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arxiv: 2606.28629 · v1 · pith:C7RM3WTInew · submitted 2026-06-26 · 🧮 math.DS

Paper Fortune Tellers in Julia sets of Generalized McMullen maps II: Sidecars and Zippers

Suzanne Boyd , Kelsey Brouwer This is my paper

Pith reviewed 2026-06-30 00:27 UTC · model grok-4.3

classification 🧮 math.DS
keywords Generalized McMullen mapsJulia setshyperbolic componentsquadratic Julia setsMandelbrot setexternal rayscombinatorial modelrabbit Julia set
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The pith

Generalized McMullen maps admit hyperbolic Julia sets containing homeomorphic copies of rabbit, aeroplane, or Kokopelli quadratic Julia sets.

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

This paper extends a combinatorial model previously developed for Generalized McMullen maps whose Julia sets contain basilica copies to additional quadratic types. It shows that the same ray-landing identifications produce hyperbolic Julia sets with infinitely many homeomorphic copies of rabbit, aeroplane, or Kokopelli quadratic Julia sets when the underlying c-value lies in a bulb attached to the main cardioid or in the main cardioid of a principal baby Mandelbrot set. The construction requires no renormalizations. A sympathetic reader would care because the result enlarges the catalog of explicitly describable hyperbolic components in this rational family by a concrete combinatorial rule.

Core claim

The combinatorial model and ray-landing identifications developed for the basilica case extend directly to rabbit, aeroplane, and Kokopelli cases, yielding hyperbolic Julia sets of Generalized McMullen maps that contain infinitely many homeomorphic copies of these quadratic Julia sets, with the c-value taken from any bulb attached to the main cardioid of the Mandelbrot set or from the main cardioid of any principal baby Mandelbrot set.

What carries the argument

The combinatorial model of Julia sets containing infinitely many homeomorphic copies of a chosen quadratic Julia set, realized by an algorithm that changes a finite number of pairs of external ray landing point identifications starting from the basilica or its generalizations.

If this is right

  • Hyperbolic components exist in parameter space for each such baby quadratic type.
  • The Julia sets contain the additional sidecar and zipper structures described by the same finite-modification algorithm.
  • Infinitely many distinct homeomorphic copies appear for each admissible c-value without further restrictions.
  • The classification covers all principal baby Mandelbrot sets attached to the main cardioid.

Where Pith is reading between the lines

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

  • The same extension technique may apply to still other quadratic Julia sets not covered in this catalog.
  • The pattern suggests a uniform description of all hyperbolic components whose baby sets arise from principal Mandelbrot copies.
  • Explicit parameter bounds or pictures of the new components could be computed directly from the ray identifications.

Load-bearing premise

The combinatorial model and ray-landing identifications developed for the basilica case extend directly to rabbit, aeroplane, and Kokopelli cases without requiring new conditions or renormalizations.

What would settle it

A concrete parameter pair (a, b) for which the baby Julia set is a rabbit but the predicted ray-landing pattern fails to produce the expected homeomorphic copies inside the Generalized McMullen Julia set.

Figures

Figures reproduced from arXiv: 2606.28629 by Kelsey Brouwer, Suzanne Boyd.

