On iterates of rational functions with maximal number of critical values

Let F be a rational function of one complex variable of degree m ≥ 2. The function F is called simple if for each z ∈ CP¹ the preimage P−1{z} contains at least m − 1 points. We show that if F is a simple rational function of degree m ≥ 4 and F ^◦l = Gr ◦Gr−1 ◦· · · ◦G1, l ≥ 1, is a decomposition of an iterate of F into a composition of indecomposable rational functions, then r = l, and there exist M¨obius transformations µi, 1 ≤ i ≤ r − 1, such that Gr = F ◦µr−1, Gi = µi⁻¹◦F ◦µ_i−1, 1 < i < r, and G1 = µ⁻¹1◦F. As an application, we provide explicit solutions of a number of problems in complex and arithmetic dynamics for “general” rational functions.