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For cluster-algebra maps with pq>4 the dynamical degree exceeds 1, so they admit no conserved quantity and no invariant fibration.

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2026-07-10 12:39 UTC pith:VLCLXDA6

load-bearing objection Clean answer to the open integrability question for μp,q via explicit dynamical degrees and superattracting points; the calculations check out.

arxiv 2607.08125 v1 pith:VLCLXDA6 submitted 2026-07-09 math.DS math-phmath.MP

Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations

Andrei Grigorev , Krishna Chaitanya Kalidindi , Andres Quintero Santander , Roland Roeder This is my paper
classification math.DS math-phmath.MP MSC 37F1014E0513F6037A35
keywords birational mapsdynamical degreecluster algebrasinvariant fibrationalgebraic stabilitysuperattracting pointspositive entropy measurestropicalization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies a two-parameter family of birational maps of the plane that arise as compositions of cluster-algebra mutations. Earlier works asked whether these maps possess a conserved quantity (or, more generally, an invariant fibration). The authors answer that question by computing the first dynamical degree of each map. When the product of the parameters is larger than four the degree is strictly greater than one; a classical theorem then forces the absence of any invariant fibration and therefore of any rational first integral. The same conclusion is reached for most negative-parameter maps by an analogous computation or by the discovery of superattracting periodic points. Once an algebraically stable model is constructed and the dynamical degree is known, standard ergodic-theory results supply invariant measures of positive entropy and positive Lyapunov exponents. The work therefore converts a question from discrete integrable systems into a precise statement in complex dynamics and settles it for the generic parameter range.

Core claim

When p and q are positive integers with pq>4 the dynamical degree of the mutation map equals (pq-2+sqrt((pq-2)^2-4))/2, which is greater than one; consequently the map admits neither an invariant fibration nor a rational conserved quantity. Parallel statements hold for most pairs of negative integers.

What carries the argument

Algebraically stable lift obtained by a finite sequence of point blow-ups; the dynamical degree is then the spectral radius of the induced linear action on the Picard group of the blown-up surface.

Load-bearing premise

That a finite sequence of point blow-ups always produces an algebraically stable model on which the pull-back operator can be computed.

What would settle it

Exhibit a single pair of integers p,q with pq>4 for which the lifted map on every finite blow-up of the plane still possesses a destabilizing orbit, or for which an explicit rational first integral can be written down.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Maps with pq>4 cannot be integrated by any rational conserved quantity and therefore lie outside the classical integrable regime of cluster algebra mutations.
  • Each such map carries a unique measure of maximal entropy equal to the logarithm of its dynamical degree, together with positive Lyapunov exponents.
  • Superattracting fixed points or period-3 cycles appear after the blow-ups, giving an independent geometric obstruction to invariant fibrations.
  • The same tropical and Picard-group techniques apply immediately to the remaining open cases of mixed-sign parameters.

Where Pith is reading between the lines

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

  • The explicit closed-form dynamical degree may allow one to decide arithmetic-degree questions for rational points under iteration of these maps.
  • The KAM-like pictures observed numerically for certain negative parameters suggest that the positive-entropy measure coexists with large regions of bounded, nearly integrable motion.
  • Once the mixed-sign parameter plane is treated by the same methods, the entire two-parameter family will be classified by dynamical degree.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies the birational maps μp,q of the plane arising from rank-2 cluster mutations (and their sign-reversed analogues). After compactifying to CP2 and constructing explicit algebraically stable models by finitely many point blow-ups (Xp,q for p,q≥1, Y for p,q≤-2), the authors compute the induced pull-back operators on Picard groups and extract the dynamical degrees: for p,q≥1 and pq>4 one has λ1=(pq-2+√((pq-2)2-4))/2>1 (Theorem A); for p,q≤-2, λ1 is the largest real root of a degree-4 polynomial and is likewise >1 (Theorem C). Consequently the maps admit no invariant fibration and in particular no rational conserved quantity (Theorems B,D), answering questions of Machacek–Ovenhouse and Chen–Li. Superattracting fixed points or 3-cycles supply an independent obstruction to fibrations. With algebraic stability and λ1>1 in hand, standard ergodic results yield measures of maximal entropy with positive Lyapunov exponents (Theorem E). Affine cases pq=4 are integrated explicitly via conserved quantities and Hamiltonian flows.

