We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$ , then the set of all nonnegative integers n such that $\Phi ^n(\alpha )\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $ , $\alpha $ and V) such that for each positive integer M, $$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$

中文翻译：

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正特征的动态莫德尔-朗猜想的稀疏结果

我们证明了支持动力学 Mordell-Lang 猜想的定量部分结果（也称为DML 猜想) 在积极的特点。更准确地说，我们展示了以下内容：给定一个字段ķ有特色的p, 一个半阿贝尔变体X在有限子域上定义ķ并具有规律的自我映射$\Phi :X{\longrightarrow } X$定义在ķ， 一个点$\alpha \in X(K)$和一个子品种$V\子集 X$, 那么所有非负整数的集合n这样$\Phi ^n(\alpha )\in V(K)$是有限多个算术级数与子集的并集小号具有存在正实数的性质一种（仅取决于X,$\披$,$\阿尔法$和五) 使得对于每个正整数米,$$\begin{align*}\scriptsize\#\{n\in S\冒号 n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align* }$$