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 $d\geq 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$ has a factor of genus zero or one. In particular, we show that if $A$ is not a “generalized Lattès map”, then this condition is satisfied if and only if there exists a rational function $V$ such that $U\circ V=A^{\circ l}$ for some $l\geq 1$. 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 $K$.