Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the only such functions are those of the form $cx^j(\psi(x))^m$ with $\psi \in K(x)$, and for $m \leq 4$ we show the only additional cases are certain Lattes maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n \geq 0 : \phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ is a union of finitely many arithmetic progressions, where $\phi^{n}$ denotes the $n$th iterate of $\phi$ and $\lambda \in K(x)$ is any map Mobius-conjugate over $K$ to $x^m$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell-Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^m = \phi^{n}(x)$. We describe all $\phi$ for which these curves have an irreducible component of genus at most 1, and show that such $\phi$ must have two distinct iterates that are equal in $K(x)^*/K(x)^{*m}$.

中文翻译：

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有理函数轨道上的幂：算术动力学 Mordell-Lang 猜想的例子

令 $K$ 是特征为零的有限生成域。对于固定的 $m \geq 2$，我们研究了在 $K$ 上定义的有理函数 $\phi$，其具有包含无限多个不同的 $m$th 次幂的 $K$ 轨道。对于 $m \geq 5$ 我们展示的唯一这样的函数是 $cx^j(\psi(x))^m$ 和 $\psi \in K(x)$ 的形式，对于 $m \leq 4$ 我们展示的唯一附加情况是某些 Lattes 映射和四个有理函数族，它们的特殊性质似乎以前没有被研究过。通过额外的分析，我们表明索引集 $\{n \geq 0 : \phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ 是有限多个算术级数，其中 $\phi^{n}$ 表示 $\phi$ 的第 $n$ 个迭代，$\lambda \in K(x)$ 是 $K$ 到 $x 的任何映射莫比乌斯共轭^米$。当索引集为无穷大时，我们给出了所涉及的等差数列的数量和模数的界限。这些结果与动力学 Mordell-Lang 猜想相似，并激发了关于轨道与态射值集的交点的新猜想。我们证明中的一个关键要素是研究曲线 $y^m = \phi^{n}(x)$。我们描述了所有 $\phi$，其中这些曲线具有至多 1 的不可约分量，并证明这样的 $\phi$ 必须有两个不同的迭代，它们在 $K(x)^*/K(x) 中相等^{*m}$。