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Similarity between the Multibrot set and the Julia set of correspondences at Misiurewicz points

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Pith's one-line read At Misiurewicz points of the correspondence family z ↦ (z^p+c)^(1/q), the Multibrot set and the filled Julia set are asymptotically similar, sharing a common scaling factor and limit model up to a nonzero constant; for (p,q)=(4,2) the resul

desk verdict A genuine extension of Tan Lei's similarity theorem to algebraic correspondences, with a solid transversality proof, but the Multibrot–Julia similarity hinges on an unproven density claim that the paper itself flags. read the letter →

arxiv 2509.07266 v1 pith:2DMWODRM submitted 2025-09-08 math.DS

Carlos Siqueira This is my paper · ORCID
classification math.DS MSC 37F0537F1037F32
keywords MisiurewiczpointsMultibrotsetJuliaalgebraiccorrespondencesasymptoticself-similaritytransversalityconditionholomorphicmotionsunicriticalfamily
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

This paper studies the parameter space of a multi-valued iterated map family, f_c(z)= the q-th root of (z^p + c), with p > q > 0. The key claim is that at a Misiurewicz point—a parameter where the critical point 0 has exactly one bounded orbit, eventually periodic—the Multibrot set (parameters for which 0 has a bounded orbit) and the filled Julia set (points with at least one bounded orbit) are asymptotically similar: under magnification by powers of a single complex scaling factor, both sets converge to the same limit shape up to a nonzero constant. This extends the classical 1990 result for quadratic polynomials by replacing polynomials with algebraic correspondences, which are multi-valued maps. The proof works whenever a natural transversality condition holds; the paper proves that condition algebraically for the family (p,q)=(4,2), and gives experimental evidence for other exponents.

What carries the argument

The key objects are the (p:q) algebraic correspondences f_c(z)=(z^p+c)^(1/q), which send a point to q images and admit p preimages, and their Misiurewicz parameters: parameter values where the critical point 0 has exactly one bounded orbit and that orbit is strictly preperiodic, landing on a repelling cycle. The argument's engine is a holomorphic motion of that cycle: Lemma 3.1 shows the cycle, and the univalent branches along it, persist holomorphically as c varies, so the preperiodic orbit continues as a unique bounded orbit ξ(c). The transversality condition—the requirement that w(c)=h_c(g_c(c))−g_c(c), with g_c the branch composition to the cycle and h_c the cycle composition, satisfy w'

What would settle it

Take any Misiurewicz parameter a for the family (1.1) with (p,q) not (4,2), and compute w'(a) = d/dc [h_c(g_c(c))−g_c(c)] at c=a by following the branches explicitly. A vanishing derivative for one such parameter would disprove the transversality conjecture and remove the hypothesis of Theorem B for that point. Alternatively, numerically magnify M_{p,q} and K_a about a by powers of λ(a) and measure the Hausdorff distance between the scaled sets; if that distance does not tend to zero, the asymptotic similarity is false.

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Extended reading notes

Core claim

The paper's central claim is Theorem B (Theorem 5.2 in the text): let a be a Misiurewicz parameter for the correspondence family (1.1), and suppose the transversality condition holds at a. Then both the Multibrot set M_{p,q} and the filled Julia set K_a are asymptotically self-similar about a with the same scaling factor λ(a), and their limit models coincide up to multiplication by a nonzero complex constant. Here λ(a) is the multiplier of the repelling cycle on which the critical orbit lands. When (p,q)=(4,2), the transversality condition is proved in full (Theorem 3.1), so the similarity becomes an unconditional theorem (Corollary 5.1). The paper also proves an independent statement (Theor

Load-bearing premise

The argument rests on a density property of the Julia set—that periodic points which repel nearby orbits are dense—which the paper cites from an unpublished preprint; if that property fails, the dense holomorphic motions required by the similarity theorem cannot be built.

