A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic

TL;DR

This paper investigates the intersection patterns of sequences generated by rational maps on algebraic varieties over fields of positive characteristic, revealing structured behaviors and connections to algebraic tori.

Contribution

It introduces a fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic, showing structured sets of indices and characterizing sequences via algebraic tori.

Findings

The set of indices where the sequence hits a finitely generated subgroup is a finite union of arithmetic progressions plus a density zero set.

Sequences entirely contained in the subgroup correspond to algebraic tori and iterates of rational maps.

Results have various applications in algebraic dynamics and number theory.

Abstract

We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a (irreducible) quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. We show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and the $\varphi$ orbit of $x$ is Zariski dense in $X$ then {there is} a multiplicative torus $\mathbb{G}_m^d$ and maps $\Psi:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ \Psi^n(y)$ for some $y\in \mathbb{G}_m^d$. We then describe various applications of our results.