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

Abstract

We show that for a general rational function $A$ of degree $m\ge 2$, every decomposition of an iterate $A^{\circ n}$, $n\ge 1$, into a composition of indecomposable rational functions is equivalent to the decomposition $A\circ \cdots \circ A$ with $n$ factors. As an application, we prove that for every $m\ge 2$ there exists a non-empty Zariski open subset $U\subset {\mathrm{Rat}}_m$ such that, for any $A_1,A_2\in U$, the endomorphism of ${\mathbb{CP}}^1\times {\mathbb{CP}}^1$ given by $(z_1,z_2)\mapsto (A_1(z_1),A_2(z_2))$ admits an irreducible periodic algebraic curve that is neither a vertical nor a horizontal line if and only if $A_1$ and $A_2$ are conjugate.