Invariant curves for endomorphisms of $\mathbb P^1\times \mathbb P^1$

Abstract

Let $A_1, A_2\in \mathbb C(z)$ be rational functions of degree at least two that are neither Latt\`es maps nor conjugate to $z^{\pm n}$ or $\pm T_n.$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of $(\mathbb P^1(\mathbb C))^2$ of the form $(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$ In particular, we show that if $A\in \mathbb C(z)$ is not a "generalized Latt\`es map", then any $(A,A)$-invariant curve has genus zero and can be parametrized by rational functions commuting with $A$. As an application, for $A$ defined over a subfield $K$ of $ \mathbb C$ we give a criterion for a point of $(\mathbb P^1(K))^2$ to have a Zariski dense $(A, A)$-orbit in terms of canonical heights, and deduce from this criterion a version of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits. We also prove a result about functional decompositions of iterates of rational functions, which implies in particular that there exist at most finitely many $(A_1, A_2)$-invariant curves of any given bi-degree $(d_1,d_2).$