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

TL;DR

The paper proves that for general rational functions, their iterates have unique indecomposable decompositions, and characterizes when certain endomorphisms admit non-trivial periodic curves.

Contribution

It establishes the uniqueness of indecomposable decompositions of iterates for general rational functions and characterizes periodic curves in product endomorphisms.

Findings

Decompositions of iterates are essentially unique for general rational functions.

Periodic curves in product endomorphisms occur only when the functions are conjugate.

Characterization of when endomorphisms admit non-trivial periodic curves.

Abstract

We show that for a general rational function $A$ of degree $m \geq 2$, any decomposition of its iterate $A^{\circ n}$, $n \geq 1$, 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.