# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Bell, J.P., A generalised skolem – mahler – lech theorem for affine varieties, J. London math. soc. (2), 73, 2, 367-379, (2006) · Zbl 1147.11020 [2] R.L. Benedetto, D. Ghioca, P. Kurlberg, T.J. Tucker, The dynamical Mordell-Lang conjecture, submitted for publication, 2007, available online at http://arxiv.org/abs/0712.2344 · Zbl 1285.37021 [3] J.P. Bell, D. Ghioca, T.J. Tucker, The dynamical Mordell-Lang problem for étale maps, submitted for publication, 2008, available online at http://arxiv.org/abs/0808.3266 · Zbl 1230.37112 [4] Bourbaki, N., Lie groups and Lie algebras. chapters 1-3, (), translated from the French, reprint of the 1989 English translation · Zbl 0672.22001 [5] Bruin, Nils, Chabauty methods using elliptic curves, J. reine angew. math., 562, 27-49, (2003) · Zbl 1135.11320 [6] Denis, L., Géométrie diophantienne sur LES modules de Drinfel’d, (), 285-302 · Zbl 0798.11022 [7] Denis, L., Géométrie et suites récurrentes, Bull. soc. math. France, 122, 1, 13-27, (1994) · Zbl 0795.14008 [8] Fakhruddin, N., Questions on self maps of algebraic varieties, J. Ramanujan math. soc., 18, 2, 109-122, (2003) · Zbl 1053.14025 [9] Faltings, G., The general case of S. Lang’s conjecture, (), 175-182 · Zbl 0823.14009 [10] Flynn, E. Victor; Wetherell, Joseph L., Finding rational points on bielliptic genus 2 curves, Manuscripta math., 100, 4, 519-533, (1999) · Zbl 1029.11024 [11] D. Ghioca, T.J. Tucker, Mordell-Lang and Skolem-Mahler-Lech theorems for endomorphisms of semiabelian varieties, unpublished, 2007, 15 pp., available online at http://arxiv.org/pdf/0710.1669 [12] D. Ghioca, T.J. Tucker, p-Adic logarithms for polynomial dynamics, unpublished, 2007, 11 pp., available online at http://arxiv.org/pdf/0705.4047 [13] Ghioca, D.; Tucker, T.J., A dynamical version of the mordell – lang theorem for the additive group, Compos. math., 164, 2, 304-316, (2008) · Zbl 1169.11024 [14] Ghioca, D.; Tucker, T.J.; Zieve, M., Intersections of polynomial orbits, and a dynamical mordell – lang conjecture, Invent. math., 171, 2, 463-483, (2008) · Zbl 1191.14027 [15] Hartshorne, R., Algebraic geometry, (1977), Springer-Verlag New York · Zbl 0367.14001 [16] Herman, M.; Yoccoz, J.-C., Generalizations of some theorems of small divisors to non-Archimedean fields, (), 408-447 [17] Hungerford, T.W., Algebra, Grad. texts in math., vol. 73, (2003), Springer-Verlag New York, reprint of the 1974 original [18] Jost, J., Riemannian geometry and geometric analysis, Universitext, (2002), Springer-Verlag Berlin · Zbl 1034.53001 [19] Lang, S., Algebra, (2002), Springer-Verlag New York · Zbl 0984.00001 [20] Lech, C., A note on recurring series, Ark. mat., 2, 417-421, (1953) · Zbl 0051.27801 [21] Mahler, K., Eine arithmetische eigenshaft der Taylor-koeffizienten rationaler funktionen, Proc. kon. nederlandsche akad. wetenschappen, 38, 50-60, (1935) · JFM 61.0176.02 [22] Rivera-Letelier, J., Dynamique des fonctions rationnelles sur des corps locaux, Geometric methods in dynamics. II, Astérisque, 287, 147-230, (2003) · Zbl 1140.37336 [23] Robert, A.M., A course in p-adic analysis, Grad. texts in math., vol. 198, (2000), Springer-Verlag New York · Zbl 0947.11035 [24] Skolem, T., Ein verfahren zur behandlung gewisser exponentialer gleichungen und diophantischer gleichungen, C. R. 8 congr. scand. Stockholm, 163-188, (1934) · JFM 61.1080.01 [25] Vojta, P., Integral points on subvarieties of Semiabelian varieties. II, Amer. J. math., 121, 2, 283-313, (1999) · Zbl 1018.11027 [26] Yu, K., Linear forms in p-adic logarithms. II, Compos. math., 74, 1, 15-113, (1990) · Zbl 0723.11034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.