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Ghioca–Tucker dynamical Mordell–Lang conjecture

Let XX be a quasi-projective variety defined over C\mathbb C, let f:X→Xf:X\rightarrow X be an endomorphism, and let VV be any subvariety of XX. For a point x∈X(C)x\in X(\mathbb C), define its return set to VV by

{n∈N∣fn(x)∈V(C)}.\{n\in\mathbb N\mid f^n(x)\in V(\mathbb C)\}.

An arithmetic progression is a set of the form {an+b∣n∈N}\{an+b\mid n\in\mathbb N\} with a,b∈Na,b\in\mathbb N. Ghioca–Tucker dynamical Mordell–Lang conjecture. For every xx, the return set is a union of at most finitely many arithmetic progressions.

This is a dynamical analogue of the Mordell–Lang conjecture and is known in several cases, including étale maps. The source also notes that it concerns forward orbits rather than backward orbits.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Ghioca–Tucker dynamical Mordell–Lang conjecture

    Let kk be an algebraically closed field of characteristic zero. Let XX be a quasi-projective variety over kk, and let f:X⇢Xf:X\dashrightarrow X be a dominant rational self-map. Let V⊆XV\subseteq X be a Zariski closed subset, and let x∈X(k)x\in X(k) be a point whose forward ff-orbit is well-defined. Ghioca–Tucker's dynamical Mordell–Lang conjecture. The set

    {n∈Z≥0:f∘n(x)∈V}\left\lbrace n\in\mathbb{Z}_{\geq0}: f^{\circ n}(x)\in V\right\rbrace

    is a finite union of arithmetic progressions. This is the dynamical analogue of the classical Mordell–Lang conjecture for subvarieties of semi-abelian varieties. The conjecture is open in this generality.

    source: Geng-Rui Zhang, “Rank-two recurrence results for polynomials and questions of dynamical Mordell–Lang type”, arXiv:2605.27058 (2026).

Sources & referencesView supporting material

Primary source

Junyi Xie, “Algebraic dynamics of the lifts of Frobenius”, arXiv:1602.04253 (2018).

algebraic-dynamicsarithmetic-dynamicsarithmetic-geometrydynamical-mordell-lang

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