= Möbius conjugacy normal forms
{c}
{title2=$z\mapsto\mu z\ (\mu\ne0)\quad\text{or}\quad z\mapsto z+1$}
A nonidentity <Möbius transformation> has one or two distinct <fixed points> on the <Riemann sphere>. If there are two, send them to zero and infinity; the conjugated map fixes both and is a nonzero scaling. If there is one, send it to infinity; the resulting affine map cannot have slope different from one, since that would give another fixed point. It is a nonzero translation, which a scaling conjugates to $z+1$. The identity is the scaling $\mu=1$. Fixed-point sets are carried bijectively under conjugacy, so a translation is not conjugate to a scaling.
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