Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded

Kelsey Brouwer; Suzanne Boyd

REVIEW 3 major objections 5 minor 16 references

Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read In generalized McMullen maps, the position of the second critical value inside a preimage baby basilica forces a precise, finite set of external-ray re-identifications in the next preimage — and the same altered shape then repeats…

desk verdict Genuinely new conditional lamination combinatorics for preimages of baby basilicas, but the headline 'large class of maps' has no proved example. read the letter →

arxiv 2501.07545 v1 pith:J6HYP72U submitted 2025-01-13 math.DS

Suzanne Boyd , Kelsey Brouwer This is my paper · ORCID

classification math.DS MSC 37F1037F20

keywords generalizedMcMullenmapsJuliasetscriticalorbitsexternalrayslaminationpolynomial-likebasilicacombinatorialdynamics

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

Generalized McMullen maps $F(z)=z^n+\frac{a}{z^n}+b$, with $n\geq 3$, have only two critical values $v_+$ and $v_-$ even though they have $2n$ critical points. The paper claims that when both critical orbits stay bounded and $v_+$ sits in a baby basilica (a degree-2 polynomial-like copy of the Julia set of $z^2-1$), the position of $v_-$ inside a preimage copy $K_-$ completely controls the combinatorics of the preimage $J_0$ of $K_-$. If $v_-$ is in the expected component, $J_0$ is an ordinary basilica copy; if $v_-$ is $N$ components away from that spot, the lamination of $J_0$ differs from the basilica lamination by exactly $2N+1$ leaves, according to an explicit re-identification rule. The result is a family of altered quadratic Julia sets, obtained by breaking a finite number of external-ray pairings and gluing them differently, that occur alongside infinitely many homeomorphic copies of quadratic Julia sets. If the assumed parameter values exist, this gives a complete combinatorial model for the new shapes observed in numerical images.

What carries the argument

The machinery is the external-angle assignment $\Gamma\colon S^1\to J^*$ constructed in Theorem 3.1 on the tree of all preimages of a baby quadratic Julia set, using the polynomial-like mapping theorem to initialize the angles on the baby Julia set itself. On a preimage copy containing a critical point, $\Gamma$ makes $F$ conjugate to angle doubling ($t\mapsto 2t$); on a preimage copy without a critical point, $\Gamma$ makes $F$ preserve the angle. The lamination $L_\gamma$ (the chord diagram of identified external angles) attached to each preimage is then compared with the basilica lamination. The fortune-teller step is a local re-gluing: two chords bounding the preimages of a component are broken and reconnected in the crossed pairing, which is exactly the operation that merges two Fatou components into one critical-point component while splitting the previous central component.

What would settle it

A single rigorously verified parameter triple satisfying A1 through A5, with $v_-$ at a known distance $N$, whose preimage lamination does not show exactly the $2N+1$ leaf changes of Theorem 4.4, would refute the combinatorial rule. Concretely, one could take the numerically suggested Type 1-1 candidate $n=11$, $a=0.2075$, $b=-0.004483852355144613+0.0032857624406795257i$, prove it lies in the intended hyperbolic and polynomial-like setting, and test whether the predicted reidentifications appear on $J_0$. In the other direction, an exhaustive proof that no such parameters exist would make the described class empty and the theorem vacuous.

Watch

Extended reading notes

Core claim

The central discovery is a combinatorial rule, Theorem 4.4, that describes exactly how a preimage of a baby basilica Julia set changes when the free critical value sits in an unexpected Fatou component. Let $K_+$ be a baby basilica and $K_-$ a preimage copy containing $v_-$; write $N$ for the number of steps from the expected component $L$ to the component $U_-$ containing $v_-$ along the shortest chain of adjacent Fatou components. Then the lamination $L_0$ of $J_0=\partial F^{-1}(K_-)$ is the basilica lamination with $2N+1$ leaves changed: for each step $i$, the identifications $a_i^1\sim b_i^2$ and $a_i^3\sim b_i^4$ are replaced by $a_i^1\sim b_i^4$ and $b_i^2\sim a_i^3$, except that if the first step passes through the central component $M$, the exception $1/6\sim 1/3$ and $2/3\sim 5/6$ applies. Geometrically, two Fatou components merge into the critical-point component and the old central component splits in two; the paper compares this to opening a paper fortune teller in the other direction. Every earlier preimage in the tree is then homeomorphic to this altered set, so the Julia set contains infinitely many altered copies alongside infinitely many true quadratic copies.