Figure 1
Figure 1. Figure 1: The Julia set of Fn,a,b for n = 3, a = 0.05855 − 0.01282i, b = 0.02 + 0.03i. The baby quadratic Julia set is on the positive real axis. Its n − 1 = 2 rotationally symmetric preimages are apparent, as are several smaller deeper-level preimages. An “altered” preimage is several preimages deep, so it is too small to see in detail without zooming in to the area outlined by the red square. homeomorphic copies o… view at source ↗
Figure 2
Figure 2. Figure 2: Portions of the Julia set of Fn,a,b for n = 5, a = 0.1317 − 0.0073i, b = 0.03 + 0.02i. The right image is homeomorphic to a quadratic Julia set, as it is a preimage copy under Fn,a,b of a baby quadratic Julia set in J(Fn,a,b) associated with the critical value v+. The left is a preimage of the right under Fn,a,b, and is what we refer to as “altered”. The red dot in the center of the right image marks the l… view at source ↗
Figure 3
Figure 3. Figure 3: On the right is a schematic of the right of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Julia set for rn,a, n = 4, a = 0.16 + 0.026i showing bifurcation as v− leaves K−. v±, 0 are marked in red, with v− on the left. See [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Standard basilica rays and diagram naming some Fatou components of the basilica Julia set in terms of rays. We refer to two Fatou components of the filled Julia set as adjacent if their boundaries touch. If two Fatou components share a boundary point, that point is the location of an angle identification. A simple example is that M = ( 1/3 ↶ 1/6 2/3 À 5/6 ) is adjacent to L = ( 5/12 ↶ 1/3 7/12 À 2/3 ) beca… view at source ↗
Figure 6
Figure 6. Figure 6: Counting rotations on a sample path from a U+ to a U2 = U− in a 4-rabbit. Here, U+ = U0 is 1 rotation from U1, and U1 is 3 rotations from U− = U2. A(ℓ1 + 1) 1 i ∼ ⋅ ⋅ ⋅ ∼ A(ν) 1 i and a 1 i ∼ b 1 i ∼ ⋅ ⋅ ⋅ ∼ A(ℓi) 1 i ∼ A(ℓi + 1) 2 i ∼ ⋅ ⋅ ⋅ ∼ A(ν) 2 i if Ui lies ℓi counterclockwise rotations from Ui−1, with all other identifications left unchanged. To ease the reading of this notation, consider the subscr… view at source ↗
Figure 7
Figure 7. Figure 7: A diagram showing the identified angles on a 3-rabbit taken from the 1 7 -bulb of M. increasing order, we see that 1 28 < 1 14 < 11 14 < 23 28 , so the component should be labeled B2 = ( 1/14 ↶ 1/28 11/14 À 23/28). To demonstrate the fourth case, we must instead take c to be from a 6 7 -bulb 3- rabbit. In this Julia set, the attracting cycle occurs between the central component and the two lower components… view at source ↗
Figure 8
Figure 8. Figure 8: A diagram showing the external angle identifications on an altered 1 7 -bulb 3-rabbit when U− = ( 9/28 ↶ 2/7 15/28 À 4/7 ) . there are now components which meet at the same component junctions as U 1 1 and M or U 2 1 and M, but are not involved in this splitting and rejoining. We see that these components appear to stay attached at these component junctions, keeping their original identified angle names an… view at source ↗
Figure 9
Figure 9. Figure 9: Both for n = 5, b = 0.01 + 0.03i, and a = 0.17353 + 0.04181i, K− with v− identified inside of T2 (right), and its altered preimage K0 (left). Example 3.10. Suppose Fn,a,b satisfies Assumption 3.7, and that v− is in the central component of J−, which we refer to M as in “middle” or “main”; that is, let U− = M = ( 1/7 ↶ 1/14 4/7 À 9/14). This component is also adjacent to U+ = T1, sharing the same common bou… view at source ↗
Figure 10
Figure 10. Figure 10: A diagram showing the external angle identifications present on an altered 3-rabbit from the 1 7 -bulb when U− = M = ( 1/7 ↶ 1/14 4/7 À 9/14). This component is one rotation away from T1, so this example is also of Type N = 1 with a single rotation. The preimages of U− in K(Pc) meet M at the points with identified angles 9 112 ∼ 11 112 ∼ 15 112 and 65 112 ∼ 67 112 ∼ 71 112 and are colored orange in [PITH… view at source ↗
Figure 11
Figure 11. Figure 11: Both for n = 3, b = 0.01 + 0.04i, and a = 0.1 − 0.76i, K− with v− identified inside of M (right), and its altered preimage K0 (left) [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left: A diagram showing the identified angles on an altered 3-rabbit taken from the 1 7 -bulb where U− = T1T1 = ( 37/224 ↶ 9/56 43/224 À 11/56). Right: An example of left; specifically, K0 where n = 5, b = 0.01 + 0.03i, and a = 0.17357 + 0.041914i [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Left: A diagram showing the identified angles on an altered 3-rabbit taken from the 1 7 -bulb where U− = MR1 = ( 23/224 ↶ 11/112 29/224 À 15/112). Right: An example of left; specifically, K0 where n = 5, b = 0.01 + 0.03i, and a = 0.17356 + 0.041864i. Example 3.12. Suppose that Fn,a,b satisfies Assumption 3.7, and that U− is the upper of the largest set of ears* on the right side of M, that is U− = MR2 = (… view at source ↗
Figure 14
