F50; Secondary 30D05, 37F10