Periodic curves for general endomorphisms of $\mathbb{C}{\mathbb{P}}^1\times \mathbb{C}{\mathbb{P}}^1$

We show that for a general rational function $A$ of degree $m\ge 2$, any decomposition of its iterate $A^{\circ n}$, $n\ge 1$, into a composition of indecomposable rational functions is equivalent to the decomposition $A^{\circ n}$ itself. As an application, we prove that if $(A_1,A_2)$ is a pair of general rational functions, then the endomorphism of $\mathbb{C}{\mathbb{P}}^1\times \mathbb{C}{\mathbb{P}}^1$ given by $(z_1,z_2)\mapsto (A_1(z_1),A_2(z_2))$ admits a periodic curve that is neither a vertical nor a horizontal line if and only if $A_1$ and $A_2$ are conjugate.