标题： 动力学族的正则多项式自映射在 $\mathbb{P}^2$ 上具有许多周期曲线 显示英文标题

标题： Dynamical families of regular polynomial endomorphisms on $\mathbb{P}^2$ with many periodic curves

摘要： 在本文中，我们证明了对于$\mathbb{P}^2$上的正次数常规多项式自映射，包含周期曲线的 Zariski 稠密集的曲线族在该自映射的某个迭代下是不变的。 这一设定与 DeMarco 和 Mavraki 最近提出的相对动力学 Manin-Mumford 猜想密切相关，该猜想涉及参数化的自映射和代数簇族。 我们的结果证明了该猜想的一个较弱版本，其中自映射是在$\mathbb{P}^2$上的常规多项式自映射，在族中保持固定，且该曲线族包含周期曲线的稠密集。 这一结果也可以看作是在除子模空间上的动力学 Manin-Mumford 类型陈述，并在更强假设下证明了动力学 Manin-Mumford 猜想的一个特例。 此外，我们的结果特别意味着在常规多项式自映射作用下，族中一个一般的曲线集合具有统一的次数稳定陈述。 我们证明了一个更一般的关于$\mathbb{P}^K$中正维度子簇族在自映射族作用下的次数稳定陈述是由相对动力学 Manin-Mumford 猜想预测的。 然后我们证明了当$K=2$时，在某些关于无穷远线上分支的限制条件下，该结论对常规多项式自映射族成立。 最后，我们将我们的结果应用于分类所有接受无限多个有界次数周期曲线的常规多项式自映射。 显示英文摘要

摘要： In this paper, we prove that for a regular polynomial endomorphism of positive degree on $\mathbb{P}^2$, a family of curves containing a Zariski dense set of periodic curves is invariant under some iterate of the endomorphism. The setting is closely related to the Relative Dynamical Manin-Mumford Conjecture, recently proposed by DeMarco and Mavraki, which concerns a parametrized family of endomorphisms and varieties. Our result proves a weaker version of the conjecture where the endomorphism is a regular polynomial endomorphism on $\mathbb{P}^2$ that remains fixed in the family, and the family of curves contains a dense set of periodic curves. This result can also be viewed as a Dynamical Manin-Mumford type statement on the moduli space of divisors, and it proves a special case of the Dynamical Manin-Mumford Conjecture with a stronger assumption. Moreover, our result specifically implies a uniform degree stabilization statement for a generic set of curves in a family under the transformation of a regular polynomial endomorphism. We demonstrate that a more general degree stabilization statement for a family of positive dimension subvarieties in $\mathbb{P}^K$ under the transformation of a family of endomorphisms is predicted by the Relative Dynamical Manin-Mumford Conjecture. We then prove that it is true when $K=2$ for families of regular polynomial endomorphisms under certain restrictions on the ramifications at the line at infinity. Finally, we demonstrate an application of our result to classify all regular polynomial endomorphisms that admit infinitely many periodic curves of bounded degree.