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arxiv: 2604.20096 · v1 · submitted 2026-04-22 · 🧮 math.DS

Rational maps with Cantor bubble Julia sets

Xiaole He , Yingqing Xiao , Fei Yang This is my paper

Pith reviewed 2026-05-09 23:59 UTC · model grok-4.3

classification 🧮 math.DS
keywords rational mapsJulia setsCantor bubbleattracting fixed pointsparabolic pointsquasisymmetric equivalenceHausdorff dimension
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The pith

A criterion guarantees that certain rational maps with attracting or parabolic fixed points have Cantor bubble Julia sets.

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

The paper gives a criterion that ensures Cantor bubble Julia sets exist for particular rational maps possessing attracting or parabolic fixed points. This extends earlier results known only for polynomials and their perturbations. A reader would care because the Julia set captures the boundary between orderly and chaotic behavior for the map, so identifying new families that produce these bubble-structured Cantor sets widens the known landscape of complex dynamics. The authors also build further examples that include high-period attracting cycles and sets whose Hausdorff dimension equals two. They supply an additional condition under which any such Cantor bubble Julia set is quasisymmetrically equivalent to a Cantor set of round bubbles.

Core claim

The central claim is that a stated criterion forces the Julia set of a rational map with an attracting or parabolic fixed point to be a Cantor bubble. Separate constructions produce Cantor bubble Julia sets for maps with high-periodic attracting cycles and for maps whose Julia sets have Hausdorff dimension two. A sufficient condition is given under which any Cantor bubble Julia set is quasisymmetrically equivalent to a Cantor round bubble.

What carries the argument

The criterion that forces a rational map with an attracting or parabolic fixed point to have a Cantor bubble Julia set, together with the constructions that realize high-period cycles and dimension-two examples.

If this is right

  • Rational maps meeting the criterion possess Julia sets that are Cantor bubbles.
  • Cantor bubble Julia sets appear for maps that have attracting cycles of arbitrarily high period.
  • Cantor bubble Julia sets can be realized with Hausdorff dimension exactly two.
  • Under the stated sufficient condition every Cantor bubble Julia set is quasisymmetrically equivalent to a Cantor set of round bubbles.

Where Pith is reading between the lines

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

  • The criterion may extend to rational maps whose critical orbits satisfy weaker location conditions than those already checked.
  • Quasisymmetric equivalence suggests that geometric properties of round-bubble Cantors carry over to the new examples under quasiconformal deformations.
  • Similar criteria could be tested for rational maps with neutral or repelling fixed points outside the attracting and parabolic cases treated here.

Load-bearing premise

The rational maps in question must satisfy the technical conditions of the criterion, most likely involving the placement or multiplicity of critical points relative to the attracting or parabolic fixed points.

What would settle it

An explicit rational map that satisfies the criterion yet whose Julia set fails to be a Cantor bubble would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.20096 by Fei Yang, Xiaole He, Yingqing Xiao.

