Arithmetic Degrees are Cohomological Lyapunov Multipliers

Jiarui Song; Junyi Xie; She Yang

REVIEW 2 major objections 4 minor 21 references

Arithmetic Degrees are Cohomological Lyapunov Multipliers

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For a surjective endomorphism of a normal projective variety over a finitely generated field of characteristic zero, if the orbit of a point is Zariski dense, then the arithmetic degree of the point must be one of the cohomological…

desk verdict Real strengthening of Kawaguchi–Silverman with a repairable gap in the canonical-height step; worth refereeing seriously. read the letter →

arxiv 2507.17643 v1 pith:KIHPA7QD submitted 2025-07-23 math.DS math.AGmath.NT

Jiarui Song , Junyi Xie , She Yang This is my paper · ORCID

classification math.DSmath.AGmath.NT MSC 37P3037P5514G40

keywords arithmeticdegreecohomologicalLyapunovmultiplierdynamicalKawaguchi–SilvermanconjectureZariskidenseorbitMordell–LangMoriwakiheightendomorphismofprojectivevariety

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a precise restriction on arithmetic degrees in algebraic dynamics. For a surjective endomorphism $f$ of a normal projective variety over a finitely generated field of characteristic zero, if the orbit of a point $x$ is Zariski dense, then its arithmetic degree $\alpha_f(x)$ must be one of the cohomological Lyapunov multipliers $\mu_i(f)=\lambda_i(f)/\lambda_{i-1}(f)$, the ratios of consecutive dynamical degrees. Since the first multiplier $\mu_1(f)=\lambda_1(f)$ is the dynamical degree, the theorem says that dense orbits grow at one of a short list of intrinsic rates rather than at an arbitrary eigenvalue of the numerical pull-back. A corollary settles the Kawaguchi–Silverman conjecture whenever the two largest dynamical degrees differ, and another corollary gives a dynamical Mordell–Lang statement by comparing growth rates of two dense orbits.

What carries the argument

The carrying object is the cohomological Lyapunov multiplier $\mu_i(f)=\lambda_i(f)/\lambda_{i-1}(f)$, the ratio of consecutive dynamical degrees of $f$, which are real eigenvalues of the numerical pull-back $f^*:N^1(X)_{\mathbb{R}}\to N^1(X)_{\mathbb{R}}$. The key steps are a lifting lemma that lifts the generalized eigenspaces for multipliers above the spectral radius of $f^*$ on $\mathrm{Pic}^0(X)$ to $\mathrm{Pic}(X)_{\mathbb{R}}$ in an $f^*$-equivariant way, and a big-cone property from a companion paper ensuring that a big divisor class lies in the span of the eigenspaces of the multipliers. Heights along the orbit are then controlled by canonical heights built from Jordan blocks, while the part belonging to smaller multipliers grows too slowly to match $\alpha$.

What would settle it

Compute, for a concrete surjective endomorphism of a normal projective variety over a number field, both the arithmetic degree of a Zariski dense orbit, for instance via heights of iterates, and the list $\{\mu_1,\dots,\mu_d\}$; find an orbit whose $\alpha$ is a real eigenvalue of $f^*$ on $N^1(X)_{\mathbb{R}}$ but is not among the $\mu_i$. Alternatively, exhibit an endomorphism for which the claimed identity $\{\mu_i\}=\{\alpha:\mathrm{Im}(f^*-\alpha)\cap\mathrm{Big}(X)=\emptyset\}$ fails, since the proof imports that identity.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: under the Zariski-density assumption, $\alpha_f(x)\in\{\mu_1(f),\dots,\mu_d(f)\}\cap\mathbb{R}_{\geq 1}$. The proof assumes for contradiction that $\alpha$ lies strictly between $\mu_\ell$ and $\mu_{\ell+1}$, splits a big divisor class coming from the intersection of the big cone with the sum of generalized eigenspaces of $f^*$ corresponding to multipliers above $\alpha$, and shows that one part of the height grows at rate $\alpha$ while the other part grows too fast or too slow, forcing a contradiction. The corollary is that if $\lambda_1(f)>\lambda_2(f)$, then $\alpha_f(x)=\lambda_1(f)$ for every Zariski dense orbit, exactly as the Kawaguchi–Silverman conjecture predicts.

Load-bearing premise

The proof rests on two cited structural theorems about cohomological Lyapunov multipliers: that they are exactly the real eigenvalues $\alpha$ of the numerical pull-back whose image misses the big cone, and that the sum of the corresponding generalized eigenspaces meets the big cone; if either statement fails under the stated hypotheses, the conclusion no longer follows.