Significance. The work cleanly settles the integrability question for the family μp,q by importing the well-developed toolkit of complex surface dynamics (algebraic stability, dynamical degree, superattracting points, Diller–Favre/Dinh–Nguyen criteria). The dynamical-degree formulae are obtained by two independent methods (Picard spectrum and the Alonso–Suris–Wei local-index recurrence) that agree, and the algebraically stable models are constructed and verified by hand rather than merely invoked. The resulting non-existence statements answer concrete questions from the cluster-algebra literature, while the ergodic consequences illustrate the broader dynamical richness of these maps. The explicit tropical analysis and the integration of the affine cases further strengthen the contribution.

minor comments (6)
  1. The date line reads “July 10, 2026”; this is presumably a typographical error and should be corrected before publication.
  2. In the abstract and introduction the phrase “question posted by” should be “question posed by”.
  3. Section 1.3, Remark 1.3: the numerical evidence for λ1(μ-1,q)>1 when q=-3,-4 is useful but could be stated more precisely (e.g., approximate growth rates after a fixed number of iterates) so that a reader can reproduce the experiment.
  4. Figure 1 and the subsequent critical-behaviour diagrams would benefit from a short caption explaining the colour coding of the critical curves and the meaning of the arrows.
  5. Appendix B is valuable as an independent verification, yet a one-sentence pointer in the main text (near Theorem 6.5) that the same closed form is recovered by the local-index method would help the reader locate the cross-check.
  6. A few minor notational inconsistencies appear (e.g., ν p,q versus μ-p,-q; occasional use of both “superattracting” and “super-attracting”). Standardising them would improve readability.

Circularity Check

0 steps flagged

No significant circularity: dynamical degrees are spectral radii of explicitly constructed pull-back operators on Picard groups of concrete blow-up models, cross-checked by independent local-indices recurrences and superattracting points.

full rationale

The central claims (Theorems A–D) rest on two independent, fully explicit calculations. First, the authors construct algebraically stable models Xp,q (semi-simple and non-semi-simple cases, §§5.3–5.4) and Y (negative pairs, §8.1.1) by a finite sequence of point blow-ups whose charts and critical-curve images are written down; algebraic stability is verified by direct inspection that no critical curve or exceptional divisor lands on a destabilizing orbit (Prop. 5.12, Prop. 8.4, Cor. 2.13–2.14). The induced pull-back µ* on the free abelian group Pic of finite rank is then computed by intersection multiplicities read off the defining equations (Props. 6.1, 6.3); its spectral radius is extracted by elementary linear algebra (Lems. 6.2, 6.4) and equals the claimed closed-form expression for λ1. An independent route (App. B) recovers the same recurrence for deg(µ^n) via the local-indices method of Alonso–Suris–Wei, again without free parameters. Non-existence of invariant fibrations follows either from λ1>1 (Prop. 1.1, classical) or from the superattracting fixed/periodic points that the authors locate by direct differentiation in the same charts (Lems. 5.6, 5.11, 8.6 + Prop. 1.2, proved in the paper). Tropicalization (Sec. 7) and the ergodic consequences (Thm. E) are applications of external theorems once λ1>1 is known; they do not feed back into the degree computation. No parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work as a black box, and no quantity is redefined in terms of the quantity being “predicted.” The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside the standard toolkit of complex surface dynamics and algebraic geometry. No free parameters are fitted; the only external theorems invoked are classical (Diller-Favre algebraic stability, Dinh-Nguyen-Truong dynamical-degree comparison, Diller-Dujardin-Guedj ergodic theory). The surfaces Xp,q and the tropical fans are constructed, not postulated.

axioms (4)
  • standard math A birational map of a smooth projective surface admits an invariant fibration if and only if its first dynamical degree equals 1 (Diller-Favre + Dinh-Nguyen-Truong).
    Proposition 1.1; used to convert λ1>1 into non-existence of conserved quantities.
  • standard math Existence of a finite sequence of point blow-ups rendering any birational surface map algebraically stable (Diller-Favre).
    Theorem 2.8; the entire stable-model construction of Sections 5 and 8 rests on it.
  • standard math For an algebraically stable birational map, λ1 equals the spectral radius of the induced pull-back on Pic.
    Lemma 2.10; converts matrix computations into dynamical degrees.
  • standard math Algebraically stable separating maps with λ1>1 admit an invariant measure of maximal entropy with the stated Lyapunov bounds (Diller, Diller-Dujardin-Guedj).
    Theorems 9.2-9.3; used for Theorem E.

