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A dynamical version of the Mordell-Lang conjecture for the additive group. (English) Zbl 1169.11024
The main result of the paper is Theorem 2.5: Let $$\phi_1,\ldots,\phi_g$$ be Drinfeld $${\mathbb F}_q[t]$$-modules over a finite extension $$K$$ of $${\mathbb F}_q(t)$$. Let $$(x_1,\ldots,x_g)\in{\mathbb G}_a^g$$ and $$\Gamma\subseteq{\mathbb G}_a^g(K)$$ the cyclic $$(\phi_1,\ldots,\phi_g)$$-submodule generated by $$(x_1,\ldots,x_g)$$. Let $$V\subseteq{\mathbb G}_a^g$$ be an affine subvariety defined over $$K$$. Then $$V(K)\cap\Gamma$$ is a finite union of cosets of $$(\phi_1,\ldots,\phi_g)$$-submodules of $$\Gamma$$.
A version where $$V$$ is a curve and $$\Gamma$$ is a rank one $$(\phi_1,\ldots,\phi_g)$$-submodule is also proved (Theorem 4.2).
A more general version of these results had been proposed by L. Denis [Ohio State Univ. Math. Res. Inst. Publ. 2, 285–302 (1992; Zbl 0798.11022)] as a function field analog of the Mordell-Lang conjecture. He also proved some partial results.
The word dynamical in the title of the current paper comes from the fact that Theorem 2.5 can also be interpreted from the point of view of polynomial dynamics as a statement about the orbit of $$(x_1,\ldots,x_g)$$.
Versions of the above theorems where $$K$$ has transcendence degree at least $$2$$ over $${\mathbb F}_q$$ were proved by the first author in [Int. Math. Res. Not. 53, 3273–3307 (2005; Zbl 1158.11030)] and in “Towards the full Mordell-Lang conjecture for Drinfeld modules”, Can. Math. Bull. 53, No. 1, 95–101 (2010; Zbl 1219.11086).

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14K12 Subvarieties of abelian varieties 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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