Tame rational functions: Decompositions of iterates and orbit intersections

Let $A$ be a rational function of degree at least two on the Riemann sphere. We say that $A$ is tame if the algebraic curve $A(x)-A(y)=0$ has no factors of genus zero or one distinct from the diagonal. In this paper, we show that if tame rational functions $A$ and $B$ have orbits with infinite intersection, then $A$ and $B$ have a common iterate. We also show that for a tame rational function $A$ decompositions of its iterates $A^{\circ d},$ $d\geq 1,$ into compositions of rational functions can be obtained from decompositions of a single iterate $A^{\circ N}$ for $N$ big enough.