Let $f:D(0,1) \to D(0,1)$ be a holomorphic map where $f$ has two distinct fixed points. Could anyone advise me how to prove $f$ is identity map? Do I use Schwarz-Pick somewhere?
Let $g(z)=f(z)-z.$ Then $ |zg(z)|\leq \dfrac{a-z}{1-\overline{a}z}, \dfrac{b-z}{1-\overline{b}z},$ where $a,b$ are the distinct fixed points. How should I proceed from here?
Thank you.
Best Answer
Denote the fixed points by $\mu$ and $\eta$ ($\mu\ne\eta$) and define $$\varphi_{\mu}(z)=\frac{\mu-z}{1-\bar\mu z},$$then $\varphi_\mu$ is an automorphism of $D(0,1)$ such that $\varphi_{\mu}(0)=\mu$, $\varphi_{\mu}(\mu)=0$ and $\varphi_{\mu}\circ \varphi_{\mu}=\rm{Id}_{D(0,1)}$. (You can check this or see it in Stein's book.)
Now, it follows that $$\varphi_{\mu}\circ f\circ\varphi_{\mu}(0)=0\quad \text{and}\quad\varphi_{\mu}\circ f\circ \varphi_{\mu}\circ\varphi_{\mu}(\eta)=\varphi_{\mu}(\eta).$$
Since $\varphi_{\mu}\circ f\circ\varphi_{\mu}$ is a holomorphic map from $D$ to $D$ and $\varphi_{\mu}(\eta)\ne 0$, by Schwarz Lemma, it follows that $\varphi_{\mu}\circ f\circ\varphi_{\mu}=\rm{Id_{D}}$, hence $f=\rm{Id_{D}}$.