Editorial extensions

If this is right

The arithmetic degree of every Zariski dense orbit is forced into a finite list of dynamical invariants, so existence of the limit is paired with a strong membership statement.
When $\lambda_1(f)>\lambda_2(f)$, the Kawaguchi–Silverman conjecture holds: every Zariski dense orbit has arithmetic degree $\lambda_1(f)$.
If the multiplier lists of two endomorphisms are disjoint, no positive-dimensional subvariety of the product can contain a dense intersection of the product orbit, because the two coordinates grow at incompatible rates (Corollary 1.5).
The result strengthens the earlier description of $\alpha_f(x)$ as an eigenvalue of $f^*$ on $N^1(X)_{\mathbb{R}}$ by selecting the specific eigenvalues that are Lyapunov multipliers.
Since the arguments run over finitely generated fields via Moriwaki heights, the theorem applies over arbitrary characteristic-zero fields, including $\mathbb{C}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same height-splitting strategy might be adapted to dominant rational maps, where Zariski dense orbits are harder to control but the multiplier list is still defined.
A testable computational extension is to compute the multiplier list and the arithmetic degree for concrete low-dimensional examples, such as endomorphisms of abelian surfaces or cyclic covers, and check the equality $\alpha=\mu_i$.
For the dynamical Mordell–Lang application, the disjointness hypothesis is stronger than needed: since the proof only uses $\alpha_f(x)\neq\alpha_g(y)$, any condition implying different arithmetic degrees would give the same conclusion, and Theorem 1.2 suggests checking the multiplier lists is the natural route.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main result is a genuine step forward: for a surjective endomorphism of a normal projective variety over a finitely generated characteristic-zero field, the arithmetic degree of a Zariski dense orbit is a cohomological Lyapunov multiplier, and when lambda_1 > lambda_2 the Kawaguchi–Silverman conjecture follows. That goes beyond KS16a (eigenvalue only) and Mat25 (generic orbits). The proof idea is attractive: lift the relevant part of N^1 into Pic using a spectral radius bound on Pic^0, then use the big-cone intersection of eigenspaces to build a big divisor whose height growth pins down the arithmetic degree. Lemma 3.1 is a nice construction, and the DML corollary is a useful extra.

The weak spot that matters sits in the contradiction argument. E is defined as the sum of generalized eigenspaces for mu_i(f) > sqrt(lambda_1(g)). When the Albanese is trivial, sqrt(lambda_1(g)) = 0, so E includes every positive multiplier, including mu_i(f) < 1. For a surface automorphism with lambda_1 > 1, the eigenvalue 1/lambda_1 < 1 is there. The proof then applies [KS16a, Theorem 5] to the full Jordan matrix Lambda and concludes that h_L ∘ f^n Lambda^{-n} converges to a canonical height vector. For a coordinate with 0 < mu < 1, the height along the orbit is bounded, so dividing by mu^n blows up; that theorem is only valid for eigenvalues above 1. So the canonical height vector used later is not justified in those cases. The repair is straightforward: take E = direct sum over mu_i(f) > alpha_f(x) instead. In the nontrivial branch alpha_f(x) > sqrt(lambda_1(g)), so the lifting and growth estimates still work, and all eigenvalues in E are >1. But the proof as written does not do this.

There is also a dependency on [Xieb], an unpublished preprint of the same group, for Theorems 2.2 and 2.3. Those are load-bearing and should be checked carefully. This is a real citation-pattern soft spot, not a fatal one.

Overall, the strategy is plausible, the main theorem is significant, and the flaws I see are repairable rather than fatal. I would send this to a serious referee, with specific instructions to verify the canonical-height step and the hypotheses on [Xieb].

Referee Report

2 major / 4 minor

Summary. This paper studies surjective endomorphisms f:X→X of normal projective varieties over finitely generated fields of characteristic zero. It proves (Theorem 1.2) that for any point x whose f-orbit is Zariski dense, the arithmetic degree α_f(x) is one of the cohomological Lyapunov multipliers μ_i(f)=λ_i(f)/λ_{i-1}(f), and in particular α_f(x)≥1. Corollary 1.3 deduces the Kawaguchi–Silverman conjecture when λ_1(f)>λ_2(f). Corollary 1.5 applies Theorem 1.2 to a dynamical Mordell–Lang type statement for product endomorphisms. The proof uses Moriwaki heights, a lift of the generalized eigenspaces of f^* on N^1(X)_R to Pic(X)_R, and the big-cone characterization of Lyapunov multipliers imported from [Xieb], then runs a height-growth comparison on a big divisor along a subsequence.

