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We study an open question at the interplay between the classical and thedynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be analgebraically closed field of positive characteristic, let $G$ be a finitelygenerated subgroup of the multiplicative group of $K$, and let $X$ be a(irreducible) quasiprojective variety defined over $K$. We consider $K$-valuedsequences of the form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colonX\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational mapsdefined over $K$ and $x_0\in X$ is a point whose forward orbit avoids theindeterminacy 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 ofupper 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} amultiplicative 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 variousapplications of our results.

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