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 every z ∈ ℂℙ 1 the preimage F −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 = G r ◦ G r −1 ◦ ⋯ ◦ G 1 , l ≥ 1, is a decomposition of an iterate of F into a composition of indecomposable rational functions, then r = l and there exist Möbius transformations μ i , 1 ≤ i ≤ r − 1, such that G r = F ◦ μ r −1 , G i = μ i −1 ◦ F ◦ μ i −1 , 1 < i < r , and G 1 = μ 1 −1 ◦ F . As applications, we solve a number of problems in complex and arithmetic dynamics for “general” rational functions.