Dynamical Uniform Bounds for Fibers and a Gap Conjecture

Dragos Ghioca; Jason Bell; Matthew Satriano

arxiv: 1906.08683 · v1 · pith:SV6OZCUVnew · submitted 2019-06-20 · 🧮 math.NT · math.AG· math.DS

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

Jason Bell , Dragos Ghioca , Matthew Satriano This is my paper

Pith reviewed 2026-05-25 19:19 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.DS

keywords dynamical Mordell-Lang conjectureétale endomorphismsuniform boundsheight gapsorbitsquasi-projective varietiesnumber fieldsWeil height

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The pith

For étale endomorphisms, finite fiber preimages in an orbit have sizes bounded uniformly across all target points, and arbitrary endomorphisms satisfy a logarithmic height growth gap.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when an étale endomorphism acts on a quasi-projective variety over a characteristic zero field, and a morphism maps the orbit to another variety, the sets of times when the orbit hits any given point in the target are finite only if their sizes are bounded by a constant independent of the point. It also proves a gap principle for heights: along any orbit under an arbitrary endomorphism, the heights of the images either take finitely many values or grow at least like the logarithm of the number of iterations. This strengthens the dynamical Mordell-Lang conjecture to a uniform statement for étale maps. A sympathetic reader would care because it gives effective control over how often orbits can return to fibers without depending on the specific point.

Core claim

Assume X is quasi-projective over K of char 0 with étale endomorphism Φ, f morphism to Y quasi-projective over K. For x in X(K), if every S_y = {n in N | f(Φ^n(x)) = y} is finite, then there is N with #S_y <= N for all y in Y(K). For the second result, over number field K with rational map f to P^1 and arbitrary endomorphism Φ, either f of the orbit is finite or the limsup of h(f(Φ^n(x))) / log n is positive.

What carries the argument

The sets S_y of return times to fibers, for which finiteness implies uniform bounded cardinality when Φ is étale; the limsup height growth rate for the gap result.

If this is right

The number of solutions to f(Φ^n(x)) = y is bounded uniformly in y, when finite.
Heights in orbits grow at least logarithmically or the projected orbit is finite.
These hold for quasi-projective varieties in characteristic zero.
The results apply to rational maps to projective line for the height gap.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the uniform bound holds then all hits can be found by iterating only up to that bound.
The height gap implies that sub-logarithmic growth forces the image of the orbit to be finite.
Similar uniform statements might require étaleness or characteristic zero assumptions.
The gap principle separates finite-image cases from those with positive growth rate.

Load-bearing premise

The endomorphism Φ is required to be étale on a quasi-projective variety over a field of characteristic zero.

What would settle it

An example of an étale endomorphism and a morphism where the cardinalities of the finite sets S_y are unbounded as y varies.

read the original abstract

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for \'etale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our first result, we assume $X$ is a quasi-projective variety defined over a field $K$ of characteristic $0$, endowed with the action of an \'etale endomorphism $\Phi$, and $f\colon X\to Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then for any $x\in X(K)$, if for each $y\in Y(K)$, the set $S_y:=\{n\in \mathbb{N}\colon f(\Phi^n(x))=y\}$ is finite, then there exists a positive integer $N$ such that $\#S_y\le N$ for each $y\in Y(K)$. For our second result, we let $K$ be a number field, $f:X\dashrightarrow \mathbb{P}^1$ is a rational map, and $\Phi$ is an arbitrary endomorphism of $X$. If $\mathcal{O}_\Phi(x)$ denotes the forward orbit of $x$ under the action of $\Phi$, then either $f(\mathcal{O}_\Phi(x))$ is finite, or $\limsup_{n\to\infty} h(f(\Phi^n(x)))/\log(n)>0$, where $h(\cdot)$ represents the usual logarithmic Weil height for algebraic points.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a uniform bound for étale dynamical Mordell-Lang and a height gap for general endomorphisms.

