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

Abstract

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 $A^{\circ d}(x)-U(y)=0$ 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 $l\ge 1.$ 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 $K.$

Extended and polished version

References in corpus (2)

• Finiteness theorems for commuting and semiconjugate rational functions
• On generalized Lattès maps