Rational maps with constant Thurston pullback mapping

Guizhen Cui; Yiran Wang

REVIEW 2 major objections 5 minor 14 references

Rational maps with constant Thurston pullback mapping

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Every non-trivial regular or mixing CTP polynomial factors as a Belyi map composed with a rational map.

desk verdict A significant classification result with a genuine missing step in Lemma 5.5; worth reviewing, but not ready as is. read the letter →

arxiv 2507.04215 v1 pith:C5UW3DXT submitted 2025-07-06 math.DS

Guizhen Cui , Yiran Wang This is my paper · ORCID

classification math.DS MSC 37F2037F3430C10

keywords constantThurstonpullbackmappingCTPmapsMcMullen'sconditionBelyimarkedrationalmonodromygroupbranchedtreespolynomialdynamics

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

CTP maps are marked rational maps whose Thurston pullback mapping is constant. This paper proves that every non-trivial polynomial CTP map whose marked set is regular or mixing—that is, whose regular set $E = A \setminus f^{-1}(V_f)$ is either all of $A$ or non-empty but not all of $A$—satisfies McMullen's condition: it can be written as $g \circ s$ with $s$ a Belyi map and $\#(s(A) \cup V_s) = 3$. The proof reduces the statement to an equality between the monodromy stabilizer of a regular marked point and the common stabilizer of the whole regular set, then converts that equality into a rotational power symmetry that produces the Belyi factor. The paper also constructs a new mixing CTP map and shows it satisfies neither McMullen's nor Saenz's condition.

What carries the argument

The object that carries the argument is the monodromy group $\operatorname{Mon}(f,b)$ acting on the fiber $f^{-1}(b)$ over $b = f(E)$, together with the stabilizers $\operatorname{Stab}(a)$ for $a \in E$ and their common intersection $\operatorname{Stab}_*(E)$. The load-bearing identity is $\operatorname{Stab}(a) = \operatorname{Stab}_*(E)$ for every $a \in E$. For CTP Belyi polynomials ($\#V_f = 3$) it is proved from the branched tree $T = f^{-1}(I)$, where $I$ joins the two finite critical values: Lemma 5.3 locates a unique vertex $c_0$ through which all marked edges pass, and Lemma 5.4 converts that geometry into the stabilizer equality. For $\#V_f \geq 4$, the proof performs a pinching operation on a Jordan domain $D$ — replacing $f$ on each preimage component of $D$ by a branched covering with no critical points outside $A$ — to obtain a CTP Belyi polynomial, shows the induced monodromy homomorphism is injective, and pulls the equality back. With the equality in hand, lifts $\lambda_{i,j}$ of $f$ moving one regular point to another are conformal automorphisms; they generate a finite cyclic group, giving a rotation and the power factorization $f = g \circ P$.

What would settle it

A concrete test is to search for a non-trivial regular or mixing CTP polynomial with $\# V_f = 3$ for which $\operatorname{Stab}(a) = \operatorname{Stab}_*(E)$ holds but the conformal automorphism $\lambda$ demanded in Lemma 5.5 does not exist for some generic $w'$ with $|w'| > |P(E)|$. If such a polynomial also fails McMullen's condition, the theorem is false; if it satisfies McMullen's condition, the proof of Lemma 5.5 is invalid and needs repair. A computational version is to check, for an explicit CTP polynomial satisfying the hypotheses, whether the regular set $E$ lies on a round circle centered at the rotation center $c$ and whether every such $w'$ is reached by a conformal automorphism preserving $A\setminus E$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1: if $(f,A)$ is a non-trivial polynomial CTP map with regular or mixing marked set, then $f$ satisfies McMullen's condition, i.e. there exist a Belyi map $s$ and a rational map $g$ with $f = g \circ s$ and $\#(s(A) \cup V_s) = 3$. In concrete terms, the marked polynomial is forced to be a post-composition of a Belyi map: all the rigidity of the constant pullback collapses to a three-point critical-value configuration. The proof shows first that the regular set maps to one point $b$, then that the stabilizer of each point of $E$ in the monodromy group equals the common stabilizer $\operatorname{Stab}_*(E)$, and finally that this equality yields a cyclic group of conformal symmetries giving a power map $P(z)=(z-c)^d$ with $f = g \circ P$.

