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The dynamical Mordell-Lang problem for étale maps. (English) Zbl 1230.37112
Let $$X$$ be a quasiprojective complex variety, let $$\Phi\:X\to X$$ be a morphism, and let $$V$$ be a subvariety of $$X$$. For integers $$i\geq0$$, let $$\Phi^i$$ denote the $$i$$th iterate $$\Phi\circ\dots\circ\Phi$$, and for any $$\alpha\in X(\mathbb C)$$ let $$\mathcal O_\Phi(\alpha)$$ denote the (forward) $$\Phi$$-orbit $$\{\Phi^i(\alpha):i\geq0\}$$ of $$\alpha$$. The dynamical Mordell-Lang conjecture is the question of characterizing the set $$\{i\geq0:\Phi^i(\alpha)\in V\}$$ for fixed $$\alpha$$ and $$V$$.
The present paper considers the case in which $$\Phi$$ is étale, and shows that under this assumption, if $$\alpha\in X(\mathbb C)$$ then the intersection $$\mathcal O_\Phi(\alpha)\cap V(\mathbb C)$$ is a union of finitely many orbits of the form $$\mathcal O_{\Phi^N}(\Phi^\ell(\alpha))$$ for $$N,\ell\in\mathbb N$$.
This theorem gives a new proof of the classical (arithmetic) Mordell-Lang conjecture in the special case of cyclic subgroups.
It also provides a positive answer to Question 11.6 in the paper [Duke Math. J. 126, No. 3, 491–546 (2005; Zbl 1082.14003)] by D. S. Keeler, D. Rogalski and J. T. Stafford. Specifically, it shows that if $$X$$ is an integral projective scheme over an algebraically closed field of characteristic zero, and if $$\Phi$$ is an automorphism of $$X$$, then every Zariski-dense orbit has the property that all of its infinite subsets are also Zariski-dense.
The theorem is proved by $$p$$-adic analytic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. (The methods are made to work over $$\mathbb C$$ by first reducing to a subring $$R$$ of finite type over $$\mathbb Z$$, and then embedding $$R$$ into $$\mathbb Z_p$$ for some suitable $$p$$, so that the geometric properties of the situation are preserved.)

##### MSC:
 37P20 Dynamical systems over non-Archimedean local ground fields 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37P55 Arithmetic dynamics on general algebraic varieties
##### Keywords:
dynamical Mordell-Lang problem; Skolem’s method
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