Critical points of iterates of a rational map ($\mathrm{Crit}(R^n)={\bigcup }_{j<n}{R}^{-j}\mathrm{Crit}(R)$)

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. ConsequentlyIn 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 $(P^n{)}^{\prime }(z)={\prod }_{j=0}^{n-1}{P}^{\prime }(P^j(z))$.