Load-bearing premise

The load-bearing premise is the existence assertion in Lemma 5.5: for a non-trivial CTP polynomial satisfying the stabilizer equality, the CTP property itself is claimed to produce, for every sufficiently large $|w'|$, a conformal automorphism $\lambda$ of the sphere with $\lambda(A\setminus E) = A\setminus E$ and $\lambda(E) \subset P^{-1}(w')$; if that assertion fails, the rotation symmetry forcing $A\setminus E \subset \{c,\infty\}$ does not follow.

Editorial extensions

If this is right

Every non-trivial regular or mixing CTP polynomial has a factorization $f = g \circ s$ in which $s$ is a Belyi map and the set $s(A) \cup V_s$ has exactly three points.
For such maps the regular set $E$ is mapped by $f$ to a single point, and the complement $A \setminus E$ has at most two points (Lemma 3.4).
Composing a CTP map with any rational map on the left produces another CTP map (Lemma 4.2), so the new examples generate infinite families.
The new mixing CTP map $R(z) = -((z^2-1)(z^2+3)/(4z^2))^3$ with $A = E \cup \{\infty\}$ is a CTP map that satisfies neither McMullen's nor Saenz's condition.
The stabilizer tests distinguish the classes: McMullen's condition forces $\operatorname{Stab}(a) = \operatorname{Stab}_*(E)$, while Saenz-type examples have elements of $\operatorname{Stab}(a)$ that move other regular points.

Reading between the lines

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

Editorial inference: because the proof ultimately runs on the monodromy action and a branched tree, the same stabilizer equality could be checked algorithmically for any explicitly given polynomial CTP map; whether rational CTP maps admit the distinguished vertex of Lemma 5.3 is the natural next question.
Editorial inference: the paper leaves the branched case $E = \emptyset$ open, and the new mixing example suggests that the obstruction to McMullen's condition lives in the quotient of a monodromy stabilizer by $\operatorname{Stab}_*(E)$; checking that quotient for other examples may either extend or clamp the classification.
Editorial inference: the orbit $\{\tau(E) : \tau \in \operatorname{Mon}(R,b)\}$ computed in Section 4.3 has exactly the eight sets listed there, so that list can be used as a test pattern: any CTP candidate whose orbit shape differs would be a new non-McMullen example.

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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

I read the paper with the stress-test note in hand, and the note lands. The main theorem is a real step: if correct, it closes the classification question for regular and mixing CTP polynomials, and the new example family in §4.2 looks genuinely new. The proof strategy is sensible—reduce to Belyi case via pinching, then use monodromy/stabilizer analysis. I believe the authors know their material. The citation pattern is fine, and I see no parameter fitting or circularity.

The soft spot is Lemma 5.5. After reducing to f = g ∘ P, the proof says "Since (f,A) is a CTP map, there exists a conformal map λ of C such that λ(A\E)=A\E and λ(E) ⊂ P^{-1}(w')." I cannot see where this comes from. Theorem A gives a purely topological condition, and Lemma 3.5 yields only an isotopy, not a Möbius transformation. The conclusion A\E ⊂ {c,∞} rests entirely on this λ. As written, the proof of the #E≥3 case is incomplete. This is not cosmetic; it is the heart of Theorem 2.1.

There are smaller gaps. Lemma 3.7 asserts that a certain cross-ratio is constant "by considering the asymptotic behavior" as a critical value approaches another; the constancy itself is not derived. Theorem 4.3's monodromy check is summarized as "it is easy to check" and the appendix helps, but full verification of the eight sets and the intersection counts is still laborious. Those are presentation issues, not conceptual ones.

None of this suggests the theorem is false. The architecture is plausible and the examples pass explicit checks. But the missing Lemma 5.5 argument is load-bearing, and a referee should demand it before the paper can be accepted.

Who is this for? Specialists in Thurston's pullback map and rational maps without completely stable multicurves. The result matters to them, and the new examples expand the known zoo. I would bring it to a reading group only if the group is already working in this area. My own work is elsewhere, so I probably would not cite it in the next year.

