论文标题
多项式动态和小轨道的本地分析
Polynomial dynamics and local analysis of small and grand orbits
论文作者
论文摘要
我们证明了Manin-Mumford猜想在数字字段上用于多项式动力学系统的类似物。在我们的设置中，扭转点的作用是由点$α$的小轨道占据的。 McMullen和Sullivan在研究理性地图动力学的研究中引入了一个点的小轨道，其中\ begin {align*} \ mathcal {s}_α= \ {phation = \ in \ in \ in \ mathbb {c} c} c}; f^{\ circ n}（β）= f^{\ circ n}（α）\ text {对于某些} n \ in \ mathbb {z} _ {\ geq 0} \}。我们的主要定理是对代数关系的分类，当在代数数字上定义了所有内容，而$ f $ $ f $ d $ d $ d $至少在$ f $上定义了一切时，至少有2 $的证明是对本地化的仔细证明，与此相比，相比，$ f $的证明是相比，在$ f $上定义了一切时，它在$ \ mathcal {s}_α$中都存在。在$ k $的无限位置，我们使用已知的Fatou和Levin的刚性定理来证明这种新事物。这些可能对复杂的动态具有独立的兴趣。在有限的地方，我们介绍了新的非架构方法，以研究可能适用于其他算术环境中的养生问题。我们在有限位置的方法使我们能够对大轨道\ begin {align*} \ mathcal {g}_α= \ {β\ {β\ in \ mathbb {c}中无限对成对的所有代数关系进行分类; f^{\ circ n}（β）= f^{\ circ m}（α）\ text {对于某些} n，m \ in \ mathbb {z} _ {z} _ {\ geq 0} \ geq 0} \} \ end end end {align {align {align {align {align {align {有限的地方$ v $良好的还原联合总理至$ d $。这是多项式动力学有限等级组的Mordell-Lang猜想的类似物。
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.