Figure 14. Figure 14: A diagram showing the angle identifications on a Julia set taken from the 9 31 -bulb of M. is, if you follow the attracting periodic cycle, each Fatou component is two coun￾terclockwise rotations from the previous one. We will not discuss 5-rabbits from the two lower bulbs, as each is functionally a reflection of its corresponding upper bulb. Assumptions 3.13. Fn,a,b satisfies Assumptions 2.3, and is conj… view at source ↗
Figure 15
Figure 15. Figure 15: Top: A diagram of an altered 9/31-bulb 5-rabbit where U− = T1. Bottom: K0 where n = 3, b = 0.03 + 0.02i, and a = 0.07008 + 0.02525i. counterclockwise rotations away from the standard critical value location of U+ = T2, the upper of the green ears in [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Top: A diagram of an altered 9/31-bulb 5-rabbit where U− = T1T1 = ( 165/992 ↶ 41/248 195/992 À 49/248). Bottom: K0 where n = 3, b = 0.03 + 0.02i, and a = 0.070054 + 0.025373i. spines. For both aeroplane and Kokopelli Julia sets, all angles that are identified are identified in pairs (rather than larger sets like 3-rabbits which have angles identified in sets of three, etc.). When an aeroplane baby Julia s… view at source ↗
Figure 17
Figure 17. Figure 17: Top: A quadratic 3-aeroplane, Pc for c ≈ −1.755. Bottom: A quadratic 4-Kokopelli, Pc, c ≈ −0.1565 + 1.032i.; a set of 3 arrows points at a spine junction. In both images, critical orbit Fatou components are circled, and arrows indicate the cycles. The name compares the angles splitting from their original identifications to the unzipping of a zipper, where the reidentification is zipping two pieces from s… view at source ↗
Figure 18
Figure 18. Figure 18: A diagram showing the external angles on an unaltered 3-aeroplane. the paper fortune teller alteration occurring over and over again in sequence. Thus we provide no formal proof of this theorem. This Theorem is best understood through the use of examples, which we provide in the following sub-sections; e.g., Example 4.6. 4.1. Zippers: Aeroplanes. Before providing some altered aeorplane examples, we begin … view at source ↗
Figure 19
Figure 19. Figure 19: A diagram showing the angles in the 3-aeroplane which split and reidentify when the preimages of U− lie on the real line. Top: J0, Bottom: J−. the center of the spine S 0 which connects U 1 1 and U 2 1 . Therefore we consider each angle α ∈ [1/7, 3/14] which lands on S 0 . In this case, since S 0 terminates in 0 and 1 2 , we have on J− that α ∼ (1 − α) and ( 1 2 − α) ∼ ( 1 2 + α). On J0, these pairings ch… view at source ↗
Figure 20
Figure 20. Figure 20: Top: A diagram of an altered 3-aeroplane where U− = ( 2/7 ↶ 3/14 5/7 À 11/14). Bottom: K0 where n = 3, b = 0.02 + 0.03i, and a = 0.01949 − 0.01126i. case will be considered to be of Type N = 2. Observe that the two preimages of U− under Pc are U 1 2 = R1T = ( 53/448 ↶ 13/112 59/448 À 15/112) and U 2 2 = L1B = ( 277/448 ↶ 69/112 283/448 À 71/112), colored purple, which do not lie on the same spine in the u… view at source ↗
Figure 21
Figure 21. Figure 21: Top: A diagram of an altered 3-aeroplane where U− = MT = ( 53/224 ↶ 13/56 59/224 À 15/56). Bottom: K0 where n = 3, b = 0.02 + 0.03i, and a = 0.019491 − 0.011214i. considered further alterations to this already altered 3-aeroplane, and in [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: An altered 3-aeroplane of Type N = 4. 4.2. Zippers with Spine Sidecars: Kokopellis. Next, we proceed to analyze alterations of Kokopelli Julia sets. The quadratic Julia sets typically referred to as Kokopellis are spawned from the main cardioid of the largest baby Mandelbrot set connected by a spine to the upper 3-rabbit ( 1 7 -) bulb, which gives them an attracting cycle of period 4, but we expand our ca… view at source ↗
Figure 23
Figure 23. Figure 23: A diagram showing the external angles on an unaltered Kokopelli Julia set. Example 4.10. As a simplest example, let U− = L2 = ( 19/120 ↶ 3/20 11/40 À 17/60). The two preimages of U− = U1 under Pc are U 1 1 = R1 = ( 19/240 ↶ 3/40 51/80 À 77/120) and U 2 1 = L1 = ( 17/120 ↶ 11/80 23/40 À 139/240), colored pink in [PITH_FULL_IMAGE:figures/full_fig_p035_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Top: A diagram showing the identified angles on an altered Kokopelli baby Julia set where U− = ( 19/120 ↶ 3/20 11/40 À 17/60). Bottom: K0 where n = 3, b = 0.02 + 0.02i, and a = 0.084593 + .061765i. central component are where angle identifications have been altered, except for the spine sidecars in black which are unaltered. A diagram detailing these changes is given in [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 25
Figure 25. Figure 25: An altered Kokopelli baby Julia set of Type N = 3 [PITH_FULL_IMAGE:figures/full_fig_p037_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: K0 where n = 3, b = 0.02 + 0.02i, a = 0.06740764 + 0.028731855i. 5. Future Work To finish the catalog of potential Julia alterations, our next case study involveS alterations of the external angles on all remaining hyperbolic components of M [PITH_FULL_IMAGE:figures/full_fig_p037_26.png] view at source ↗
read the original abstract

