Abstract
We show that for any given complex number „ with modulus at most one and any given real number fi, there exists a cubic Blaschke product such that the point at infinity is its fixed point with multiplier „ and its restriction on the unit circle is a critical circle map with rotation number fi. Moreover if the given real number fi is irrational of bounded type, then a modified Blaschke product is quasiconformally conjugate to some quadratic rational function with a Siegel disk whose boundary is a quasicircle containing its critical point and the point at infinity is its fixed point with multiplier „.
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