For peer review: yes, send it out. The claim is important enough and the method serious enough to warrant referee time. But my verdict would be major revision, conditional on a real proof of the conformal-map existence in Lemma 5.5.

Referee Report

2 major / 5 minor

Summary. The manuscript studies marked rational maps with constant Thurston pullback (CTP). The main result, Theorem 2.1, states that every non-trivial regular or mixing CTP polynomial satisfies McMullen's condition, i.e. f = g ∘ s for a Belyi map s and a rational map g with #(s(A) ∪ V_s) = 3. The proof separates into a Belyi case with #V_f = 3 (Section 5) and a general case reduced by a pinching construction (Section 6). The paper also constructs a new CTP example (R,A) and proves that it is CTP but satisfies neither McMullen's nor Saenz's condition, with an elementary verification in Appendix A.

Significance. If the main theorem is correct, it provides a concrete algebraic characterization of non-trivial CTP polynomials: outside the regular Saenz-type family, such maps factor through a Belyi map with three post-critical values. The tree and monodromy machinery in Section 5 is a useful new tool, and the paper gives an explicit, checkable family of CTP examples with a complete description of the monodromy orbits. However, the proof as written contains a load-bearing assertion about the existence of a conformal map in Lemma 5.5 that is not established; the same type of assertion recurs in Section 4.3. These gaps prevent the main theorem and the example classifications from being fully justified in the present form.

major comments (2)

[§5.3, Lemma 5.5] In the case #E ≥ 3, the proof asserts: 'Since (f,A) is a CTP map, there exists a conformal map λ of C such that λ(A\E) = A\E and λ(E) ⊂ P^{-1}(w′).' This is not a consequence of the CTP property as defined in Section 2 or as used through Theorem A and Lemma 3.5. The CTP hypothesis gives lifts φ1 that are isotopic to the identity rel A, and Lemma 3.5/Corollary 3.6 give only homeomorphisms or equivalences of marked maps; no conformal automorphism with a prescribed image of E is produced. The following sentence, 'the round circle containing E maps to the round circle containing P^{-1}(w′) by λ', and the final deduction A\E ⊂ {c,∞} depend entirely on this λ. A proof of the existence of such a conformal map, or a different argument that #(P(A) ∪ V_P) = 3, is required for Theorem 2.1.
[§4.3, Lemma 4.5(3) and Lemma 4.6(b)] The same unsupported inference appears in the proof of Lemma 4.5(3): 'Since (S,A) is a CTP map, there is a conformal map λ of C such that λ(a) = a, λ(a1) = a1, λ(a2) = a3 and λ(a3) = a2.' Neither the definition of CTP nor the topological characterization in Theorem A provides such a conformal automorphism with prescribed values on a marked quadruple. Lemma 4.6(b) repeats this argument for the example (R,A). Since these lemmas are used to conclude that the examples do not satisfy McMullen's or Saenz's conditions, the example classification is not yet established without an additional proof or reference.

minor comments (5)

[Throughout] There are several typographical errors: 'Uniformalization' should be 'Uniformization', 'Corollay' should be 'Corollary', 'Stablizer' should be 'Stabilizer', 'definations' should be 'definitions', and 'for any other else point' should be 'for any other point'.
[§4.1, Theorem 4.1] The sentence 'Since deg_c S = 3 for c = 0,1,∞, S does not satisfy McMullen's condition' is too terse; a reader needs at least one explanatory sentence, or a reference, for why local degree 3 at all critical values rules out the factorization f = g ∘ s with #(s(A) ∪ V_s) = 3.
[§4.2, Theorem 4.3] The verification in the proof depends on the labeling a1,...,a12 shown in Figure 3, but the text does not define these points explicitly; the elementary proof in Appendix A gives formulas for a1,a2,a3, but the remaining labels are only implicit in the figure. Adding the definitions would improve reproducibility.
[§5.1] The notation T(a1,...,an) = T[a1,...,an] \ {endpoints} is introduced, but the phrase 'endpoints of T[a1,...,an]' should be clarified as the vertices of degree one in that subtree; the current wording may be ambiguous when a1,...,an are themselves endpoints.
[§6.1] In the pinching construction, the assertion that the Uniformization Theorem yields a homeomorphism φ such that g = eg ∘ φ^{-1} is a polynomial needs a brief justification, since not every branched covering of the sphere is topologically equivalent to a rational map; the fact that eg has a unique pole at ∞ is the relevant point and should be stated.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the main theorem is proved from an external topological characterization and monodromy/tree arguments; the one questionable conformal-map assertion in Lemma 5.5 is a proof gap rather than a circular reduction.

