 Critical points of iterates of a rational map

ID: critical-points-of-iterates-of-a-rational-map

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

Crit (R^{n}) = ⋃_{j = 0}^{n - 1} R^{- j} (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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