Load-bearing premise

The entire construction is conditional on the existence of parameter values $(n,a,b)$ satisfying assumptions A1 through A5, in particular that $v_+$ lies in a baby basilica and $v_-$ lies in a specified Fatou component of a preimage copy of it; the paper states in its final paragraph that this existence is not established.

Editorial extensions

If this is right

For any placement of $v_-$ that is $N$ components from the expected location, the lamination of the immediate preimage $J_0$ is fully determined by the $2N+1$ leaf changes, so the shape of $J_0$ and of its entire preimage tree is known in advance.
If $v_-$ sits in the expected component $L$, no alteration occurs and every preimage is an ordinary baby basilica; the altered phenomenon is strictly tied to the free critical value being off-schedule.
Each altered set is homeomorphic to a quadratic Julia set with finitely many external-ray landing pairings changed, so some Fatou components merge and others split, and these altered copies form an infinite tree of preimages.
The induction in Theorem 4.4 shows that longer, backtracking paths do not change the final lamination, so the shortest path from $L$ to $U_-$ is the only data that matters.
The paper suggests the same splitting-and-reidentifying mechanism should extend to other quadratic baby Julia sets, such as rabbits, where altered preimages are already visible numerically.

Reading between the lines

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

The paper's own final paragraph concedes that no parameter triple is proven to satisfy assumptions A1 through A5; closing this gap is the difference between a conditional model and an instantiated one. A concrete next step is to verify hyperbolicity and the polynomial-like condition for the numerically reported Type 1-1 and Type 2 examples.
The explicit angle replacements suggest a fast numerical test: compute which external rays land together on a numerically drawn $J_0$; the pattern $a_i^1\sim b_i^4$, $b_i^2\sim a_i^3$ should be visible whenever $v_-$ is off-schedule, and a single verified counterexample to that pattern would pinpoint a flaw in the induction.
The $2N+1$ leaf count could serve as a combinatorial invariant measuring how far $v_-$ has wandered from its expected location; in principle, one could locate the free critical value inside a deep preimage by counting leaf differences in its preimage lamination.
The theorem's restriction to the basilica is probably not essential: translating the fortune-teller re-gluing to other quadratic laminations would give a broader family of altered Julia sets, with the same linear leaf-count growth in the distance from the expected component.

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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

Colleague,

The bottom line: this paper has a genuinely new combinatorial idea—the 'fortune teller' re-identification of external ray pairs in preimages of baby basilica Julia sets—and it proves a clean conditional theorem about how the lamination changes with the location of the second critical value. But the abstract overreaches: the 'large class of maps' is never shown to be nonempty. The paper itself, in the final paragraph of Section 4, says that establishing parameters satisfying A1–A5 is future work. So the main phenomenon is uninstantiated as of now.

What's new and good: Theorem 3.1 assigns external angles to all preimages of a baby quadratic Julia set in a way that respects the dynamics (angle doubling at critical preimages, identity elsewhere). That's a useful construction. Then Propositions 4.1–4.3 and Theorem 4.4 work out, in careful lamination detail, the exact changes: if v− is N Fatou components away from the 'expected' location, the lamination of the preimage J0 differs from the basilica lamination by 2N+1 leaves, with a specific re-identification rule (with the one exceptional case when the first step is the central component M). The proofs are combinatorial and appear internally consistent. The figures help, and the paper is honest about what is conditional.

The soft spot is the one I already flagged: Assumptions A1–A5 are never verified for any parameter. The cited papers [BM23, BH24] prove A3 for some parameters and give bounded-orbit regions, but the full conjunction—including hyperbolicity and v− landing in the right preimage copy of the baby basilica, in the specified Fatou component—is not established. The paper concedes this explicitly. That means the headline claim of 'a large class of maps' is a conjecture about parameter space, not a theorem. It also means the Type 1-1, 1-2, and Type N classifications are conditional on a nonempty configuration. If the configuration is inconsistent, the lamination theorem is vacuous.