full rationale

The paper's central claim is that every non-trivial regular or mixing CTP polynomial satisfies McMullen's condition. The proof chain uses Theorem A from Buff-Epstein-Koch-Pilgrim [3] to translate the CTP hypothesis into a non-essential-preimage condition, then uses monodromy, branched trees, stabilizers, and pinching to arrive at the factorization f = g∘P with #(P(A) ∪ VP) = 3. McMullen's condition is the target of the proof, not an input: the paper proves the implication one way, while the converse (McMullen condition implies CTP) is stated as a known example and is not used to prove Theorem 2.1. There is no fitted parameter called a prediction and no uniqueness theorem imported from the authors' own prior work. The self-citations in the introduction concern hyperbolic rational maps, stable multicurves, and renormalization; they are contextual and not load-bearing for the main theorem. The only passage that warrants attention is Lemma 5.5, where the text asserts that because (f,A) is a CTP map, there exists a conformal map λ preserving A\E and moving E into a P-fiber, without giving a proof. This assertion is load-bearing for the final conclusion A\E ⊂ {c,∞}, and if it cannot be justified the proof of that case is incomplete. However, this is a potential mathematical gap, not a circularity: the asserted λ is not defined in terms of the McMullen factorization, nor is the desired conclusion used as a hypothesis to produce it. The derivation does not reduce to its own inputs by construction, so the circularity score remains essentially zero; the modest score of 1 reflects the unresolved proof gap rather than any circular step.

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

The paper introduces no fitted constants or ad hoc parameters. Its extra assumptions (e.g., #Vf = 3 in the Belyi case, #Vf ≥ 4 in the pinching case) are hypotheses of intermediate lemmas, not free parameters. The only new object is the pinching construction, which is a construction method rather than a postulated entity.

assumptions (4)

standard math Uniformization Theorem and the existence of the Thurston pullback map on Teichmüller space
Used in Section 2 to define the Thurston pullback mapping and in Lemma 3.1.
standard math Thurston's topological characterization of post-critically finite rational maps (Douady-Hubbard proof)
Background behind the Thurston pullback mapping, introduced in Section 1 and used throughout.
domain assumption Theorem A of Buff-Epstein-Koch-Pilgrim: a marked rational map is CTP if and only if preimages of curves are non-essential
This prior theorem is the main tool used throughout, e.g., in Lemma 3.4 and Theorem 4.3.
standard math A conformal automorphism of the Riemann sphere fixing three points is the identity
Used in Lemma 3.1 to pass from isotopy rel A0 and rel A1 to isotopy rel their union.

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Cite this review

Pith. "Pith review of Rational maps with constant Thurston pullback mapping." pith.science (2026). https://pith.science/paper/C5UW3DXT

@misc{pith2026250704215,
  author       = {Pith},
  title        = {Pith review of: Rational maps with constant Thurston pullback mapping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C5UW3DXT}},
  note         = {Machine review of arXiv:2507.04215}
}

read the original abstract

In this paper, we study CTP maps, that is, marked rational maps with constant Thurston pullback mapping. We prove that all the regular or mixing CTP polynomials satisfy McMullen's condition. Additionally, we construct a new class of examples of CTP maps.

Figures

Figures reproduced from arXiv: 2507.04215 by the authors.