read the letter

The two main results are a uniform bound on those S_y sets for étale endomorphisms and a height gap for arbitrary ones. The uniform version of dynamical Mordell-Lang is new here. It says that if each y is hit only finitely many times by the orbit under f, then there's a bound N that works for all y at once. The gap result says the heights either stabilize to a finite set or grow at least like log n in the limsup. The paper does well by keeping the hypotheses clear: char zero, quasi-projective varieties, étale for the uniform part. That matches what the reader noted as the weakest assumption, and it seems necessary. Soft spots: the proofs are not visible in the abstract, so I can't check the details, but nothing in the statement looks circular or overclaimed. The results build on prior work in the area without obvious gaps in the logic. This paper is for people in arithmetic dynamics. Anyone looking at orbit finiteness or height growth in dynamical systems over number fields would find the statements useful. It deserves peer review because the claims are specific enough to be checked and the topic is active. Recommendation: yes, send it out for refereeing.

Referee Report

0 major / 3 minor

Summary. The paper proves two results in arithmetic dynamics. For the first: Let X be quasi-projective over a char-0 field K, Φ an étale endomorphism of X, f: X → Y a morphism to another quasi-projective Y over K, and x ∈ X(K). If S_y = {n ∈ ℕ | f(Φ^n(x)) = y} is finite for every y ∈ Y(K), then there exists N such that #S_y ≤ N for all y. For the second: Let K be a number field, Φ an arbitrary endomorphism of X, f: X ⇢ ℙ¹ a rational map. Then either f(O_Φ(x)) is finite or limsup_{n→∞} h(f(Φ^n(x)))/log(n) > 0, where h is the logarithmic Weil height.

Significance. If correct, the first result supplies a uniform bound strengthening the Dynamical Mordell-Lang conjecture precisely under the étale hypothesis already present in the literature; the second supplies a clean height-growth dichotomy for orbits under arbitrary endomorphisms. Both statements are stated with explicit hypotheses and could serve as tools for further work on finiteness and canonical heights in dynamical systems.

minor comments (3)

The abstract and introduction should explicitly reference the sections containing the proofs of the two theorems (e.g., “Theorem 1.1 is proved in §3”); currently the reader must hunt for the derivations.
Notation: the forward orbit is denoted O_Φ(x) in the second result but never defined in the provided abstract; add a sentence in the introduction clarifying the notation.
The statement of the gap result uses “arbitrary endomorphism” while the uniform result requires étale; a short remark comparing the two hypotheses would help the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our results on the uniform Dynamical Mordell-Lang conjecture under the étale hypothesis and the logarithmic gap for heights along orbits. We appreciate the recommendation of minor revision. No major comments were provided in the report, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under explicit hypotheses

full rationale

The paper's central results are stated as theorems under explicit, independent hypotheses (étale endomorphism Φ on quasi-projective X over char-0 field for the uniform fiber bound; arbitrary endomorphism for the height-gap result). These strengthen the Dynamical Mordell-Lang conjecture and a standard height-growth dichotomy without any quoted equations or claims reducing predictions to inputs by construction, self-definitions, or load-bearing self-citations. The setup invokes no ansatz smuggled via prior work by the same authors, no fitted parameters renamed as predictions, and no uniqueness theorems imported from overlapping citations as external facts. The derivation chain remains self-contained against external benchmarks in arithmetic dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established background results from algebraic geometry and arithmetic dynamics; no free parameters or invented entities are visible in the abstract.

axioms (2)

standard math Standard functorial properties of Weil heights on projective varieties
Invoked for the height gap statement
standard math Étale morphisms are locally isomorphisms and preserve certain finiteness properties
Central hypothesis for the uniform bound

pith-pipeline@v0.9.0 · 5825 in / 1209 out tokens · 34565 ms · 2026-05-25T19:19:26.088382+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

[1]

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J. P. Bell, A generalised Skolem-Mahler-Lech theorem for affine varieties, J. London Math. Soc. (2) 73 (2006), no. 2, 367--379

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S. Bosch, U. G\"untzer, and R. Remmert, Non-Archimedean analysis: A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 261. Springer-Verlag, Berlin, 1984. xii+436 pp

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J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell--Lang problem for \' e tale maps , Amer. J. Math. 132 (2010), no. 6, 1655--1675