I don't think this kills the paper. The conditional combinatorial result is a real contribution for people working on rational maps with two bounded critical orbits, and the angle-assignment framework may be reusable. But the abstract and introduction should be reworded to say 'we provide a combinatorial model for a family of maps satisfying A1–A5, and we conjecture such maps exist,' or the authors need to prove existence. I'd send it to a serious referee, with the expectation of heavy revision.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the family of generalized McMullen maps F_{n,a,b}(z)=z^n+a/z^n+b, with n≥3 fixed. Under a list of assumptions A1–A5 (hyperbolicity, both critical orbits bounded and in different Fatou components, existence of a baby basilica K+ associated with v+, and placement of the other critical value v− in a preimage copy K− of K+), the authors construct an external-angle assignment Γ on the tree of preimages of K+ (Theorem 3.1). They then analyze the lamination of a preimage J0 of J−=∂K−: if v− lies N Fatou components away from the "expected" component, they claim that the lamination L0 differs from the basilica lamination by exactly 2N+1 leaves, with explicit changed ray identifications (Theorem 4.4 and Propositions 4.1–4.3). The resulting objects are called "altered quadratic Julia sets" or "paper fortune teller" sets. The paper is primarily combinatorial and conditional: it contains many numerical images of purported examples, but it does not prove that any parameter satisfies all of A1–A5.

Significance. If the constructive combinatorial picture is correct, the paper offers a new and concrete phenomenon: within a single rational map, baby quadratic Julia sets coexist with infinitely many copies of a modified lamination obtained by finitely many split/reidentification steps. The angle-assignment construction in Theorem 3.1 and the explicit alteration algorithm in Theorem 4.4 are the paper's main contributions, and the accompanying images are suggestive. However, the abstract's headline claim of a "large class of maps" is not backed by an existence proof for the assumed parameter configuration, and the validity of the iterated lamination procedure in Theorem 4.4 is asserted rather than rigorously verified. Both gaps are load-bearing for the central claim, although they appear fixable in a revision.

major comments (3)

[Assumptions (Section 2) and Future Explorations (Section 4)] The abstract's "large class of maps" is not shown to be nonempty. Assumption A5 postulates a very specific configuration: the second critical value v− lies in a preimage copy K− of the baby basilica K+, in the same attracting basin as v+ but not in its immediate basin, in a specified Fatou component of that preimage. The final paragraph of Section 4 explicitly concedes: "actually establishing that there are parameters for which F_{n,a,b} satisfies Assumptions A1 through A5 would be a good avenue for future study." Since no parameter is proved to satisfy all five assumptions, Theorems 3.1 and 4.4 are conditional statements about a possibly empty family. The paper should either prove existence of at least one parameter satisfying A1–A5 (ideally with an open neighborhood of such parameters), or explicitly restate the abstract and main results as conditional on that existence. Numerical images are not a substitute for this proof.
[Theorem 4.4, proof] The proof never verifies the central combinatorial hypothesis that the iterative split/reidentification procedure produces a valid lamination without chord crossings at every intermediate step. The sentence "unless those two Fatou components that will join are adjacent to M, we can't just make that one change in the lamination diagram, as it would create chord intersections" is an assertion, not a proof, and the induction step similarly assumes that after k changes the next two preimage components are adjacent to the current central component. This is load-bearing, because the "altered quadratic Julia set" is defined by the lamination obtained from this procedure; without a rigorous no-crossing argument the object may not exist. A finite chord-counting argument in the basilica lamination, or an explicit recursive description of leaf coordinates, would close the gap.
[Theorem 4.4, statement and notation] The statement's notation is not fully formal. The labels (b^i_1 ↶ a^i_1 / ... ) refer to preimage components in Kc, but after the first reidentification the "current central component" is a new merged object, so it is unclear which chords in the intermediate lamination the next step acts on. Relatedly, the exception for U1=M is stated with different angle pairings (1/6∼1/3 and 2/3∼5/6) than the general rule; the relationship between these pairings and the general formula should be spelled out. Without this, Theorem 4.4 is not yet a precise combinatorial algorithm that a reader could implement, even granting the existence of the parameter configuration in A5.

minor comments (5)

[Section 2, parameter notation] The text writes "C∗ = C/{0}"; this should be C∖{0}, since the slash suggests a quotient rather than a punctured plane.
[Proposition 4.1] The statement says v− lies in the component "L=(1/3↶1/6 / 2/3↶5/6)", but according to the conventions in §4.1 that label is M, not L; the expected component should be L=(5/12↶1/3 / 7/12↶2/3).
[Throughout] The matrix notation for Fatou components is typeset with broken arrow symbols such as "a2/r⇣urve⇡rrowr⫯g⊸tb2"; this notation should be cleaned up, since the reader cannot reliably distinguish the two identified pairs without a diagram.
[Figure 1 and Example 1] The caption of Figure 1 gives a=0.1317−0.0073i, while Example 1 says n=5, a=0.0137−0.0073i, b=0.03+0.02i; the parameter values should be harmonized or the discrepancy explained.
[Theorem 3.1, Step 0] The choice of z0 versus z1/2 in the angle assignment is described as arbitrary but it sets an orientation for Jm,j; the dependence of later lamination comparisons on this choice should be stated explicitly, or shown to be canonical up to the 180° symmetry described in Corollary 3.3.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorem 4.4's lamination changes are derived from local degree and the assumed location of v−; the unproved existence of parameters satisfying A1–A5 is a limitation, not a circularity.

