Algebraic curves $A^{\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functions

We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\ge 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$hasafactorofgenuszeroorone\mathrm{.}Inparticular,weshowthatif$A$ is not a generalized Latt\es map’, then this condition is satisfied if and only if there exists a rational function $V$ such that $U\circ V=A^{\circ l}$forsome$l\geq 1.$ We also prove a version of the dynamical Mordell-Lang conjecture, concerning intersections of orbits of points from ${\mathbb{P}}^1(K)$ under iterates of $A$ with the value set $U({\mathbb{P}}^1(K))$, where $A$ and $U$ are rational functions defined over a number field $K\mathrm{.}$