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A case of the dynamical Mordell-Lang conjecture. (English) Zbl 1285.37021
This paper concerns the following form of the dynamical Mordell-Lang conjecture, due to S.-W. Zhang [Surv. Differ. Geom. 10, 381–430 (2006; Zbl 1207.37057)] and generalized by D. Ghioca and T. J. Tucker [J. Number Theory 129, No. 6, 1392–1403 (2009; Zbl 1186.14047)]:
Conjecture: Let \(X\) be an algebraic variety over \(\mathbb C\), let \(\Phi: X\to X\) be an endomorphism of \(X\) over \(\mathbb C\), let \(V\subset X\) be a closed subvariety, and let \(\alpha\in X(\mathbb C)\) be an arbitrary point. Then the set \(\{n\in\mathbb N:\Phi^{\circ n}(\alpha)\in Y(\mathbb C)\}\) is a finite union of points and arithmetic progressions.
This paper proves some special cases of this conjecture. To describe them, first recall that if \(K\) is a field and if \(\phi\in K(t)\) is a rational function, then a point \(\alpha\in\mathbb P^1(\overline K)\) is a periodic point for \(\phi\) if \(\phi^{\circ n}(\alpha)=\alpha\) for some integer \(n>0\), and such a periodic point is said to be superattracting if \((\phi^{\circ n})'(\alpha)=0\). A point \(\alpha\in\mathbb P^1(\overline K)\) is said to be exceptional if its backward orbit is finite.
This paper proves the following special case of the dynamical Mordell-Lang conjecture:
Theorem: Let \(\phi\in\overline{\mathbb Q}(t)\) be a rational function with no superattracting periodic points other than exceptional points. Let \(\Phi:=\phi\times\dots\times\phi\) act on \((\mathbb P^1)^g\) (coordinatewise). Then for each point \(\alpha\in(\mathbb P^1)^g(\overline{\mathbb Q})\), and for each curve \(C\subseteq(\mathbb P^1)^g\) defined over \(\overline{\mathbb Q}\), the set of integers \(n\) such that \(\Phi^{\circ n}(\alpha)\in C(\overline{\mathbb Q})\) is a finite union of points and arithmetic progressions.
Several variations on this theorem are also proved, involving different ground fields \(\mathbb C\) and \(\mathbb Q\), and in some cases replacing \(C\) with a subvariety of arbitrary dimension (with correspondingly stronger or weaker hypothesis on \(\phi\) or \(\Phi\)).
The proofs use the \(p\)-adic method of Skolem, Mahler and Lech. The proofs of the results over \(\mathbb C\) use in addition a specialization argument, together with recent results in model theory from an early version of a paper of A. Medvedev and T. Scanlon [Ann. Math. (2) 179, No. 1, 81–177 (2014; Zbl 1347.37145)].
A novel aspect of the paper is that it succeeds in applying the method to ramified maps \(\phi\), within limits on the ramification (no superattracting points).
The paper concludes with an appendix by Umberto Zannier, giving a stronger variation of a result used to find a suitable non-archimedean place at which to apply the Skolem-Mahler-Lech method.

MSC:
37P55 Arithmetic dynamics on general algebraic varieties
37P20 Dynamical systems over non-Archimedean local ground fields
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