Why is the Dynamical Mordell-Lang conjecture interesting?

The gist of the Dynamical Mordell-Lang conjecture is:

Let's say you have a set $S$. Inside $S$, you have a point $x$ where you start off at. Now, you also have a function $f: S \rightarrow S$ (is that how I correctly write "maps from something in $S$ to something else in $S$"?). So each time you apply $f$ to $x$, you land at another point inside $S$ ("duh"), but what if you keep landing inside a closed subset of $S$, $Y$? Not just once ("duh"), a few times ("hrm"), or a billion times ("hmmm"), but an infinite number of times?

Clearly, $Y$ must have some special relationship to $f$? Some special "structure"? Recently, it was proved that that there is some number $k \in \mathbb{N}$ such that $Z \subset Y$ is invariant under the "$k$th iterant" of $f$, $f^{\circ k}$ (i.e. $f$ applied $k$ times to $x$).

Why is Dynamical Mordell-Lang interesting? Which open questions might eventually benefit from having such a result? What new mathematical vistas does it expose? Or, to what long program of work does it add?

I recall that the speaker mentioned that there was is an interesting application of this conjecture to the theory of differential equations, and that in fact, someone working on differential equations about 30 or 40 years ago conjectured that there should be a "Dynamical Mordell-Lang"...perhaps this tidbit, if it isn't too vague, might be a helpful starting point for answers?

See this meta.MO question for background on this MO question

Maybe a brief history would provide an answer. The original Mordell conjecture asserted that if a curve $C:f(X,Y)=0$ given by a polynomial with rational coefficients has genus at least 2, then it has only finitely many points with rational coordinates. (More generally, this should be true for any number field $K$.) This was a bold general conjecture about solutions to Diophantine equations, a subject dating back to antiquity.

Algebraic geometers and complex analysts studying the geometry of (smooth projective) curves discovered that a curve $C$ of genus $g$ can be embeded into an abelian variety $J$ of dimension $g$ called its Jacobian variety. So one might hope to study $C(K)$ by first analyzing the group $J(K)$ and then seeing which of those points lie on $C$. Enter the famous Mordell-Weil Theorem: The abelian group $J(K)$ is finitely generated.

So now we have the group $J(\mathbb C)$, which looks like $\mathbb C^g/\text{(lattice)}$, a finitely generated subgroup $\Gamma=J(K)\subset J(\mathbb C)$, and a curve $C(\mathbb C)\subset J(\mathbb C)$. If $g\ge2$, then just by dimension count, it seems reasonable to suppose that $C(K) = \Gamma \cap C(\mathbb C)$ is finite.

Lang realized that when phrased as an intersection in this way, there was no reason to restrict to curves. So let $A$ be an abelian variety (which is a variety that's also a group), let $Y\subset A$ be a subvariety, let $\Gamma\subset A(\mathbb C)$ be a finitely generated group. Mordell-Lang Conjecture The intersection $\Gamma\cap Y$ is finite unless $Y$ contains a translate of an abelian subvariety of $A$.

Let's consider the special case that $\Gamma$ has rank $1$, say $\Gamma=nP_0$ for some fixed $P_0\in A$ and $n\in\mathbb Z$. Then we can describe $\Gamma$ by iterating the "translation-by-$P_0$-map" $T:A\to A$ defined by $T(P)=P+P_0$. Thus a special case of the Mordell-Lang conjecture says if the forward orbit $\{T^{\circ n}(O) : n\ge1\}$ of the iterates of $T$ intersects $Y$ infinitely often, then $Y$ contains a translate of an abelian subvariety of $A$. And now you can begin to see how to generalize the original Mordell-Lang conjecture to a more general dynamical system.

Thus let $X$ be an algebraic variety, let $f:X\to X$ be a morphism, let $x_0\in X$ be a starting point, let $Y\subseteq X$ be a subvariety, and let $\mathcal O_f(x_0)=\{f^{\circ n}(x_0):n\ge0\}$ be the forward orbit of $x_0$ under iteration of $f$. Dynamical Mordell-Lang Conjecture (Version 1) If $\mathcal O_f(x_0)\cap Y$ is infinite, then there is a positive dimensional subvariety $Z\subset Y$ such that $f(Z)\subseteq Z$.

Here is an alternative formulation, which is often useful for proofs: Dynamical Mordell-Lang Conjecture (Version 2) The set $\{n : f^{\circ n}(x_0)\in Y\}$ is the union of a finite set and a finite collection of one-sided arithmetic progressions.

The original Mordell conjecture for curves was proven by Faltings. Vojta then gave a very different proof, and Faltings adapted and extended Vojta's ideas to prove the original Mordell-Lang conjecture for abelian varieties. There are various special cases of the dynamical Mordell-Lang conjecture that are known, but it is still open (as far as I'm aware) in full generality, even for morphisms $\mathbb P^2\to\mathbb P^2$.

• Thanks very much for your answer! However, I do not understand it :( I think a lot of it is me not getting the vocabulary you are using, and my lack of familiarity with algebraic geometry. Would you be willing to simplify/explain in simpler terms starting from the second paragraph? – user89 Apr 15 '17 at 19:54