full rationale

The derivation chain runs from Assumptions A1–A5 to the external-angle construction in Theorem 3.1 and then to the leaf-reidentification rules in Propositions 4.1–4.3 and Theorem 4.4. The key step is not fitted: given v− in a named Fatou component U−, the conclusion that the preimage lamination L0 differs by the pair swaps a_i^1∼b_i^4 and b_i^2∼a_i^3 is derived from the fact that K0 must contain a critical point mapping 2:1 onto K− and from the angle-doubling conjugacy in Theorem 3.1. No equation in the paper defines the conclusion into the hypotheses. The self-citations [BM23, BH24] are used only to assert that Assumptions A3 and A4 occur in some parameter regions; they are published results and are not the mechanism by which Theorem 4.4 is proved. The paper's final paragraph explicitly concedes that proving parameters satisfying A1–A5 is future work: "However, actually establishing that there are parameters for which Fn,a,b satisfies Assumptions A1 through A5 would be a good avenue for future study." This is a genuine limitation for the abstract's claim of a "large class of maps" — the class is only conditionally nonempty — but it is not circular reasoning, because the conditional combinatorial statements remain well-defined and independently grounded. Score 1 reflects the minor self-citational and existence caveat, not a circular derivation.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

The central derivation relies on standard polynomial-like and lamination machinery, plus the paper's assumptions A1-A5. No free parameters are fitted to data. The main unproven premise is the existence of parameters satisfying all assumptions, which the paper explicitly leaves to future work. The "altered" object is introduced by definition and lacks independent proof of instantiation.

assumptions (8)

standard math Douady-Hubbard straightening theorem for polynomial-like maps (Theorem 2.2) and external ray/lamination theory for locally connected Julia sets.
Used to identify the baby Julia set K+ with a quadratic basilica and to assign external angles to its boundary.
domain assumption Assumption A1: F is hyperbolic.
Assumed to imply local connectivity of the Julia set and to support the angle assignment construction; stated in Section 2 Assumptions.
domain assumption Assumption A2: both critical values have bounded orbits and lie in different Fatou components.
Invoked to use the XQY14 result that all Fatou components are topological disks in this case.
domain assumption Assumption A3: F is polynomial-like of degree 2 on a region containing v+, with filled Julia set K+ conjugate to a quadratic polynomial.
Provides the baby basilica K+ and the conjugacy to Pc used throughout Sections 3 and 4.
domain assumption Assumption A4: Pc lies in the basilica bulb, so v+ is in the immediate basin of an attracting period-two cycle.
Selects the basilica case for the detailed lamination analysis.
domain assumption Assumption A5: the other critical value v- lies in the interior of a preimage copy K- of K+.
Defines the altered preimage situation; the specific Fatou component of v- determines the type of alteration.
domain assumption Without loss of generality, v- lies on the "left" side of K- or in a smaller decoration attached to M.
Stated in Section 4.2 as a simplification; justified by reversing the top/bottom choice in the angle assignment construction, but the decoration case is only sketched.
ad hoc to paper The sequence of splits and reidentifications in Theorem 4.4 produces a valid lamination without chord crossings at every intermediate step.
The proof of Theorem 4.4 assumes this validity; it is justified by diagrams and informal adjacency arguments rather than by a fully formal combinatorial proof.

invented entities (1)

Altered quadratic Julia set (paper fortune teller set)
purpose: Describes the topological type of preimage baby Julia sets when v- is not in the expected Fatou component.
The object is defined by the paper's own splitting and reidentification algorithm. Numerical images are suggestive, but no parameter existence proof is given, so the independent evidence outside the paper's stated assumptions is not established.

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Pith. "Pith review of Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded." pith.science (2026). https://pith.science/paper/J6HYP72U

@misc{pith2026250107545,
  author       = {Pith},
  title        = {Pith review of: Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6HYP72U}},
  note         = {Machine review of arXiv:2501.07545}
}

read the original abstract

For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for, some new dynamical behavior. In particular, we describe a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets and infinitely many subsets homeomorphic to a set which is obtained by starting with a quadratic Julia set, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we will describe.

