Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose $f: \mathbb{D} \rightarrow \mathbb{D}$ is analytic. Furthermore, suppose $f \circ f = g$, and $g$ is a linear fractional map. Does this guarantee that $f$ is linear fractional? I know it would be true if the domain was the sphere instead of the disk, or if $g$ was onto the disk (as we already have that $f$ is injective because $g$ is). But I want this to be true in general. Any insight?

share|improve this question

1 Answer 1

Yes. Consider the sequence of open disks $$\mathbb D\subset g^{-1}(\mathbb D)\subset g^{-2}(\mathbb D)\subset \dots . $$ Let $\mathbb D'$ be the union. This is either an open disk or the complement of a point in the Riemann sphere. You can extend $f$ to $\mathbb D'$ by defining $f(z)=g^{-n}(f(g^n(z)))$ for sufficently large $n$. Now $f$ maps $\mathbb D'$ to itself in a one to one and onto fashion, so it is a linear fractional map.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.