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

TL;DR

This paper characterizes pairs of complex rational functions whose iterated difference curves have genus at most one, linking their structure to special maps called generalized Lattès maps and exploring a dynamical Mordell-Lang conjecture.

Contribution

It provides a complete description of such rational function pairs and establishes a version of the dynamical Mordell-Lang conjecture for these functions over number fields.

Findings

Characterization of rational functions with genus-zero or one iterated difference curves.

Identification of conditions under which $A$ and $U$ are related via a rational function $V$.

Proof of a dynamical Mordell-Lang conjecture for rational functions over number fields.

Abstract

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 $K.$