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The positive-characteristic dynamical Mordell–Lang conjecture

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Let XX be a quasiprojective variety defined over a field KK of characteristic pp, endowed with an endomorphism Φ\Phi. For a point α∈X(K)\alpha\in X(K) and a subvariety Y⊆XY\subseteq X, define

S:={n∈N0 ⁣:Φn(α)∈Y(K)}.S:=\left\{n\in\mathbb{N}_0\colon \Phi^n(\alpha)\in Y(K)\right\}.

Dynamical Mordell–Lang conjecture. The set SS is a finite union of arithmetic progressions together with finitely many sets of the form

{∑i=1rdipkini ⁣:ni∈N0},\left\{\sum_{i=1}^r d_i p^{k_i n_i}\colon n_i\in\mathbb{N}_0\right\},

for some r∈Nr\in\mathbb{N}, rational numbers did_i, and nonnegative integers kik_i for i=1,…,ri=1,\dots,r.

This is a positive-characteristic variant of the dynamical Mordell–Lang conjecture, motivated by the classical characteristic-zero formulation and by work on dynamics in characteristic pp. The supplied text does not state whether this formulation is known or remains open.

Equivalent formulations 1Other wordings

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

  1. The positive-characteristic dynamical Mordell–Lang conjecture

    Let XX be a variety over an algebraically closed field KK of characteristic p>0p>0, let ff be a rational self-map of XX, let x∈X(K)x\in X(K) be a closed point whose orbit Of(x)\mathcal{O}_{f}(x) is well-defined, and let V⊆XV\subseteq X be a closed subvariety. Positive-characteristic dynamical Mordell–Lang conjecture. The return set

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

    is a pp-normal set in N\mathbb{N}. This was proposed as a positive-characteristic replacement for the complex dynamical Mordell–Lang statement because the latter fails in positive characteristic. The paper disproves this original formulation, so the conjecture is refuted.

    source: Junyi Xie and She Yang, “On the dynamical Mordell-Lang conjecture in positive characteristic”, arXiv:2403.09181 (2024).

References

Primary source

Jason Bell and Dragos Ghioca, “A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic”, arXiv:2205.02644 (2022).

arithmetic-dynamicsdynamical-mordell-langpositive-characteristic
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