Editorial extensions

If this is right

  • At every Misiurewicz point satisfying transversality, M_{p,q} and K_a have the same asymptotic scaling: magnifying either about a by powers of λ(a) produces the same limit set up to a nonzero complex constant.
  • For (p,q)=(4,2), the transversality check is algebraic, so the similarity theorem is unconditional for that family: K_a and M_{4,2} are asymptotically similar at every Misiurewicz point.
  • For the Julia-side result (Theorem 4.1), no transversality is needed: the filled Julia set K_c is asymptotically λ(c)-self-similar about every point of the preperiodic orbit and of the repelling cycle.
  • The paper reduces the full similarity theorem to a single condition: prove transversality at a Misiurewicz point and the whole similarity result follows by the same argument.
  • If the transversality conjecture holds for all integer exponents, the similarity theorem would hold for every rational exponent p/q > 1 in the family.

Reading between the lines

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

  • If the density claim cited from the preprint weakens, the similarity theorem may still hold for the filled Julia set but would need a different dense set; Remark 5.1 already shows the natural replacement Y(c) does not work, so this is a concrete open problem.
  • The 2-adic valuation proof of transversality for (p,q)=(4,2) is algebraic and may transfer to other even-exponent families where the sign-cancellation structure is analogous; testing p=6,q=2 at a Misiurewicz point would give a cheap check.
  • A quantitative computational experiment—recording Hausdorff distances between λ(a)^n-scaled M_{p,q} and K_a at, say, p/q=3/2—would estimate whether the conjectured transversality holds and with what convergence rate; the paper's figures are suggestive but not numerical.
  • The theorem's 'up to a nonzero complex constant' allows the parameter-space and dynamical-plane models to differ by a rotation-dilation; identifying that constant µ_a explicitly for small periods may reveal a scaling-universality statement across the parameter and dynamical planes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the unicritical family of algebraic correspondences f_c(z)= q√(z^p+c). For a Misiurewicz parameter a (unique bounded critical orbit, strictly pre-periodic), it proves (Theorem 4.1) that the filled Julia set K_c is asymptotically self-similar about the points of the associated pre-periodic orbit. The central result (Theorem 5.2) asserts that, under a transversality condition w'(a)≠0, the Multibrot set M_{p,q} and the Julia set K_a are asymptotically similar about a, with common scale λ(a) and limit models coinciding up to a constant. For (p,q)=(4,2), Theorem 3.1 proves transversality by a 2-adic valuation argument, giving an unconditional Corollary 5.1.

Significance. If correct, this is a substantial extension of Tan Lei's theorem to algebraic correspondences. The reduction of similarity to transversality and the algebraic proof of transversality for the (4,2)-semigroup are original and potentially useful. The paper is honest about its dependence on the unpublished preprint [19] and includes Remark 5.1 flagging the exact difficulty. However, the main theorem is only as strong as the unproved density of repelling periodic points in K_a, so the significance is contingent.

