It's a problem from my complex dynamics lesson.
Prove that $f:\mathbb{T}\to \mathbb{T}$ is a homoemorphism iff $\lvert a\rvert\geq 3$,where $\mathbb{T}$ is the unit circle and $$ f(z)=z^2\frac{z-a}{1-\overline{a}z}$$
This is the third part of a problem,and the first two questions are following:
(1). Let $R$ be arational map satsfying $R(\mathbb{T})\subset \mathbb{T}$.Prove that $R$ must be the following form $$ R\left( z \right) =e^{i\theta}\prod_{i=1}^d{\frac{z-a_i}{1-\overline{a_i}z}}\qquad where~a_i\in \mathbb{C}\backslash \mathbb{T}~and~\theta \in R $$ (2).If $\lvert a_i\rvert\leq 1$ for all $1\leq i\leq d$,what is the Julia set of $R$?
I have solved the first two problems. The first mainly utilizes the principle of maximum modulus. The second answer is $\mathcal{J}(R)=\mathbb{T}$ or Cantor set on $\mathbb{T}$.
I don't know how the third question is related to the first two questions, nor do I know what kind of technology should be used to solve it. Maybe it only needs the method of complex analysis to solve it.