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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.

37P55 Arithmetic dynamics on general algebraic varieties
37P20 Dynamical systems over non-Archimedean local ground fields
Full Text: DOI
[1] Amerik, E., Bogomolov, F., Rovinsky, M.: Remarks on endomorphisms and rational points, pp. 22 preprint, available at arxiv.org/abs/1001.1150
[2] Amerik, E.: Existence of non-preperiodic algebraic points for a rational self-map of infinite order, pp. 6, preprint, available at arxiv.org/abs/1007.1635 · Zbl 1241.14011
[3] Beardon A.F.: Symmetries of Julia sets. Bull. Lond. Math. Soc. 22(6), 576–582 (1990) · Zbl 0694.30023 · doi:10.1112/blms/22.6.576
[4] Beardon, A.F.: Iteration of rational functions. In: Complex analytic dynamical systems. Graduate Texts in Mathematics, vol. 132. Springer, New York (1991) · Zbl 0742.30002
[5] Bell J.P.: A generalised Skolem–Mahler–Lech theorem for affine varieties. J. Lond. Math. Soc. (2) 73(2), 367–379 (2006) · Zbl 1147.11020 · doi:10.1112/S002461070602268X
[6] Bell J.P., Ghioca D., Tucker T.J.: The dynamical Mordell-Lang problem for étale maps. Am. J. Math. 132(6), 1655–1675 (2010) · Zbl 1230.37112
[7] Benedetto R., Ghioca D., Kurlberg P., Tucker T.J.: A gap principle for dynamics. Compos. Math. 146, 1056–1072 (2010) · Zbl 1209.37115 · doi:10.1112/S0010437X09004667
[8] Bombieri, E., Gubler, W.: Heights in diophantine geometry. In: New Mathematical Monographs, vol. 4. Cambridge University Press, Cambridge (2006) · Zbl 1115.11034
[9] Denis L.: Géométrie et suites récurrentes. Bull. Soc. Math. France 122(1), 13–27 (1994)
[10] Evertse J.-H., Schlickewei H.P., Schmidt W.M.: Linear equations in variables which lie in a multiplicative group. Ann. Math. (2) 155(3), 807–836 (2002) · Zbl 1026.11038 · doi:10.2307/3062133
[11] Faltings, G.: The general case of S. Lang’s conjecture. In: Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., no. 15, Academic Press, San Diego, CA, pp. 175–182 (1994) · Zbl 0823.14009
[12] Ghioca D., Tucker T.J.: Periodic points, linearizing maps, and the dynamical Mordell-Lang problem. J. Number Theory 129, 1392–1403 (2009) · Zbl 1186.14047 · doi:10.1016/j.jnt.2008.09.014
[13] Ghioca, D., Tucker, T.J., Zhang, S.: Towards a Dynamical Manin-Mumford conjecture. Internat. Math. Res. Notices (to appear) · Zbl 1267.37110
[14] Ghioca D., Tucker T.J., Zieve M.E.: Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture. Invent. Math. 171, 463–483 (2008) · Zbl 1191.14027 · doi:10.1007/s00222-007-0087-5
[15] Ghioca, D., Tucker, T.J., Zieve, M.E.: The Mordell–Lang question for endomorphisms of semiabelian varieties (submitted) · Zbl 1256.14046
[16] Ghioca, D., Tucker, T.J., Zieve, M.E.: Complex polynomials having orbits with infinite intersection (submitted)
[17] Hrushovski, E.: Proof of Manin’s theorem by reduction to positive characteristic. In: Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, pp. 197–205. Springer, Berlin (1998) · Zbl 0925.03167
[18] Jones R.: The density of prime divisors in the arithmetic dynamics of quadratic polynomials. J. Lond. Math. Soc. (2) 78(2), 523–544 (2008) · Zbl 1193.37144 · doi:10.1112/jlms/jdn034
[19] Lang S.: Fundamentals of Diophantine Geometry. Springer, New York (1983) · Zbl 0528.14013
[20] Laurent M.: Equations diophantiennes exponentielles. Invent. Math. 78, 299–327 (1984) · Zbl 0554.10009 · doi:10.1007/BF01388597
[21] Lech C.: A note on recurring series. Ark. Mat. 2, 417–421 (1953) · Zbl 0051.27801 · doi:10.1007/BF02590997
[22] Mahler K.: Eine arithmetische Eigenshaft der Taylor-Koeffizienten rationaler Funktionen. Proc. Kon. Nederlandsche Akad. v. Wetenschappen 38, 50–60 (1935) · JFM 61.0176.02
[23] Medvedev, A., Scanlon, T.: Polynomial dynamics, pp. 67 (submitted). Available on http://arxiv.org/abs/0901.2352 · Zbl 1347.37145
[24] Morton P., Silverman J.H.: Rational periodic points of rational functions. Internat. Math. Res. Notices 2, 97–110 (1994) · Zbl 0819.11045 · doi:10.1155/S1073792894000127
[25] Raynaud M.: Courbes sur une variété abélienne et points de torsion. Invent. Math. 71(1), 207–233 (1983) · Zbl 0564.14020 · doi:10.1007/BF01393342
[26] Raynaud, M.: Sous-variétés d’une variété abélienne et points de torsion, Arithmetic and geometry, vol.\(\sim\)I, Progr. Math., vol. 35, pp. 327–352. Birkhäuser, Boston, MA, (1983)
[27] Rivera-Letelier, J.: Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287, 147–230 (Geometric methods in dynamics. II) (2003) · Zbl 1140.37336
[28] Robert, A.M.: A course in p-adic analysis. In: Graduate Texts in Mathematics, vol. 198. Springer, New York (2000) · Zbl 0947.11035
[29] Serre, J.-P.: Lectures on the Mordell-Weil theorem, 3rd edn. In: Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig (1997) (Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre)
[30] Siegel, C.L.: Über einige Anwendungen Diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys. Math. Kl., pp. 41–69 (1929) [Reprinted as pp. 209–266 of his Gesammelte Abhandlungen I. Springer, Berlin (1966)]
[31] Silverman J.H.: Integer points, Diophantine approximation, and iteration of rational maps. Duke Math. J. 71(3), 793–829 (1993) · Zbl 0811.11052 · doi:10.1215/S0012-7094-93-07129-3
[32] Skolem, T.: Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, C. r. 8 congr. scand. à Stockholm, pp. 163–188 (1934) · JFM 61.1080.01
[33] Ullmo E.: Positivité et discrétion des points algébriques des courbes. Ann. Math. (2) 147(1), 167–179 (1998) · Zbl 0934.14013 · doi:10.2307/120987
[34] Vojta P.: Integral points on subvarieties of semiabelian varieties. I. Invent. Math. 126(1), 133–181 (1996) · Zbl 1011.11040 · doi:10.1007/s002220050092
[35] Zhang S.: Equidistribution of small points on abelian varieties. Ann. Math. (2) 147(1), 159–165 (1998) · Zbl 0991.11034 · doi:10.2307/120986
[36] Zhang, S.: Distributions in algebraic dynamics. In: Survey in Differential Geometry, vol. 10, pp. 381–430. International Press, Boston (2006) · Zbl 1207.37057
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