major comments (3)
  1. [§5, Lemma 5.6] The proof invokes 'By Theorem??, this set is dense in J_a=K_a'. This theorem is not identified; the only external source offered is the unpublished preprint [19, Theorem D]. The density of repelling periodic points in K_a is exactly what makes X'(a) dense in X(a) and is therefore load-bearing for hypothesis (ii) of Theorem 5.1. Without it, Theorem 5.2 and Corollary 5.1 do not follow. Remark 5.1 explicitly concedes that this density cannot currently be shown. This must be repaired by a proof or a precise, verifiable reference.
  2. [§6, Lemma 6.1] In case (b) (and similarly (d)) the proof asserts 'v(ζ−η) equals min{v(ζ),v(η)}'. This is not a valuation identity: equality holds only when the valuations differ; otherwise cancellation can make the left side larger. The congruence (6.3), and hence the proof of Theorem 3.1, depends on this step. The conclusion may be true, but the argument as written is incomplete and needs a correct valuation estimate.
  3. [§5, Eq. (5.8)] The displayed computation of F_c(a,z_ℓ) is invalid: adding the term (F(a,z_ℓ)−F(c,z_ℓ(c)))/(c−a) changes the limit by a generally nonzero quantity, so the chain of equalities does not hold. The intended formula F_c(a,z_ℓ)=−z'_ℓ(a) follows instead by differentiating the identity F(c,z_ℓ(c))=0 with respect to c. As printed, the proof of u'(a)=w'(a)/(λ(a)−1) is not correct. This is a local error, but it occurs in the central theorem.
minor comments (5)
  1. [§5, Lemma 5.6] The placeholder 'Theorem??' should be replaced by an actual citation; please also clarify the status of [19, Theorem D] for the equality K_a=J_a.
  2. [§4, Eq. (4.4)] The equation contains a duplicated expression '=φ_c(V_{c,r}∩K_c)∩D_{λ(c)r}'; likely a typo.
  3. [Proof of Theorem 3.1] In the second case, 'F'_{ℓ+n-1}(a)−F'_ℓ(a)' should probably be 'F'_{ℓ+n}(a)−F'_ℓ(a)' to match the first case and the definition of w(c).
  4. [§6.2] The arrow in 'ˇz_1 = a fc → ˇz_2' should be 'f_a' for clarity.
  5. [Figure 1 caption] The text mentions magnifications of 10^3 and 10^5, but the caption is not fully explicit about the scale factor for each panel; please make it consistent.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 5.2's Multibrot–Julia similarity is load-bearing on the unproved equality K_a=J_a / density of repelling cycles, cited only to the author's unpublished [19, Theorem D] and an unresolved 'Theorem??'; the paper's Remark 5.1 concedes the premise is not established.

  1. self citation load bearing [Section 5, Lemma 5.6 (with Definition 5.1 and the Remark after Theorem C)]
    "Let R_a denote the set of all repelling periodic points of f_a. By Theorem??, this set is dense in J_a = K_a."

    Lemma 5.6 constructs the dense subset X'(a) = φ_a ∘ g_a(R_a ∩ Ω_a^∘) ∪ ∂D_r that is needed to satisfy hypothesis (ii) of Tan Lei's Theorem 5.1, the engine of Theorem 5.2. Since J_a is defined (Definition 2.1) as the closure of repelling cycles, 'R_a dense in J_a = K_a' is exactly the assertion K_a = J_a. The paper does not prove this assertion; Definition 5.1 says '[19, Theorem D] ensures that K_a = J_a', the Remark after Theorem C calls it 'a crucial ingredient', and Lemma 5.6 refers only to an unresolved 'Theorem??'. The paper's own Remark 5.1 concedes that the needed density cannot be shown without further analysis. If K_a contains an open Fatou-like component, then R_a is not dense in K_a, Lemma 5.6 fails, and Theorem 5.2, together with the unconditional Corollary 5.1, does not follow.

full rationale

The paper does not exhibit an equation-restatement or fitted-input circularity: Theorem B is explicitly conditional on the transversality condition, and the reduction of similarity to transversality is a legitimate mathematical reduction. The algebraic proof of transversality for (p,q)=(4,2) in Section 6 is self-contained and independent. The serious problem is different: the proof of Theorem 5.2 requires a dense set of repelling periodic points in K_a, which is equivalent to the equality K_a = J_a because J_a is defined as the closure of repelling cycles. That equality is not derived in the present paper; it is imported from the author's own unpublished preprint [19, Theorem D] and from an unresolved 'Theorem??' in Lemma 5.6. The manuscript itself flags the gap in Remark 5.1, saying that without further analysis of the structure of K_a the density cannot be shown. If that premise fails, Lemma 5.6 cannot construct the dense set X'(a), Tan Lei's Theorem 5.1 does not apply, and the central similarity theorem does not follow. Because the surrounding argument has substantial independent content and the missing premise may well hold, the appropriate circularity score is 4 rather than higher.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof adds no fitted constants or new entities. It rests on prior results, several from the author's own work: [19] supplies K_a=J_a and repelling-cycle density, [17] supplies the escaping-radius framework, and Tan Lei's theorem supplies the parameter-space self-similarity machinery. The (4,2) transversality proof in Section 6 is self-contained.

