Dynamical Mordell–Lang conjecture

Remarks

A set $\set{ak + b : k \in \N}$ with $a = 0$ is the single point $\set{b}$, so finite sets are allowed. An equivalent form concerns the orbit $\set{\Phi^n(x) : n \in \N}$ of $x$. If the orbit meets $V$ in infinitely many points, then $V$ contains a positive-dimensional subvariety $W$ with $\Phi^a(W) \subseteq W$ for some $a \ge 1$ and $\Phi^b(x) \in W$ for some $b$. The conjecture is open in general. Bell, Ghioca and Tucker proved it when $\Phi$ is étale. When $X = \C^N$ and $\Phi$ is linear, it is the Skolem–Mahler–Lech theorem on the zeros of a linear recurrence. When $\Phi$ is a translation on a semiabelian variety, it follows from the Mordell–Lang theorem of Faltings and Vojta, which gives the name. The main tool is $p$-adic. On suitable residue classes of $n$, the map $n \mapsto \Phi^n(x)$ is interpolated by a $p$-adic analytic function. A nonzero analytic function on $\Z_p$ has finitely many zeros. Over fields of positive characteristic the statement fails. For example, over $\F_p(t)$ let $\Phi(x, y) = (tx, (1 + t)y)$ on $\mathbb{G}_m^2$, let $V$ be the line $y = x + 1$ and start at $(1, 1)$. Then $\Phi^n(1, 1) = (t^n, (1 + t)^n)$ lies in $V$ exactly when $(1 + t)^n = t^n + 1$, that is, when $n$ is a power of $p$.