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arxiv: 2606.24520 · v1 · pith:3BESHAYWnew · submitted 2026-06-23 · 🧮 math.DS

Rigidity of McMullen Julia sets

Yan Gao , Luxian Yang , Jinsong Zeng This is my paper

Pith reviewed 2026-06-25 22:06 UTC · model grok-4.3

classification 🧮 math.DS
keywords quasisymmetric classificationMcMullen mapsJulia setspostcritically finitequasisymmetric rigiditydihedral symmetrySierpiński carpetnecklace Julia set
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The pith

The Julia sets of postcritically finite McMullen maps are quasisymmetrically classified, each with exactly its natural finite dihedral symmetry group.

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

The paper classifies the Julia sets of the maps f_λ(z) = z^n + λ/z^n that are postcritically finite, for n at least 2 and λ nonzero complex, up to quasisymmetric equivalence. It proves that the only quasisymmetries of each such Julia set are those generated by the obvious rotational and reflectional symmetries of the map itself, forming a finite dihedral group. The classification covers every topological type that occurs in the family: Sierpiński-like carpets, necklaces, and clusters. A reader would care because the result gives the first known rigid Julia sets inside each of the three classes and shows that quasisymmetric rigidity holds uniformly across the entire family.

Core claim

The authors establish a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps f_λ(z)=z^n+λ/z^n and prove that the quasisymmetry group of each such Julia set coincides exactly with the finite dihedral group generated by the natural symmetries of the map. This holds uniformly for the Sierpiński-like carpet, necklace, and cluster topological classes.

What carries the argument

The quasisymmetric equivalence relation on these Julia sets, together with the explicit identification of each equivalence class's symmetry group as the finite dihedral group induced by the map.

If this is right

  • Every postcritically finite McMullen Julia set is quasisymmetrically rigid.
  • The quasisymmetry group of each such set is precisely the finite dihedral group coming from the map's natural symmetries.
  • The three topological classes (Sierpiński-like carpets, necklaces, clusters) each contain rigid examples.
  • Any two postcritically finite McMullen Julia sets are quasisymmetrically equivalent if and only if they belong to the same class and share the same symmetry type.

Where Pith is reading between the lines

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

  • Quasisymmetric conjugacy between two such maps must preserve the parameter λ up to the action of the dihedral group.
  • Small perturbations of λ that leave the postcritically finite set will generally produce Julia sets outside these rigid classes.
  • The same rigidity technique may apply to other families of rational maps whose postcritical sets are finite.

Load-bearing premise

The postcritically finite condition on the maps is assumed to produce and exhaust all three topological classes of Julia sets without leaving any class outside the classification.

What would settle it

Exhibiting either a quasisymmetric homeomorphism between two postcritically finite McMullen Julia sets from distinct topological classes or a quasisymmetric self-homeomorphism of one such set that is not an element of its dihedral symmetry group.

Figures

Figures reproduced from arXiv: 2606.24520 by Jinsong Zeng, Luxian Yang, Yan Gao.

Figure 1
Figure 1. Figure 1: Classification of PCF McMullen Julia sets. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The connectedness loci of McMullen maps. Here Λ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The construction of ξ1 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The conformal maps (7.2) at each level are the key to transferring dynamical [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the proof of Case (B). Subcase (B-1). i = i ′ = 0. Then Ω and Ω′ can be connected by B, T, and possibly some components Tk of Tk with k ≥ 1. If such a component Tk exists, write Tk ⊂ V1. By Fact 1, no components of V1 \ (T1 ∪ K(T1)) connects Ω and Ω′ . It then follows that Tk = T1; see [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗
read the original abstract

We provide a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps $f_\lambda(z)=z^n+\lambda/z^n$ with $\lambda\in\mathbb{C}^*$ and $n\geq 2$, and prove that the quasisymmetry group of each such Julia set is exactly the finite dihedral group generated by the natural symmetries of the map. These results establish quasisymmetric rigidity for all topological classes in this family, including Sierpi\'{n}ski-like carpets, necklaces, and clusters, and provide the first known examples of rigid Julia sets in each of the three classes.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript provides a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps f_λ(z)=z^n + λ/z^n with λ∈C* and n≥2. It proves that the quasisymmetry group of each such Julia set is exactly the finite dihedral group generated by the natural symmetries of the map. These results are claimed to hold uniformly across all three topological classes (Sierpiński-like carpets, necklaces, and clusters) and to supply the first rigid examples in each class.

Significance. If the classification and rigidity statements hold, the work supplies the first known rigid Julia sets in each of the three topological classes for this family and advances the program of quasisymmetric rigidity results in complex dynamics.

major comments (1)
  1. [The section (or theorem) classifying or enumerating postcritically finite parameters λ] The central claim of a 'complete' classification that covers 'all topological classes' rests on the assertion that the postcritically finite locus for each fixed n realizes every one of the three classes without gaps or omitted sub-families. The manuscript must contain an explicit enumeration or parametrization (with a dedicated theorem or subsection) showing that PCF parameters exist in each class and that the case analysis exhausts the PCF locus; without this, the uniformity statement is not load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comment below.

read point-by-point responses

  1. Referee: [The section (or theorem) classifying or enumerating postcritically finite parameters λ] The central claim of a 'complete' classification that covers 'all topological classes' rests on the assertion that the postcritically finite locus for each fixed n realizes every one of the three classes without gaps or omitted sub-families. The manuscript must contain an explicit enumeration or parametrization (with a dedicated theorem or subsection) showing that PCF parameters exist in each class and that the case analysis exhausts the PCF locus; without this, the uniformity statement is not load-bearing.

    Authors: We agree that an explicit enumeration strengthens the presentation. The manuscript already classifies PCF parameters via the possible finite critical orbit configurations for the maps f_λ, which partitions the locus into the three topological classes with no omissions (as the orbit conditions are exhaustive). To make this fully transparent as requested, the revised version will include a dedicated theorem (with a short subsection) that parametrizes the PCF locus for each fixed n≥2, confirms existence in each class, and verifies that the subsequent case analysis covers the entire locus. revision: yes

Circularity Check

0 steps flagged

No circularity: classification and rigidity proved from standard complex-dynamics tools under explicit PCF hypothesis

full rationale

The paper states a theorem classifying quasisymmetric equivalence classes of Julia sets for postcritically finite McMullen maps and proves that the quasisymmetry group equals the dihedral group generated by the map symmetries. The PCF condition is an explicit, externally verifiable hypothesis on the parameter λ; it is not fitted to data nor defined in terms of the target classification. No equations or steps in the abstract or described argument reduce a prediction to a fitted input by construction, nor does any load-bearing step rest solely on a self-citation whose content is itself unverified. The three topological classes are enumerated by direct analysis of the dynamics under the PCF assumption rather than by renaming or smuggling an ansatz. The derivation is therefore self-contained against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on the abstract; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

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