AI 中文总结
本文研究数域上规范化多项式族的轨道碰撞问题，在自然假设下给出了存在无穷多参数使得两个非预周期点的迭代轨道同时到达指定点的精确充要条件。
AI 中文摘要
设$d\ge 2$为整数，$c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$。我们考虑由$\lambda\in\bar{\mathbb{Q}}$参数化的规范化多项式族$f_\lambda(z):=z^d+\sum_{i=0}^{d-2} c_i(\lambda)\cdot z^i$；该多项式族的泛元素为$f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in \bar{\mathbb{Q}}[t][z]$。另外设$\alpha_1(t),\alpha_2(t),\beta(t)\in\bar{\mathbb{Q}}[t]$，其中对每个$i=1,2$，$\alpha_i(t)$在$f_t(z)$的作用下不是预周期的。在一些自然假设下，我们得到了存在无穷多$\lambda\in\bar{\mathbb{Q}}$使得存在依赖于$\lambda$的$m,n\in\mathbb{N}$，满足$f_\lambda^m(\alpha_1(\lambda))=f_\lambda^n(\alpha_2(\lambda))=\beta(\lambda)$的精确充要条件。
英文摘要
Let $d\ge 2$ be an integer and let $c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$. We consider the family of normalized polynomials $f_λ(z):=z^d+\sum_{i=0}^{d-2} c_i(λ)\cdot z^i$ parameterized by $λ\in\bar{\mathbb{Q}}$; the generic element of our family of polynomials is $f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in \bar{\mathbb{Q}}[t][z]$. Also, let $α_1(t),α_2(t),β(t)\in\bar{\mathbb{Q}}[t]$, where $α_i(t)$ is not preperiodic under the action of $f_t(z)$ for each $i=1,2$. Under some natural hypotheses, we obtain precise necessary and sufficient conditions for which there exist infinitely many $λ\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $λ$), we have that $f_λ^m(α_1(λ))=f_λ^n(α_2(λ))=β(λ)$.
