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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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