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J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell--Lang problem for Noetherian spaces, Funct. Approx. Comment. Math. 53 (2015), no. 2, 313--328

work page 2015
[6]

J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell--Lang conjecture, Mathematical Surveys and Monographs 210, American Mathematical Society, Providence, RI, 2016, xiii+280 pp

work page 2016
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J. W. S. Cassels, Local fields. London Mathematical Society Student Texts, 3. Cambridge University Press, Cambridge, 1986

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J. P. Bell, K. D. Nguyen, and U. Zannier, D -finiteness, rationality, and height. Preprint available online at arXiv:1905.06450

work page internal anchor Pith review Pith/arXiv arXiv 1905
[9]

R. L. Benedetto, D. Ghioca, P. Kurlberg, T. J. Tucker, A gap principle for dynamics, Compos. Math. 146 (2010), no. 4, 1056---1072

work page 2010
[10]

Ghioca and T

D. Ghioca and T. J. Tucker, Periodic points, linearizing maps, and the dynamical Mordell--Lang problem, J. Number Theory 129 (2009), no. 6, 1392--1403

work page 2009
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A. J. van der Poorten and I. Shparlinski, On linear recurrence sequences with polynomial coefficients, Glasg. Math. J. 38 (1996), 147--155

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S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433--449

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Strassman, \: U ber den Wertevorrat von Potenzreihen im Gebiet der p -adischen Zahlen

R. Strassman, \: U ber den Wertevorrat von Potenzreihen im Gebiet der p -adischen Zahlen . J. Reine Angew. Math. 159 (1928), 13--28; 65--66

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[1] [1]

Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces , Invent. Math. 97 (1989), no. 3, 613--670

work page 1989

[2] [2]

J. P. Bell, A generalised Skolem-Mahler-Lech theorem for affine varieties, J. London Math. Soc. (2) 73 (2006), no. 2, 367--379

work page 2006

[3] [3]

Bosch, U

S. Bosch, U. G\"untzer, and R. Remmert, Non-Archimedean analysis: A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 261. Springer-Verlag, Berlin, 1984. xii+436 pp

work page 1984

[4] [4]

J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell--Lang problem for \' e tale maps , Amer. J. Math. 132 (2010), no. 6, 1655--1675

work page 2010

[5] [5]

J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell--Lang problem for Noetherian spaces, Funct. Approx. Comment. Math. 53 (2015), no. 2, 313--328

work page 2015

[6] [6]

J. P. Bell, D. Ghioca, and T. J. Tucker, The dynamical Mordell--Lang conjecture, Mathematical Surveys and Monographs 210, American Mathematical Society, Providence, RI, 2016, xiii+280 pp

work page 2016

[7] [7]

J. W. S. Cassels, Local fields. London Mathematical Society Student Texts, 3. Cambridge University Press, Cambridge, 1986

work page 1986

[8] [8]

J. P. Bell, K. D. Nguyen, and U. Zannier, D -finiteness, rationality, and height. Preprint available online at arXiv:1905.06450

work page internal anchor Pith review Pith/arXiv arXiv 1905

[9] [9]

R. L. Benedetto, D. Ghioca, P. Kurlberg, T. J. Tucker, A gap principle for dynamics, Compos. Math. 146 (2010), no. 4, 1056---1072

work page 2010

[10] [10]

Ghioca and T

D. Ghioca and T. J. Tucker, Periodic points, linearizing maps, and the dynamical Mordell--Lang problem, J. Number Theory 129 (2009), no. 6, 1392--1403

work page 2009

[11] [11]

A. J. van der Poorten and I. Shparlinski, On linear recurrence sequences with polynomial coefficients, Glasg. Math. J. 38 (1996), 147--155

work page 1996

[12] [12]

S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433--449

work page 1979

[13] [13]

Strassman, \: U ber den Wertevorrat von Potenzreihen im Gebiet der p -adischen Zahlen

R. Strassman, \: U ber den Wertevorrat von Potenzreihen im Gebiet der p -adischen Zahlen . J. Reine Angew. Math. 159 (1928), 13--28; 65--66

work page 1928