We study the family of complex rational functions known as Generalized McMullen maps, F(z) = z^n + a/z^n+b, for integer n at least 3 fixed, and complex parameters a, b with a nonzero. In prior work by the same authors, we provided a combinatorial model for a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets conjugate to the ``basilica'', and infinitely many subsets homeomorphic to a set which is obtained by starting with the basilica, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we described. In this article, we generalize beyond the basilica, and provide a catalog of additional types of hyperbolic Julia sets of Generalized McMullen maps, where the ``baby'' Julia set can be any rabbit, aeroplane, or Kokopelli quadratic Julia set; that is, where the c-value can be taken from any bulb attached to the main cardioid of the Mandelbrot set, or from the main cardioid of any principal baby Mandelbrot set (no renormalizations).

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

2 major / 2 minor

Summary. The paper extends prior combinatorial models for Julia sets of the family F(z) = z^n + a/z^n + b (n ≥ 3) from the basilica case to hyperbolic components whose 'baby' Julia sets are rabbits, aeroplanes, or Kokopelli quadratics (c-values from bulbs off the main cardioid or principal baby Mandelbrot sets, without renormalizations). It catalogs additional types via sidecar and zipper constructions that modify external-ray landing identifications according to an algorithm.

Significance. If the direct extension of the ray-landing model holds, the work supplies a broader catalog of hyperbolic Julia-set types for Generalized McMullen maps, linking them to a range of quadratic combinatorial models beyond the basilica. This strengthens the combinatorial approach to these rational maps by showing that the sidecar/zipper framework transfers to different periodic-cycle structures.

major comments (2)
  1. [Introduction / combinatorial model section] The central claim that the basilica ray-landing identifications and sidecar/zipper constructions extend verbatim to rabbit (period-3), aeroplane, and Kokopelli cases requires explicit verification that the new periodic points do not alter landing pairings or impose additional combinatorial constraints; the abstract and prior-work reference provide no such check.
  2. [Section describing rabbit/aeroplane/Kokopelli extensions] The assumption of 'no renormalizations' for c-values from principal baby Mandelbrot sets needs a concrete argument that the external-ray pairings remain compatible with the higher-period cycles without introducing new periodic-cycle conditions.
minor comments (2)
  1. Clarify the precise statement of the algorithm for changing ray-landing identifications when moving from basilica to the new cases.
  2. Add a table or diagram summarizing which quadratic types are covered and which are excluded.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. We address each major comment below, indicating the revisions we will make to clarify the combinatorial extensions.

read point-by-point responses

  1. Referee: [Introduction / combinatorial model section] The central claim that the basilica ray-landing identifications and sidecar/zipper constructions extend verbatim to rabbit (period-3), aeroplane, and Kokopelli cases requires explicit verification that the new periodic points do not alter landing pairings or impose additional combinatorial constraints; the abstract and prior-work reference provide no such check.

    Authors: The full manuscript verifies the extension in the sections on rabbit, aeroplane, and Kokopelli cases by showing that the sidecar and zipper algorithms operate on the local external-ray landing data of the chosen quadratic model and are independent of the specific period of the attracting cycle. The new periodic points are incorporated directly via the choice of the quadratic Julia set (rabbit, aeroplane, or Kokopelli) without altering the global pairing rules. We agree, however, that the introduction would benefit from an explicit statement of this independence, and we will add a short paragraph outlining the verification. revision: partial

  2. Referee: [Section describing rabbit/aeroplane/Kokopelli extensions] The assumption of 'no renormalizations' for c-values from principal baby Mandelbrot sets needs a concrete argument that the external-ray pairings remain compatible with the higher-period cycles without introducing new periodic-cycle conditions.

    Authors: The no-renormalization condition is used to ensure that the baby Julia sets are obtained directly from the quadratic parameter without iterated renormalization, so that the external-ray identifications follow the quadratic combinatorics exactly. We will strengthen the relevant section by adding an explicit argument that the higher-period cycles do not impose additional constraints: the ray pairings are fixed by the internal address of the hyperbolic component, and the sidecar/zipper modifications act only on a finite set of rays that remain compatible under the higher-period dynamics. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to basilica model; generalization to rabbit/aeroplane/Kokopelli cases remains independent

full rationale

The paper cites its own prior work for the basilica combinatorial model and ray-landing identifications, then states that these extend directly to other quadratic Julia sets (rabbit, aeroplane, Kokopelli) with c-values from Mandelbrot bulbs or principal baby Mandelbrot sets, without renormalizations. This is a standard self-citation for background but is not load-bearing: the present article claims to provide the catalog of additional types and the generalization itself. No equations or steps reduce by construction to fitted parameters, self-definitions, or a chain of unverified self-citations. The derivation builds on external standard combinatorial descriptions of quadratic Julia sets rather than renaming or smuggling its own inputs. Score remains low (2) as one minor self-citation that does not force the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all such elements would be extracted from the full manuscript.

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Reference graph

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