论文标题
在全球功能领域的理性地图的循环，降低了一个不良的元素
Cycles for rational maps over global function fields with one prime of bad reduction
论文作者
论文摘要
令$ k $是特征$ p $和度数$ d $ over $ \ mathbb f_ {p}（t）$的全局函数字段。我们考虑在投影线上$ \ mathbb p^1（k）上的动态系统，这是由理性地图定义的，最多是一个不良减少的素数。主要结果是仅取决于$ p $和$ d $的周期长度的最佳界限。也给出了有限轨道基数的绑定。我们的方法是基于对属于同一有限轨道的点之间的$ \ Mathfrak P $ ADIC距离进行仔细的分析（对良好的降低量），部分是由Canci和Paladino先前的工作动机。有价值的见解由某些多项式家族提供。在这种情况下，我们还获得了有关给定程度多项式的周期点的结构和大小的大量信息。
Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The main result is an optimal bound for cycle lengths that only depends on $p$ and $D$. A bound for the cardinality of finite orbits is given as well. Our method is based on a careful analysis (for every prime of good reduction) of the $\mathfrak p$-adic distances between points belonging to the same finite orbit, in part motivated by previous work by Canci and Paladino. Valuable insight is provided by a certain family of polynomials. In this case we also gain a good deal of information about the structure and size of the set of periodic points for polynomials of given degree.
