标题： 一般自同态的周期曲线$\mathbb C\mathbb P^1\times \mathbb C\mathbb P^1$ 显示英文标题

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

摘要： 我们证明对于一个次数为$m \geq 2$的一般有理函数$A$，其迭代$A^{\circ n}$，$n \geq 1$分解为不可分解有理函数的复合形式，与分解$A^{\circ n}$本身是等价的。 作为应用，我们证明如果$(A_1, A_2)$是一对一般的有理函数，则由$(z_1, z_2) \mapsto (A_1(z_1), A_2(z_2))$给出的$\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$的自同态存在一条既不是垂直也不是水平的周期曲线，当且仅当$A_1$和$A_2$是共轭的。 显示英文摘要

摘要： 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 general pair of 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.