assumptions (6)
  • domain assumption The cycle at a Misiurewicz parameter is repelling and K_a = J_a, with repelling cycles dense in K_a ([19, Theorem D]).
    Invoked in Section 1.4, Lemma 3.2, and Lemma 5.6; supplied by the author's unpublished preprint [19], not proven in this paper.
  • standard math Koenigs linearization phi_c of the repelling fixed point z_l(c) exists and depends holomorphically on c.
    Used in Remark 4.1 and Lemma 5.1; cited to Milnor [16, p.78].
  • standard math Tan Lei's Proposition 4.1 applies in the simplified form stated as Theorem 5.1, with hypotheses (i) and (ii) sufficient.
    The paper asserts conditions (3) and (4) of the original are automatic given holomorphicity; not independently verified here.
  • domain assumption The escaping radius R_c and the properties of the basin of infinity hold as in [17, Theorem 2.1] and are stable under perturbation.
    Used in Lemma 3.2 and Lemma 5.2; cited to the author's published work [17].
  • standard math Continuity of the correspondence (c,z) -> f_c(z) in the Hausdorff topology for (c,z) near (a,z_j).
    Used in the perturbation arguments in Lemmas 3.2 and 5.2.
  • standard math The algebraic closure of Q has an extension of the 2-adic valuation; algebraic integers have nonnegative valuation.
    Used throughout Section 6 for the transversality proof; proven as Theorems 6.1-6.3 in the paper.

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Pith. "Pith review of Similarity between the Multibrot set and the Julia set of correspondences at Misiurewicz points." pith.science (2026). https://pith.science/paper/2DMWODRM

@misc{pith2026250907266,
  author       = {Pith},
  title        = {Pith review of: Similarity between the Multibrot set and the Julia set of correspondences at Misiurewicz points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DMWODRM}},
  note         = {Machine review of arXiv:2509.07266}
}
abstract

We study the fine structure of the parameter space of the unicritical family of algebraic correspondences $z^r + c$, where $r > 1$ is a rational exponent. Building on Tan Lei's result regarding the similarity between the Mandelbrot set and Julia sets in the quadratic family, we prove that the Julia set of the correspondence is asymptotically self-similar about every Misiurewicz point. Assuming that the transversality condition holds at a Misiurewicz parameter $a \in \mathbb{C}$, we prove that the associated Multibrot set (which coincides with the Mandelbrot set when $ r =2$) is asymptotically similar to the Julia set about $a$. We provide an algebraic proof of the transversality condition when the correspondence is represented by the semigroup $\langle z^2 +c, -z^2+c \rangle. $ For general exponents, experimental evidence supports the transversality condition, with infinitely many small copies of the Multibrot set accumulating at every Misiurewicz parameter.

Figures

Figures reproduced from arXiv: 2509.07266 by the authors.

Figure 1
Figure 1. The Multibrot set M5,2 and its magnifications at the Mi￾siurewicz point a = −1.027124 + 1.141048i. This phenomenon extends to general exponents p/q > 1. Heuristically, detecting a sufficiently small copy (of size about 10−5 or less) already signals the presence of a Misiurewicz point. For example, −2 is a Misiurewicz point for the family √ z 4+c, and c1 = −1.535 + 0.674i lies very close to another. In [19, Example 2… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

    math.DS 2025-11 unverdicted novelty 1.0 of 10

    Algebraic correspondences mate rational maps with Kleinian groups, and the modular Mandelbrot set is homeomorphic to the Mandelbrot set—this survey reports those results.

Reference graph

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