  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 Latt\`es map', then this condition is satisfied if
and only if there exists a rational function $V$ such that $U\circ V=A^{\circ
l}$ 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 $K.$
