= Solution
The <Blaschke product> here is taken over distinct points of the orbit; its existence is part of the hypothesis. It is not necessary to assert that every discrete orbit satisfies the <Blaschke condition>.
The modulus of a <Blaschke factor> is the <pseudohyperbolic distance>. Thus, away from the orbit,
$$
|B(z)|=\prod_{a\in G(0)}\left|\frac{z-a}{1-\overline az}\right|=\prod_{a\in G(0)}\tanh\frac{\rho(z,a)}2.
$$
The logarithms converge absolutely at such a point, by the compact convergence estimate for the <Blaschke product>. Every $T\in G$ preserves the <hyperbolic metric> and permutes the orbit. Reindexing this convergent product yields $|B(Tz)|=|B(z)|$.
The functions $B\circ T$ and $B$ have the same simple zero set, since $T$ is a <biholomorphism>. Their quotient therefore extends holomorphically and without zeros across the orbit. It has modulus one everywhere. The <open mapping theorem (complex analysis)> makes it a constant, denoted $\chi(T)$. Finally
$$
B(STz)=\chi(S)B(Tz)=\chi(S)\chi(T)B(z).
$$
Since $B$ is not identically zero, cancellation gives
$$
\boxed{\chi(ST)=\chi(S)\chi(T),\qquad |\chi(T)|=1,\qquad B(Tz)=\chi(T)B(z).}
$$
This is the <automorphy character of an orbit Blaschke product>. The equal-modulus argument is essential: equality of zero sets alone would not make the quotient of two bounded <holomorphic functions> constant.
Back to article page
