= Automorphy character of an orbit Blaschke product
{title2=$B(Tz)=\chi(T)B(z)$}
Let a <group> $G$ of <Möbius transformations> of the <unit disk> have an orbit satisfying the <Blaschke condition>, counted once per distinct orbit point. For its <Blaschke product> $B$, invariance of the <pseudohyperbolic distance> and permutation of the orbit give $|B(Tz)|=|B(z)|$. The ratio $B(Tz)/B(z)$ extends across its zeros and has modulus one, so the <open mapping theorem (complex analysis)> makes it a constant $\chi(T)$. Composition gives a <group homomorphism> $\chi:G\to\{\lambda:|\lambda|=1\}$. It is essential to compare moduli of the products: two arbitrary bounded <holomorphic functions> with identical zeros need not have constant quotient.
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