= Critical points of iterates of a rational map
{title2=$\operatorname{Crit}(R^n)=\bigcup_{j<n}R^{-j}\operatorname{Crit}(R)$}
Local degrees multiply under composition: $e_p(S\circ R)=e_p(R)e_{R(p)}(S)$, directly by substituting the two local power-series expansions. Consequently
$$
\operatorname{Crit}(R^n)=\bigcup_{j=0}^{n-1}R^{-j}(\operatorname{Crit}(R)).
$$
In particular every critical point of $R$ remains critical for each positive iterate, and every critical point of an iterate reaches a critical point of $R$ in fewer than $n$ steps. This covers infinity as well as finite points. For polynomials at finite points the same conclusion follows from $(P^n)'(z)=\prod_{j=0}^{n-1}P'(P^j(z))$.
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