On iterates of rational functions with maximal number of critical values

Let $F$ be a rational function of one complex variable of degree $m\ge 2$. 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 $m\ge 4$ and $F^{\circ l} =G_r\circ G_{r-1}\circ \dots \circ G_1$,$l\geq 1$,isadecompositionofaniterateof$F$intoacompositionofindecomposablerationalfunctions,then$r=l$ and there exist M"obius transformations ${\mu }_i,$ $1\le i\le r-1,$ 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.