On iterates of rational functions with maximal number of critical values

Abstract: Let $F$ be a rational function of one complex variable of degree $m\geq 2$. The function $F$ is called simple if for every $z\in \mathbb C\mathbb P^1$ the preimage $F^{-1}{z}$ contains at least $m-1$ points. We show that if $F$ is a simple rational function of degree $m\geq 4$ and $F^{\circ l} =G_r\circ G_{r-1}\circ \dots \circ G_1$, $l\geq 1$, is a decomposition of an iterate of $F$ into a composition of indecomposable rational functions, then $r=l$ and there exist M\"obius transformations $\mu_i,$ $1\leq i \leq r-1,$ such that $G_r=F\circ \mu_{r-1},$ $G_i=\mu_{i}^{-1}\circ F \circ \mu_{i-1},$ $1<i< r,$ and $G_1=\mu_{1}^{-1}\circ F$. As applications, we solve a number of problems in complex and arithmetic dynamics for "general" rational functions.