Book: An introduction to Teichmuller Space by Imayoshi & Taniguchi.
Let $\gamma$ be a Mobius transformation on $\Bbb C$ such that $\Bbb H\to\Bbb H$ (Automorphism of upper half plane). Suppose
$$\gamma(z) = {az+b\over cz+d},\quad ad-bc =1,c>0$$
such that $1$ is the attractive fixed point of $\gamma$, i.e., we can conjugate $\gamma$ in $\mathrm{Aut}(\Bbb H)$ so that it becomes a map $z\mapsto \lambda z$ such that $\Bbb R\ni\lambda \neq 0,1$ and if $0<\lambda<1$ then $0$ in this new map corresponds to $1$ in the original map and if $\lambda>1$ then $\infty$ in this new map corresponds to $1$in the original map (if I understand the definition correctly). Now the book say that this 'attractive' condition is equivalent to the followings:
$$a+b= c+d,\quad 0<-{b\over c}<1.$$
The first condition says $1$ is a fixed point of $\gamma$. I think the second condition is saying some attractiveness of $\gamma$ at $1$ but I don't understand why. So far I found that $-{b\over c}$ is another fixed point $\gamma$.
Using these two conditions, books concludes $\gamma(\infty) = a/c>0$ and $a+d>0$ because the middle-point $(a-d)/(2c)$ of two fixed points of $\gamma_2$ has a value less than $\gamma_2(\infty)$. I don't understand the reasoning. Could you explain? I think I don't understand the concept of attractive and repelling (fixed) points.
