Height arguments toward the dynamical Mordell-Lang problem in arbitrary characteristic
Pith reviewed 2026-05-22 22:15 UTC · model grok-4.3
The pith
If no cohomological Lyapunov multiplier of an endomorphism iteration is an integer, then the return set of a dense orbit into a curve is finite.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For an endomorphism f of a projective variety X over a field of arbitrary characteristic, if any cohomological Lyapunov multiplier of any iterate is not an integer, then for a dense orbit under f and a curve C in X, the return set {n | f^n(x) in C} is finite. Additionally, for a product endomorphism f × g on X × C with deg(g) > δ1(f), the return sets of the product system have the same form as those of (X, f).
What carries the argument
Height functions on projective varieties and the inequalities they satisfy when Lyapunov multipliers are non-integral, which bound the number of returns to subvarieties.
If this is right
- The dynamical Mordell-Lang conjecture holds in the case where multipliers avoid integers.
- Return sets for product systems reduce to the base case under the degree condition.
- Split self-maps on products of curves with pairwise distinct degrees have controlled return sets.
- In cases without the multiplier condition, return sets can be infinite and complicated even with zero entropy.
Where Pith is reading between the lines
- The results suggest that height methods can resolve more cases of dynamical Mordell-Lang in positive characteristic.
- The examples of complicated return sets on tori indicate that zero entropy alone does not imply simple dynamics for returns.
- One might test the results by computing multipliers and returns explicitly on low-dimensional varieties like elliptic curves or abelian varieties.
Load-bearing premise
Height functions on projective varieties satisfy the usual functoriality and Northcott properties even when the base field has arbitrary characteristic.
What would settle it
An explicit endomorphism of a projective variety in positive characteristic where all Lyapunov multipliers are non-integers yet some dense orbit returns infinitely often to a curve.
read the original abstract
We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer. (ii) Let $f\times g:X\times C\rightarrow X\times C$ be an endomorphism, where $f$ and $g$ are surjective endomorphisms of a projective variety $X$ and a projective curve $C$, respectively. If the degree of $g$ is greater than the first dynamical degree of $f$, then the return sets of the system $(X\times C,f\times g)$ have the same form as the return sets of the system $(X,f)$. Using the second result, we deal with the case of split self-maps of products of curves, for which the degrees of the factors are pairwise distinct. In the cases that the height argument cannot be applied, we find examples which show that the return set can be very complicated -- more complicated than experts once imagined -- even for endomorphisms of tori with zero entropy. One may compare them with the conjectures and results stated in [CGSZ21] and [XY25].
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper uses height arguments to establish two results on the dynamical Mordell-Lang problem over fields of arbitrary characteristic. Result (i) asserts that if f is an endomorphism of a projective variety and some cohomological Lyapunov multiplier of an iterate f^k is non-integral, then the return set {n | f^n(x) in C} is finite for a dense orbit and a curve C. Result (ii) states that for a product endomorphism f × g on X × C with deg(g) exceeding the first dynamical degree of f, the return sets of the product system take the same form as those of (X, f); this is applied to split self-maps of products of curves with pairwise distinct degrees. The paper also constructs examples of complicated (non-finite, non-periodic) return sets for certain zero-entropy endomorphisms of tori and split maps where the height method does not apply.
Significance. If the central claims hold, the results supply new sufficient conditions for finiteness of return sets that are valid in arbitrary characteristic and clarify the structure of return sets under degree comparisons. The explicit examples of intricate return sets even for zero-entropy maps on tori provide concrete illustrations that go beyond prior conjectures in [CGSZ21] and [XY25], helping delineate the boundaries of the dynamical Mordell-Lang problem.
major comments (2)
- [height arguments for result (i)] The derivation of the key height inequality h(f^{kn}(x)) ≥ c · |λ|^n h(x) + O(1) (used to obtain unbounded heights on the return set, contradicting Northcott finiteness on C) in the proof of result (i) invokes functoriality of Weil heights under pullback by endomorphisms and Northcott-type properties over arbitrary fields. These properties are not automatic in positive characteristic when f is inseparable or when heights arise from intersection theory on models over function fields; the inequality may fail to be strict, rendering the finiteness conclusion load-bearing on an unverified step.