Significance. If the proof is repaired, this is a significant strengthening of the known result of Kawaguchi–Silverman that α_f(x) is the modulus of an eigenvalue of f^* on N^1(X)_R; it pins the arithmetic degree to the specific cohomological Lyapunov multipliers. The corollaries are concrete and testable, and the DML application is a nice demonstration of the height method. The paper is generally well structured and the central arithmetic-degree conclusion is not assumed; it is derived from external results, the most important being the big-cone characterization in [Xieb]. The main reservation is that the proof as written has a local but load-bearing gap in the canonical-height step when the Albanese variety is trivial; this is repairable by restricting the relevant eigenspace sum to multipliers exceeding α_f(x).

major comments (2)

[Section 3, proof of Theorem 1.2, definition of E and application of [KS16a, Theorem 5]] The definition E:=⊕_{μ_i(f)>√λ1(g)}E_{μ_i(f)} includes every positive μ_i(f) when Alb(X) is trivial, because then λ1(g)=0. This can include multipliers <1; e.g. a surface automorphism with λ1(f)>1 has μ2(f)=λ2(f)/λ1(f)=1/λ1(f)<1. The proof then applies [KS16a, Theorem 5] to the full Jordan matrix Λ and states that h_L∘f^n Λ^{-n} converges pointwise to a canonical height satisfying ĥ_L∘f=ĥ_LΛ and ĥ_L=h_L+O(1). For a coordinate with eigenvalue 0<μ<1, the recurrence a_{n+1}=μa_n+b_n with b_n bounded forces a_n=h_D(f^n x) to be bounded along every orbit, so a_n μ^{-n} does not converge and typically diverges. Hence the canonical-height construction, and the formula ĥ_{M1}(f^n(x))=Σ c_{i,k}n^kμ_i^n on which the lower bound h_{eB}(f^{n_j})≥C μ_{i0}^{n_j} depends, is not justified in the trivial-Albanese case. The gap is local: because the contradiction branch has α_f(x)>λ1(g), one can replace E by ⊕_{μ_i(f)>α_f(x)}E_{μ_i(f)}; Lemma 3.1 still applies, and all remaining eigenvalues in Λ are >1, where the canonical-height step is standard. As written, however, the main theorem is not proved.
[Section 2, Theorems 2.2 and 2.3] The two key structural facts about cohomological Lyapunov multipliers—the big-cone characterization and the existence of a big class in the sum of the corresponding generalized eigenspaces—are quoted from the unpublished preprint [Xieb] (Theorems 1.3 and 1.4). Every step of the proof of Theorem 1.2 and Corollary 1.5 depends on these facts, so the main result is conditional on a source that is not yet publicly refereed. The authors should state this dependence explicitly and, if possible, include or append proofs, or at least confirm that [Xieb] is available in a citable final form. This does not by itself invalidate the argument, but it is a load-bearing incompleteness for a journal submission.

minor comments (4)

[Section 1] There is a typo in the sentence "under the weaker assumption that the the orbit is Zariski dense": "the" is repeated.
[Section 3, Lemma 3.1] The convergence estimate "the norm of (f^*)^{n-1}M Λ^{-n} is O(n^{r+2g}(ρ0/μ)^n)" is asserted from the linear-recurrence property without further detail; since this estimate is a key input to the lifting result, a short justification would improve readability.
[Section 4, proof of Corollary 1.5] The sentence "Since p1^*L1 is always big, we may choose L1 appropriately such that p1^*L1=A+E" is slightly confusing: one chooses L1 ample, then p1^*L1 is big and the decomposition into an ample part A and effective part E exists. Rephrasing would avoid the impression that L1 is being chosen after the decomposition.
[Acknowledgements and references] In the sentence about [XY25], "the second-named and third-named authors" should be "the second- and third-named authors".

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: Theorem 1.2 is proved from independent external results; the same-group preprint [Xieb] supplies cone characterizations but not the arithmetic-degree conclusion.