Figures

Figures reproduced from arXiv: 2501.07545 by the authors.

**Figure 1.** 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, 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 location of th… view at source ↗

**Figure 2.** The Julia set for the basilica P−1(z) = z 2 − 1, shown with selected external rays and letters we will use to refer to the larger Fatou components. Notice there are many situations in which multiple rays land at the same point. A concise way to express the external ray structure and these identifications for a Julia set is through Thurston’s [Thu09] lamination diagram. Using the [0, 1) parameterization of S 1 , we c… view at source ↗

**Figure 3.** Lamination diagram for the basilica P−1(z) Polynomial-like mappings. Douady and Hubbard defined polynomial-like maps to explain the existence of homeomorphic copies of polynomial Julia sets inside of other Julia sets. Definition 2.1. ([DH85]) A map f ∶ U ′ → f(U ′ ) = U is polynomial-like if ● U ′ and U are bounded, open, simply connected subsets of C, ● U ′ is relatively compact in U, and ● f is analytic and proper… view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: 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. The altered preimage is s…

**Figure 5.** Figure 5: Topological conjugacy diagram for Theorem 3.1. On J+, the map F is conjugate to a Pc on its Julia set Jc, and hence the angle assignments γc for Jc can be passed to J+. As the notation Γ∣ Jm,j may be cumbersome, we introduce γm,j ∶S 1 → Jm,j such that γm,j = Γ∣ Jm,j . …

**Figure 6.** Figure 6: A diagram demonstrating angle assignments for Jm,j if Km,j contains a critical point (Case A). Then the conjugacy statement follows directly from the definition of γm,j = Γ∣ Jm,j . Observation: Recall that by assumption, K+ contains one critical value of F. Hence if m …

**Figure 7.** Figure 7: , we use the shorthand M to refer to the central component, i.e., M = ( 1/3 ↶ 1/6 2/3 À 5/6 ), and note M is the location of the critical point in the basilica. We use L and R to refer to the largest components adjacent to M, so L = ( 5/12 ↶ 1/3 7/12 À 2/3 ) where L is…

**Figure 8.** Figure 8: Type 1-1: Lamination L0 for γ0 when v− ∈ U− = M (left), Lamination L− for γ− is the standard basilica (right) [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Type 1-1: When v− ∈ M of J−, altered shape of J0 (left) compared with J− (right). Colors have been added to show which components in J− are split or combined in J0 under the new angle identifications. Example 1 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: Type 1-2: Angles labeled on Jc where Uc = 2L. The same angles are identified on J−. When pulled back to J0, the yellow Fatou component will split in two and the blue Fatou components will combine as shown in [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

**Figure 11.** Figure 11: Type 1-2: Angles labeled on J0 when v− ∈ U− = 2L. As in [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: Type 1-2 example: Altered shape of J0 when v− ∈ U− = 2L of J− (left, n = 3, a = 0.0522 − 0.01292i, b = 0.01 + 0.03i) along with its lamination diagram L0 showing the identified angles on J0 (right). the two preimage components of U− are combined into one, that compone…

**Figure 13.** Figure 13: shows the chain of lamination changes, and [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]

**Figure 14.** Figure 14: Chain of two steps of Fatou component changes for a Type N = 2 example, when v− ∈ T. Leftmost represents J0, center is the intermediate stage and right is J− [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]

**Figure 15.** Figure 15: For Fn,a,b with n = 3, a = 0.0539 − 0.0118i, b = 0.01 + 0.03i, it appears v− lies in a component T = ( 11/48 ↶ 5/24 13/48 À 7/24) in J−, resulting in an an altered Type 2 J0. the identifications b i 1 ∼ a i 2 , bi 3 ∼ a i 4 unchanged), with one exception: if U1 = M = …

**Figure 16.** Figure 16: A J0 example of Type N = 5 is shown on the left, using n = 3, a ≈ 0.054297 − 0.012066i, b = 0.01 + 0.03i. On the right is a representation of the placement of v− within J−, v− lies in the component shaded in red. These Fatou components are not to scale, as each actual…

**Figure 17.** Figure 17: Observe both typical looking and “altered” rabbits - note the mismatched “ear” size on the larger “altered” rabbit. Here n = 3, a ≈ 0.106497 + 0.077201i, b = 0.01 + 0.05i [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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