论文标题
关于完美领域的理性功能的迭代
On iterations of rational functions over perfect fields
论文作者
论文摘要
令$ \ mathbb k $是字符$ p \ ge 0 $的完美字段，让$ r \ in \ athbb k（x）$成为一个合理的函数。本文研究了$ r^{（n）}（x）（x）=α$的数字$δ_{α，r}（n）$上的$ r^{（n）}（x）=α$上的代数关闭$ \ overline {\ mathbb k} $的$ \ \ \ \ m athbb k $ of mathbb k $ $ n $ - 折叠的$ r $与自身的成分。除了某些对$（α，r）$以外，我们证明$δ__{α，r}（n）= c_ {α，r} \ cdot d^n+o_ {α，r}（1）$ for约$ 0 <c_ {c_ {α，r} \ r} \ le 1 <d $。数字$ d $很容易从$ r $获得，我们提供了$ c_ {α，r} $的估算值。此外，我们证明了每个$ n \ ge 0 $的特殊对$（α，r）$满足$δ_{α，r}（n）\ le 2 $，我们充分描述了它们。我们还讨论了进一步的问题，并提出了一些问题，在$ \ mathbb k $有限的情况下。
Let $\mathbb K$ be a perfect field of characterstic $p\ge 0$ and let $R\in \mathbb K(x)$ be a rational function. This paper studies the number $Δ_{α, R}(n)$ of distinct solutions of $R^{(n)}(x)=α$ over the algebraic closure $\overline{\mathbb K}$ of $\mathbb K$, where $α\in \overline{\mathbb K}$ and $R^{(n)}$ is the $n$-fold composition of $R$ with itself. With the exception of some pairs $(α, R)$, we prove that $Δ_{α, R}(n)=c_{α, R}\cdot d^n+O_{α, R}(1)$ for some $0<c_{α, R}\le 1<d$. The number $d$ is readily obtained from $R$ and we provide estimates on $c_{α, R}$. Moreover we prove that the exceptional pairs $(α, R)$ satisfy $Δ_{α, R}(n)\le 2$ for every $n\ge 0$, and we fully describe them. We also discuss further questions and propose some problems in the case where $\mathbb K$ is finite.