full rationale

The paper's derivation chain is not circular. Theorem 1.2 asserts alpha_f(x) lies among the cohomological Lyapunov multipliers mu_i(f). The proof uses: (i) existence and basic bounds for arithmetic degrees from [KS16a, Sil17, Ohn22]; (ii) the characterization of mu_i(f) via the big cone from [Xieb, Theorems 1.3 and 1.4]; (iii) a lifting lemma from N^1 to Pic using the spectral radius on Pic^0; (iv) canonical-height-growth estimates from [KS16a, Theorem 5]; and (v) a contradiction using a big divisor supplied by Theorem 2.3. At no point is alpha_f(x) defined to be a mu_i(f), nor is any parameter fitted to the arithmetic degree data; Lemma 2.8 uses the independent definition of alpha_f(x) via ample heights. The main external inputs [Xieb] and [XY25] are by the same research group, and they are load-bearing preprints, but they are separate statements whose assumptions do not include the conclusion of Theorem 1.2, so the self-citation footprint is a verification/reproducibility concern rather than a circular reduction. The skeptical concern about applying [KS16a, Theorem 5] to Jordan blocks with eigenvalues mu_i(f) < 1 (e.g. in the trivial Albanese case) is a possible correctness gap, not circularity: even if the canonical-height step fails there, that would not make the theorem an input to its own proof. Overall, the central claim retains independent mathematical content and is not assumed as a premise.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a theorem-proof paper with no fitted constants. Its central claim rests on the Moriwaki height formalism, on the KS16a canonical height theorem, and especially on the Xieb preprint's characterization of cohomological Lyapunov multipliers via the big cone. The proof is only as strong as those external results.

assumptions (4)

domain assumption Theorem 2.2 from Xieb: the set of cohomological Lyapunov multipliers equals the set of real eigenvalues alpha of f^* on N^1(X)_R for which Im(f^* - alpha) does not intersect Big(X).
Used to identify the multipliers as eigenvalues of f^* and to know they are real and positive. This is the key link between the geometric multipliers and the linear algebra of the numerical pullback.
domain assumption Theorem 2.3 from Xieb: the sum of generalized eigenspaces for the multipliers mu_i(f) has nonempty intersection with Big(X).
Used in the proof of Theorem 1.2 to construct a big divisor B decomposed into a high-multiplier part and a low-multiplier part; this decomposition drives the contradiction.
standard math Moriwaki height machinery over finitely generated fields, including Northcott and projection formula.
Borrowed from Mor00 and LS24; the paper uses it to define arithmetic degrees and to control heights along orbits.
domain assumption Canonical height convergence for Jordan-block pullbacks, cited as KS16a Theorem 5.
Used to form the canonical height vector b h_L satisfying b h_L circ f = b h_L Lambda, which is essential for estimating the growth of the high-multiplier part.

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Pith. "Pith review of Arithmetic Degrees are Cohomological Lyapunov Multipliers." pith.science (2026). https://pith.science/paper/KIHPA7QD

@misc{pith2026250717643,
  author       = {Pith},
  title        = {Pith review of: Arithmetic Degrees are Cohomological Lyapunov Multipliers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KIHPA7QD}},
  note         = {Machine review of arXiv:2507.17643}
}

read the original abstract

For endomorphisms of projective varieties, we prove that the arithmetic degree of a point with Zariski dense orbit must be a cohomological Lyapunov multiplier of the dynamical system. We will apply our result to deduce a corollary towards the dynamical Mordell--Lang conjecture.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

21 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

J. P. Bell, D. Ghioca, and T. J. Tucker. The Dynamical Mordell--Lang Conjecture , volume 210 of Mathematics Surveys and Monographs . American Mathematical Society, Providence, RI, 2016

work page 2016

[2] [2]

Degrees of iterates of rational maps on normal projective varieties

Nguyen-Bac Dang. Degrees of iterates of rational maps on normal projective varieties. arXiv:1701.07760v2

arXiv

[3] [3]

Degrees of iterates of rational maps on normal projective varieties

Nguyen-Bac Dang. Degrees of iterates of rational maps on normal projective varieties. Proc. Lond. Math. Soc. (3) , 121(5):1268--1310, 2020

2020

[4] [4]

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Tien-Cuong Dinh and Vi\^et-Anh Nguy\^en. Comparison of dynamical degrees for semi-conjugate meromorphic maps. Comment. Math. Helv. , 86(4):817--840, 2011

2011

[5] [5]

Une borne sup\'erieure pour l'entropie topologique d'une application rationnelle

Tien-Cuong Dinh and Nessim Sibony. Une borne sup\'erieure pour l'entropie topologique d'une application rationnelle. Ann. of Math. (2) , 161(3):1637--1644, 2005

work page 2005

[6] [6]

Silverman

Shu Kawaguchi and Joseph H. Silverman. Dynamical canonical heights for J ordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties. Trans. Amer. Math. Soc. , 368(7):5009--5035, 2016

work page 2016

[7] [7]

Silverman

Shu Kawaguchi and Joseph H. Silverman. On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. J. Reine Angew. Math. , 2016(713):21--48, 2016

work page 2016

[8] [8]

A note on K awaguchi- S ilverman conjecture

Sichen Li and Yohsuke Matsuzawa. A note on K awaguchi- S ilverman conjecture. Internat. J. Math. , 32(11):Paper No. 2150085, 12, 2021

work page 2021