Ghioca–Tucker dynamical Mordell–Lang conjecture

About 10 years old · traced to 2016

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

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

An arithmetic progression is a set of the form $\{an+b\mid n\in\mathbb N\}$ with $a,b\in\mathbb N$ . Ghioca–Tucker dynamical Mordell–Lang conjecture. For every $x$ , 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.

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

Ghioca–Tucker dynamical Mordell–Lang conjecture
Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a quasi-projective variety over $k$ , and let $f:X\dashrightarrow X$ be a dominant rational self-map. Let $V\subseteq X$ be a Zariski closed subset, and let $x\in X(k)$ be a point whose forward $f$ -orbit is well-defined. Ghioca–Tucker's dynamical Mordell–Lang conjecture. The set
$\left\lbrace n\in\mathbb{Z}_{\geq0}: f^{\circ n}(x)\in V\right\rbrace$
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).

References

Primary source

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

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