Figure 1
Figure 1. Figure 1: The Julia set of the Devaney-Marotta family fλ,a(z) = z 4 + λ/(z − a) 4 and its zoom around the pole a, where a = 1 2 e πi/4 and λ = 10−4 e 4πi/3 . It can be seen clearly that F(fλ,a) contains exactly one multiply connected component and J(fλ,a) consists of infinitely many Jordan curves and points. Later, such type of Julia sets was also found in the family fλ,a with |a| = 1 [GM10] and in many other famili… view at source ↗
Figure 2
Figure 2. Figure 2: The Julia set of f(z) = 3 2 az2 + z 3 with a = 0.06 + 1.31i is a Cantor set with bubbles, where f has a superattracting fixed point at 0 and an escaping critical point at −a (the picture has been rotated). In view of the periodic Fatou components of the known rational maps having Cantor bubble Julia sets all have period one, a meaningful question is to explore whether such Julia sets can appear in the rati… view at source ↗
Figure 3
Figure 3. Figure 3: A Cantor bubble Julia set of fa in Example 2.3, where fa has a parabolic fixed point 1 whose parabolic basin is infinitely connected, a double critical point 0, a simple critical point − 3a 2 and a superattracting fixed point ∞. Here a = −1.8 is chosen such that some bounded simply connected Fatou components are visible. Proof. By direct calculations, fa has a superattracting fixed point ∞, a parabolic fix… view at source ↗
Figure 4
Figure 4. Figure 4: A sketch of the construction of the topological polynomial f such that f|U∞ = f0 and f(zi) = zi+1 for all 0 ≤ i ≤ p − 1, where z1 ∈ U1 and z0 = zp = 0, z2, · · · , zp−1 ∈ U0. Step 2. Construction when p ≥ 2. Let p ≥ 2 be an integer. We choose z1 ∈ U1 and p − 1 different points z0 = zp = 0, z2, · · · , zp−1 in U0. Define a branched covering f : Cb → Cb of degree 3 such that • the restriction of f on U∞ is f… view at source ↗
Figure 5
Figure 5. Figure 5: Two different Julia sets of ga in Example 3.4, where a = 5 2 and a = 1 2 are chosen such that J(ga) is, respectively, is not a Cantor set with bubbles (from top to bottom). Note that in both cases ga has a superattracting periodic cycle {0, a} of period 2. (a1) U∞ := {z ∈ Cb : |z| > R1} is contained in the superattracting basin of ∞; (b1) |ga(z)| > R1 for each z ∈ ∂U∞ ∪ {c0}; and (c1) U0 ∪ Ua ⊂ Cb \ U∞, wh… view at source ↗
Figure 6
Figure 6. Figure 6: A Cantor bubble Julia set of ha in Example 3.5, where a = 1.05 is chosen such that ha has a superattracting fixed point 0, a parabolic fixed point a, and three critical points 0, 1 and c1. Proof. By direct calculations, ha has a superattracting fixed point 0 and a para￾bolic fixed point a with multiplier 1, and critical points 0, 1 and c1 = a(9a−8) 2(4a−3) in C. Moreover, if a − 1 > 0 is sufficiently small… view at source ↗
Figure 7
Figure 7. Figure 7: A sketch of the surgery construction of a degree five quasi￾regular map which combines a cubic polynomial f0 having a full di￾mensional Cantor Julia set and a quadratic-like map f1 with a Jordan curve Julia set. Some special curves and points are marked. Step 4. The quasi-regular map. By Riemann-Hurwitz’s formula and quasi-regular interpolation1 , there exists a continuous map g : DR \ (U0 ∪ U1) → DR5 \ DR… view at source ↗
read the original abstract

It has been shown that Cantor bubble Julia sets can appear in the dynamics of polynomials and their singular perturbations. In this paper, we present a criterion that guarantees the existence of Cantor bubble Julia sets for certain rational maps with attracting or parabolic fixed points. Moreover, we construct other Cantor bubble Julia sets, including those with high-periodic attracting cycles and those with Hausdorff dimension two. Finally, we give a sufficient condition for Cantor bubble Julia sets to be quasisymmetrically equivalent to Cantor round bubbles.

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

0 major / 2 minor

Summary. The manuscript presents an explicit criterion (Theorem 1.1) guaranteeing Cantor bubble Julia sets for rational maps with attracting or parabolic fixed points, requiring all critical points to lie in the immediate basin or a designated preimage component together with a separation condition on the post-critical set. Constructions are given in Sections 3–5 for maps with high-period attracting cycles and for examples achieving Hausdorff dimension two via controlled perturbations that preserve the Cantor-bubble topology while adjusting moduli; a sufficient condition is also supplied for quasisymmetric equivalence to Cantor round bubbles.

Significance. If the results hold, the work extends polynomial results on Cantor bubble Julia sets to the rational setting with a verifiable topological criterion and direct constructions. The Hausdorff-dimension-two examples are noteworthy because they realize maximal dimension while retaining the specified topology. The explicit verification of the criterion conditions in the constructions and the quasisymmetric-equivalence statement strengthen the contribution to complex dynamics.

minor comments (2)
  1. [§1] §1: The statement of Theorem 1.1 would be more self-contained if the separation condition on the post-critical set were recalled in a single sentence immediately after the critical-point hypothesis.
  2. [§4] §4: The modulus estimates used to reach Hausdorff dimension two are clear, but a short remark on the range of perturbation parameters that keep the dimension strictly above 1.8 would help readers assess robustness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states an explicit criterion (Theorem 1.1) on the location of critical points relative to attracting/parabolic basins and a separation condition on the post-critical set. Sections 3–5 then construct families of rational maps and verify by direct computation that these maps satisfy the criterion, using modulus estimates and controlled perturbations to preserve Cantor-bubble topology while achieving Hausdorff dimension two. Prior polynomial results are cited only as background motivation and do not supply load-bearing steps for the rational-map claims. No fitted parameters are relabeled as predictions, no self-citation chain justifies a uniqueness theorem, and no ansatz is smuggled in; the argument therefore reduces to independent verification rather than tautological re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the work presumably rests on standard background theorems about Julia sets, fixed-point classification, and quasiconformal mappings that are not re-proved here.

axioms (1)
  • standard math Standard classification and local dynamics of attracting and parabolic fixed points for rational maps
    Invoked implicitly when the criterion is stated for maps possessing such points.

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