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

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 $\pm T_n\mathrm{.}$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of $({\mathbb{P}}^1(\mathbb{C}){)}^2$ of the form $(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$Inparticular,weshowthatif$A\in \mathbb C(z)$ 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 $(d_1,d_2)\mathrm{.}$