pith-pipeline@v1.1.0-grok45 · 51499 in / 2380 out tokens · 20659 ms · 2026-07-10T12:39:11.171181+00:00 · methodology

0 comments
read the original abstract

Using the methods of holomorphic dynamics we investigate planar birational mappings that arise from the theory of cluster algebras and integrable systems. Computing dynamical degrees of these mappings, many of which are greater than one, allows us to show that many of the mappings do not have a conserved quantity (nor an invariant fibration). In most of the examples, invariant fibrations can also be ruled out by finding superattracting periodic points. This answers a question posted by Machacek and Ovenhouse 2024 and by Chen and Li 2024. Moreover, having found a good algebraically stable model for the mappings and having computed the dynamical degree, we can then apply results from the ergodic theory of birational maps to produce invariant measures with positive entropy and positive Lyapunov exponents.

Figures

Figures reproduced from arXiv: 2607.08125 by Andrei Grigorev, Andres Quintero Santander, Krishna Chaitanya Kalidindi, Roland Roeder.

Figure 1
Figure 1. Figure 1: Critical behavior of µ2,3. The components Kh ω , ω ∈ R2 and Kv ζ , ζ ∈ R3 are drawn in red and blue respectively. The violet arrows show to where the components of the critical locus of µ2,3 are collapsed. Remark 5.5. Due to Lemmas 5.3 and 5.4 for a non-semi-simple pair every critical value induces a destabilizing orbit. For a semi-simple pair, the critical value [0 : 1 : 0] is fixed. This is the only crit… view at source ↗
Figure 2
Figure 2. Figure 2: Critical behavior of µ5,1 before blowing up at [0 : 1 : 0] [0:0:1] [1:0:0] {y=0} {x=0} {z=0} ”Superattracting fixed point” [0:−1:1] [ω1:0:1] [ω2:0:1] [ω3:0:1] [ω4:0:1] [ω5:0:1] [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Critical behavior of µ5,1 after blowing up. Remark 5.10. Notice that here the exceptional divisor {u2 = 0} and the proper transform {v2 = 0} of the divisor {u1 = 0} are mapped to the fixed point (0, 0). As well as in previous examples, the proper transforms of the collapsing curves Kh ω , Kv ζ are not collapsed. One can check explicitly that proper transforms of coordinate 18 [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 4
Figure 4. Figure 4: Fan of CP2 7.0.2. Toric maps. Recall the affine coordinates (x, y) =  x0 x2 , x1 x2  . Define the meromorphic two-form η = dx∧dy xy . Definition 7.2. A toric map is a rational map f : CP2 99K CP2 such that f ∗η = ρ(f)η for some constant ρ(f) ∈ C ×. Definition 7.3. For a rational toric map f : C 2 99K C 2 set (x(t), y(t)) = f(w1t a , w2t b ), for some generic (w1, w2) ∈ T 2 . Define Af (a, b) = lim t→0 … view at source ↗
Figure 5
Figure 5. Figure 5: Action of the tropicalization A of µp,q on the fan Σ(CP2 ) in the semi-simple case. Using Lemma 7.4, we get the following observations (cf. Section 5.2). • Since A(σ1) ⊂ σ1 we expect that [0 : 1 : 0] is a fixed point; see Remark 7.5. In fact it is a fixed point because there are no exceptional curves E with E ∩ T 2 ̸= ∅ passing through P, as shown in [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Action of the tropicalization A of νp, ˜ q˜ when ˜p, q˜ ≥ 1 and ˜pq >˜ 4. When using Part (3) from Lemma 7.4 in the following computations, we must always verify that the point P corresponding to the sector σ is not indeterminate. In each case, we either verify it with explicit calculations in local coordinates or by checking that there is no exceptional curve E passing through P with E ∩ T 2 ̸= ∅; See Rem… view at source ↗
Figure 8
Figure 8. Figure 8: Computer plot showing approximately the region 0 < x < 5 and 0 < y < 5. Points in green have orbit escaping to infinity. Points in black have a bounded orbit. Five different finite orbits (each of length 200) are shown in white, yellow, orange, light blue, and purple, respectively. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗

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