**Figure 1.** From a curve to an arc. Since β is homotopic to f(α) in C\Vf rel the endpoints, f −1 (β) has a component containing a0 and a1. Corollary 3.3. Let f be a rational map with deg f ≥ 2. Let a0, a1 ∈ C be two points such that f(a0) ∈/ Vf and f(a0) ̸= f(a1). Then there exists a Jordan domain Ω ⊂ C such that • f(a0), f(a1) ∈ Ω and (Ω\{f(a1)}) ∩ Vf = ∅, • ∂Ω is disjoint from Vf , • f −1 (Ω) has a component containing a0 and… view at source ↗

**Figure 2.** The picture of R−1 (S 1 ). It is easy to check that: 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Location of Ek. Let ρ∞ ∈ Mon(R, b) be induced by γ∞(t) = {2be2πit − b : 0 ≤ t ≤ 1}. Then ρ 6 ∞ = id. Set E0 = E and Ek = ρ k ∞(E) for 1 ≤ k ≤ 5. Label the points in R−1 (b) as in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The simple closed curve γi . by {ρ0, ρ1}. In particular, if e, e′ are edges of T with common endpoint c ∈ f −1 (vi) and ρ k i (e) = e, then k/ degc f is an integer and hence ρ k i (e ′ ) = e ′ . Define a norm on Mon(f) by |τ | = 0 if τ = id, and |τ | = inf{n : τ = ρ kn…

**Figure 5.** Figure 5: Chase of two edges Continuing this process successfully, we obtain a sequence {τj} in Mon(f) such that |τj | = 1, |T[e, ej ]| = j+1, T(e ′ , · · · , e′ j ) and T(e, ej−1) are contained in the two distinct components of T\{ej}, where ej = τj (ej−1) and e ′ j = τj (e ′ j…

**Figure 6.** Figure 6: Chase of edges Applying Lemma 5.1 for ˜e1 and ˜e ′ , there exists τ1 ∈ Mon(f) such that • |T[˜e1, τ1(˜e1)]| = |τ1| + 1 > 1, • τ1(˜e1) and τ1(˜e ′ ) have a common endpoint c ′ . • T[˜e1, τ1(˜e1)] ∩ T[˜e ′ , τ1(˜e ′ )] = {c ′}. Thus c ′ ̸= c and τ1(˜ej ) is not adjacent …

**Figure 7.** Figure 7: Center of marked edges There exists τ2 ∈ Mon(f) such that |τ2| = 1 and τ2(τ1(τ (e1))) and τ2(τ1(τ (e0))) are contained in the same component of T\{c ′} (see [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: Alternate pinching For any τ ∈ Stab(a), there is a sequence {i1, · · · , im} with ij ∈ {1, · · · , n} such that τ = ρim · · · · · ρi1 . There is an integer k1 ∈ Z such that τ1 := ρ k1∞ · ρi1 ∈ Stab(a). There is also an integer k2 ∈ Z such that τ2 := ρ k2∞ · ρi2 · ρ −k1…

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Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

Bogomolov and H

F. Bogomolov and H. Fu, On the PGL 2-invariant quadruples of torsion points of elliptic curves, European Journal of Mathematics 6, 1236-1253 (2020)

work page 2020

[2] [2]

X. Buff, A. Epstein and S. Koch, Eigenvalues of the Thurston operator, J. Topology 13 (3), 969-1002 (2020)

work page 2020

[3] [3]

X. Buff, A. Epstein, S. Koch and K. Pilgrim, On Thurston’s pullback map, in Complex Dynamics: Families and Friends, (A. K. Peters, Welles- ley, MA, 2009), 561-584 (2009)

work page 2009

[4] [4]

Cui and L

G. Cui and L. Tan, A characterization of hyperbolic rational maps, Invent. Math. , 183, 451-516 (2011)

work page 2011

[5] [5]

G. Cui, W. Peng and L. Tan, Renormalizations and Wandering Jordan Curves of Rational Maps, Commun. Math. Phys. 344, 67-115 (2016)

work page 2016

[6] [6]

Cui and W

G. Cui and W. Peng, On the cycles of components of disconnected Ju- lia sets, Mathematische Annalen , https://doi.org/10.1007/s00208-020- 02073-7, 2020

doi:10.1007/s00208-020- 2020

[7] [7]

G. Cui, F. Yang and L. Yang, Decomposition of rational maps by stable multicurves, Int. Math. Res. Not. 11, 1-26 (2025), https://doi.org/10.1093/imrn/rnaf147

work page doi:10.1093/imrn/rnaf147 2025

[8] [8]

Douady and J

A. Douady and J. Hubbard, A proof of Thurston’s topological charac- terization of rational functions, Acta Math. 171, 263-297 (1993)

work page 1993