Find Mobius transformations $\varphi$ satisfying $\sum (1-|\varphi_n(z)|)<\infty$ in the unit disc.

Question

Suppose that $\varphi$ is a Mobius transformation which maps the unit disc onto itself. Let $\varphi_n(z)=\varphi(\varphi_{n-1}(z))$, where $n=1,2,\ldots$ and $\varphi_0(z)=z$. Find all $\varphi$ satisfying the condition $\sum (1-|\varphi_n(z)|)<\infty$ in the unit disc if $\varphi$ has a unique fixed point on the unit circle. My thought: Suppose $\varphi(z)=\frac{az+b}{cz+d}$ with $ad-bc=1$. From the hypotheses, we know that $c \neq 0$, $a+d=\pm 2$ and the fixed point will be $\frac{\pm 1-d}{c}$. Then I don't know how to use these to determine $\varphi$. Any help?

Answer 1

I think the following solution works: Represent \begin{equation*} \varphi(z)=\lambda \cdot\frac{z-\alpha}{1-\overline{\alpha}z} \end{equation*} for some $\alpha \in D$ and $|\lambda|=1$. Let the fixed point be $\beta$. Then we can show that \begin{equation*} \varphi_n(z)=\frac{(1+nK\beta)z-nK\beta^2}{nKz+(1-nK\beta)} \end{equation*} for some nonzero $K$. As $1-|\varphi_n(0)|$ is approximately $\frac{1}{n|K|-1}$ for sufficiently large $n$, so $\varphi$ does not satisfy the condition \begin{equation*} \sum (1-|\varphi_n(z)|)<\infty \end{equation*} for all $z \in D$.