Attracting periodic point ($|(R^n)\prime (p)|<1$)

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 $|G^{\prime }(p)|$ and one. Taylor expansion gives $|G(z)-p|\le q|z-p|$ 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.

A holomorphic periodic point is superattracting if its return multiplier is zero. The return map then has local expansion $G(p+u)=p+a{u}^e+O(u^{e+1})$ with $e\ge 2$. This implies a strict contraction on a sufficiently small disc and rapid convergence of returns. Infinity is superattracting for every polynomial of degree at least two: the reciprocal-coordinate return has a zero of order equal to the polynomial degree.