On iterates of rational functions with maximal number of critical values
摘要
Let F be a rational function of one complex variable of degree m ≥ 2. The function F is called simple if for every z ∈ ℂℙ1 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 = 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 Möbius transformations μi, 1 ≤ i ≤ r − 1, such that Gr = F ◦ μr−1, Gi = μi−1 ◦ F ◦ μi−1, 1 < i < r, and G1 = μ1−1 ◦ F. As applications, we solve a number of problems in complex and arithmetic dynamics for “general” rational functions.