A sparsity result for the Dynamical Mordell-Lang Conjecture in positive characteristic

Abstract

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 $p$, given a semiabelian variety $X$ defined over a finite subfield of $K$ and endowed with a regular self-map $\Phi :X⟶X$ defined over $K$, given a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$, then the set of all non-negative 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 $N$, $\Phi$, $\alpha$, $V$) such that for each positive integer $M$, we have $$\#\left\{n\in S:n\le M\right\}\le A\cdot {\left(1+\log M\right)}^{\dim V}.$$