= Attracting periodic point
{title2=$|(R^n)\prime(p)|<1$}
A holomorphic periodic point is attracting when its return multiplier has modulus strictly less than one. For the return map $G=R^n$, choose a local coordinate at the fixed point and a number $q$ strictly between $|G'(p)|$ and one. Taylor expansion gives $|G(z)-p|\leq q|z-p|$ on a sufficiently small disc. It maps that disc into itself, and successive returns converge uniformly to $p$ there. Composing with the finitely many intervening maps gives locally uniform convergence to the periodic orbit and places its basin in the <Fatou set>.
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