证明了数域上多项式动力系统的Manin-Mumford猜想的一个类似结果。在我们的设置中，扭点的作用由点$α$的小轨道承担。小轨道的一个点介绍了麦克马伦和沙利文在他们的研究动力学的合理地图，其中一个点$α$和一个多项式$f$它是由\begin{align*} \mathcal{S}_α= \{β\in \mathbb{C}; f^{\circ n}(β) = f^{\circ n}(α) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*}我们的主要定理是一个分类的代数关系之间的无限对点在$\mathcal{S}_α$时，一切都定义在代数数和程度$d$的$f$是至少2。我们的证明依赖于对动力系统局部化的仔细研究，并遵循与这一领域以前的证明完全不同的方法。在无限的地方$K$，我们使用已知的刚性定理Fatou和莱文证明新的。这些可能是复杂动力学的独立兴趣。在有限的地方，我们引入新的非阿基米德的方法来研究丢番图问题，可能适用于其他算术环境。我们的方法在有限的地方允许我们分类所有的代数关系，持有无限对点的大轨道\begin{align*} \mathcal{G}_α= \{β\in \mathbb{C}; f^{\circ n}(β) = f^{\circ m}(α) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*}的$α$，如果$|f^{\circ n}(α)|_v \rightarrow \infty$在有限的地方$v$的良好减少互质$d$。这是一个类似的莫德尔朗猜想有限秩群多项式动力学。
英文摘要：
We prove an analogue of the Manin-Mumford conjecture for polynomial dynamical systems over number fields. In our setting the role of torsion points is taken by the small orbit of a point $α$. The small orbit of a point was introduced by McMullen and Sullivan in their study of the dynamics of rational maps where for a point $α$ and a polynomial $f$ it is given by \begin{align*} \mathcal{S}_α= \{β\in \mathbb{C}; f^{\circ n}(β) = f^{\circ n}(α) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely pairs of points in $\mathcal{S}_α$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localizations of the dynamical system and follows an entirely different approach than previous proofs in this area. At infinite places of $K$ we use known rigidity theorems of Fatou and Levin to prove new such. These might be of independent interest in complex dynamics. At finite places we introduce new non-archimedean methods to study diophantine problems that might be applicable in other arithmetic contexts. Our method at finite places allows us to classify all algebraic relations that hold for infinitely pairs of points in the grand orbit \begin{align*} \mathcal{G}_α= \{β\in \mathbb{C}; f^{\circ n}(β) = f^{\circ m}(α) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $α$ if $|f^{\circ n}(α)|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$ . This is an analogue of the Mordell-Lang conjecture on finite rank groups for polynomial dynamics.
