---
tag: 6TV7FR553H
kind: conjecture
name: Dynamical Mordell–Lang conjecture
msc: [37P55, 37P15]
uses: [EQSHJPE7H8, CDS1VWB7A3, 4YSCK1DF26, T75CZ5H47N, AYVTR2ES5B]
about: [BE4X0QG8XW]
sources:
  - url: https://en.wikipedia.org/wiki/Arithmetic_dynamics
history:
  posed_by: "Dragos Ghioca and Thomas J. Tucker"
  first_stated: "2009"
---

## Statement

Let $X$ be a [[quasi-projective variety|EQSHJPE7H8]] over $\C$ and let $\Phi \colon X \to X$ be a [[morphism|CDS1VWB7A3]].
Write $\Phi^n$ for the $n$-th [[iterate|AYVTR2ES5B]] of $\Phi$, with $\Phi^0 = \id$.
Let $V \subseteq X$ be a closed [[subvariety|4YSCK1DF26]] and let $x \in X(\C)$.
Then the set
$$\set{n \in \N : \Phi^n(x) \in V}$$
is a finite union of sets of the form $\set{a k + b : k \in \N}$ with $a, b \in \N$.

## 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|0NDVHBVBXB]].
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 $p$.

## Explanations

### L1 human

Start at a point and apply the same polynomial map again and again.
Ask at which steps the point lands on a fixed curve or surface.
The conjecture predicts that these steps are never irregular: they form finitely many arithmetic progressions, plus finitely many extra steps.
So either the orbit hits the surface only a few times, or it hits it in a regular repeating pattern.