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Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang conjecture predicts that the set of $n$ for which $\phi^n(p)$ lands in $V$ is the union of a finite set and finitely many arithmetic progressions. When $\phi$ is \'etale, the conjecture is a theorem of Bell--Ghioca--Tucker.

One can ask similar questions for other classes of maps, and there are some general results (I have in mind "The Dynamical Mordell--Lang Problem for Noetherian spaces", by the same authors).

I'm curious about the case of K\"ahler manifolds. Superficially this looks very similar to the case of projective varieties, but in the non-algebraic setting it appears that entirely different methods would be needed: the usual proof requires picking an arithmetic model and doing some $p$-adic analysis.

Has anyone thought about this, e.g. on non-projective K3s? Am I missing an easy counterexample?

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    $\begingroup$ I guess you want to replace the subvariety $V$ by a sub-Kahler manifold. I'll reveal my ignorance and ask if non-projective K3s admit interesting endomorphisms that preserve the Kahler structure. If so, then this sounds like a natural question. BTW, you should add the "arithmetic-dynamics" tag to your question. $\endgroup$
    – Joe Silverman
    Commented Jan 22, 2016 at 3:25
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    $\begingroup$ Ah, good point: yes, that's what I want $V$ to be. I don't want to ask that my maps fix a Kahler form, which would be analogous to requiring a fixed polarization in the projective case: I just want a biholomorphic map. Examples would be the things in McMullen, "Dynamics on K3 surfaces: Salem numbers and Siegel disks", though I am not especially attached to the K3 case. $\endgroup$
    – user47305
    Commented Jan 22, 2016 at 4:09

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