- [proof of result (ii) and application to split maps] Result (ii) and its application to split self-maps of products of curves likewise rely on the same height machinery to compare dynamical degrees and control return sets; any gap in the height inequalities therefore propagates to the claim that the return sets 'have the same form' as those of (X, f).
minor comments (2)
- [abstract and §1] The abstract and introduction should explicitly state the precise definition of 'cohomological Lyapunov multiplier' and the normalization of dynamical degrees used throughout.
- [examples section] Clarify whether the examples in the final section are constructed over finite fields or function fields, as this affects the interpretation of the height failures.
Simulated Author's Rebuttal
We thank the referee for their careful reading and constructive comments. Below we respond point by point to the major comments.
read point-by-point responses
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Referee: [height arguments for result (i)] The derivation of the key height inequality h(f^{kn}(x)) ≥ c · |λ|^n h(x) + O(1) (used to obtain unbounded heights on the return set, contradicting Northcott finiteness on C) in the proof of result (i) invokes functoriality of Weil heights under pullback by endomorphisms and Northcott-type properties over arbitrary fields. These properties are not automatic in positive characteristic when f is inseparable or when heights arise from intersection theory on models over function fields; the inequality may fail to be strict, rendering the finiteness conclusion load-bearing on an unverified step.
Authors: The functoriality of Weil heights under pullback holds for any endomorphism of a projective variety over an arbitrary field, as it follows directly from the definition of the height associated to an ample line bundle (the pullback multiplies the height by the degree on the Picard group, with the error term controlled by the geometry of the variety). This is standard in arithmetic geometry and does not require separability; the cohomological Lyapunov multiplier is precisely the spectral radius governing this growth, independent of characteristic. Northcott finiteness likewise applies to points of bounded height over finitely generated fields (including function fields), via the usual reduction to number fields or the absolute height. The non-integral multiplier ensures the lower bound produces unbounded heights on the return set. We will add a clarifying paragraph citing these standard facts to make the argument fully explicit in positive characteristic. revision: partial
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Referee: [proof of result (ii) and application to split maps] Result (ii) and its application to split self-maps of products of curves likewise rely on the same height machinery to compare dynamical degrees and control return sets; any gap in the height inequalities therefore propagates to the claim that the return sets 'have the same form' as those of (X, f).
Authors: Because the height inequalities are valid in arbitrary characteristic as explained above, the comparison of dynamical degrees in result (ii) proceeds without issue: the degree condition on g ensures that the height growth on the product is dominated in a way that reduces the return-set structure to that of (X, f). The application to split self-maps of products of curves with pairwise distinct degrees satisfies the hypothesis directly, so the conclusion remains unchanged. No further revision is required for this section. revision: no
Circularity Check
No circularity; results rest on external standard height theory
full rationale
The derivation chain for results (i) and (ii) proceeds from the non-integer multiplier assumption to height inequalities h(f^{kn}(x)) ≥ c·|λ|^n h(x) + O(1) via functoriality of Weil heights under endomorphisms, then invokes Northcott finiteness on the curve C. These properties are cited as holding over arbitrary fields from the broader literature rather than being defined, fitted, or proved inside the paper. The reference to [XY25] appears only for comparison with conjectures and does not supply a load-bearing uniqueness theorem or ansatz. No self-definitional step, fitted-input prediction, or self-citation reduction occurs; the argument is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Height functions on projective varieties satisfy functoriality, Northcott finiteness, and the usual inequalities under endomorphisms in arbitrary characteristic.
- domain assumption Cohomological Lyapunov multipliers and first dynamical degrees are well-defined for the endomorphisms under consideration.
discussion (0)
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