Why is it true that if a Möbius map, $f(z)$ fixes distinct $z_1,z_2\in \mathbb C_\infty$ that $f(z)$ either describes a rotation or has a pair of stable and unstable fixed points, such that iterations of $f(z)$ converges to the stable fixed point for $z\neq$ the unstable fixed point?
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The fairly standard argument is to conjugate with a Moebius transformation that sends $0$ to $z_1$ and $\infty$ to $z_2$. This gives you a Moebius transformation that fixes $0$ and $\infty$. A little algebra tells you this is just the map $z \longmapsto az$ for some complex number $a$.