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Periodic points, linearizing maps, and the dynamical Mordell-Lang problem. (English) Zbl 1186.14047
The paper begin by presenting the following
Question: Suppose that $$X$$ is a quasi-projective variety over $${\mathbb C}$$ and $$\varphi : X \rightarrow X$$ is a morphism. Let $$V$$ be a closed subvariety of $$X$$ and $$\alpha \in X({\mathbb C})$$. Are there infinitely many $$m \geq 0$$ such that $$\varphi^m(\alpha) \in V({\mathbb C})$$? Are there infinitely many of the form $$kM+l$$ for $$M>1$$ and $$l \geq 0$$?
The motivation for the question is the positive answer when we have $$\varphi^l(\alpha) \in W({\mathbb C})$$ for some periodic subvariety $$W \subset V$$. One way to attack the question is to prove that $$V \cap {\mathcal O}_{\varphi}(\alpha)$$ is at most a finite union of orbits of the form $${\mathcal O}_{\varphi^M}(\varphi^l(\alpha))$$ for some $$M,l$$. The following dynamical version of the Mordell-Lang conjecture is proposed in the paper:
Conjecture: Let $$X$$ be a quasi-projective variety defined over $${\mathbb C}$$, $$\varphi : X \rightarrow X$$ be a morphism and $$\alpha \in X({\mathbb C})$$, then for any subvariety $$V \subset X$$, the intersection $$V \cap {\mathcal O}_{\varphi}(\alpha)$$ is the union of at most finitely many orbits of the form $${\mathcal O}_{\varphi^M}(\varphi^l(\alpha))$$ for some $$M$$ and $$l$$.
Under suitable hypotheses the conjecture is proved for quasiprojective varieties defined over number fields and $${\mathbb C}_p$$. The conjecture is also proved in the case of $$X=A$$ a semi-Abelian variety defined over a finitely generated subfield $$K \subset {\mathbb C}$$ and $$\varphi : A \rightarrow A$$ defined over $$K$$.
The technique of proof is as follows: Let $$M>0$$ be an integer and suppose that $$\beta$$ is a periodic point of period dividing $$M$$, the work of M. Herman and J.-C. Yoccoz [in: Geometric Dynamics, Lect. Notes Math. 1007, 408–447 (1983; Zbl 0528.58031)] provides, for any iterate $$\varphi^l(\alpha)$$ that is close to $$\beta$$, a function $$h$$ on a neighborhood of $$\beta$$, which is $$p$$-adic analytic for a suitable $$p$$ and $$\varphi^M \circ h=h \circ A$$ for some linear function $$A$$. When $$A$$ is a homothety they apply similar techniques as used by Skolem, Mahler and Lech for linear recurrences. Based on the fact that a non-zero convergent $$p$$-adic series has at most finitely many zeros, we get that for each congruency class $$i=0,...,M-1$$ module $$M$$, either there are finitely many $$n \equiv i (\mod M)$$ such that $$\varphi^n(\alpha) \in V$$ or we have $$\varphi^n(\alpha) \in V$$ for all $$n \geq l$$ such that $$n \equiv i \pmod M$$.

##### MSC:
 14K12 Subvarieties of abelian varieties 37P35 Arithmetic properties of periodic points 37P20 Dynamical systems over non-Archimedean local ground fields 14C25 Algebraic cycles
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