arXiv:2607.29326v1

Precise asymptotics at the tip of the Mandelbrot set

Neil Dobbs, Jacek Graczyk, Nicolae Mihalache

math.DS37F1037F3537F4037F4437A1037E05

Paper metadata

Original submission: July 31, 2026
Claimed MSC: 37F10, 37F35, 37F40, 37F44, 37A10, 37E05
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 39 pages, 2 figures

Abstract

For the quadratic family $f_c(z)=z^2+c$ , the only parameters in the Mandelbrot set $\cal M$ for which the Julia set $\cal J_c$ has Hausdorff dimension $1$ are $c=0$ and $c=-2$ . Near $c=0$ , Ruelle's theory gives a real-analytic expansion of the dimension. The tip $c=-2$ of $\cal M$ , however, is a non-hyperbolic parameter and the dimension function $c\mapsto \mathrm{dim_H}(\cal J_c)$ is highly discontinuous there. We prove the sharp first-order asymptotic for the lower envelope of the Hausdorff dimension at the tip: If $c\in \cal M$ then $\mathrm{dim_H}(\cal J_c)$ lies asymptotically above $1+ Ω\sqrt{|c+2|}$ with the Jaksztas constant $Ω=\sqrt{\frac{2}{3}}\frac{1}{π\log 2}$ . This is a surprisingly precise contribution to the Yoccoz problem about unfolding attractors. The proof develops a thermodynamic formalism for degenerating families of box mappings. At each scale, for parameters $c\to -2$ , the induced dynamics exhibit a uniform property of exponential tails, generating improved control of their pressure functions.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The two advertised asymptotic statements are supported. Theorem 1 proves the normalized square-root lower envelope of hyperbolic dimension as the quadratic parameter approaches the tip. Theorem 2 proves the corresponding fixed-parameter asymptotic along the specified Misiurewicz parameters. Proposition 1.1 supplies the common pressure-to-dimension comparison.

Theorem 1Correct

Square-root lower-envelope asymptotic at the Mandelbrot tip

Pages 2–4 and 24 · Theorem 1 · arXiv:2607.29326v1

Theorem 1 identifies the normalized lower-envelope behavior of the hyperbolic dimension as the real quadratic parameter approaches the tip c=-2 through the prescribed side. The square-root scaling and numerical constant agree with the return-time normalization in the induced system. The proof supplies both inequalities: finite induced repellers give the lower bound, while uniform pressure and tail estimates bound every admissible hyperbolic set from above. The one-sided liminf in the statement is retained and is not silently replaced by a full limit.

Theorem 2Correct

Misiurewicz-parameter asymptotic

Pages 3–4 and 21–23 · Theorem 2 · arXiv:2607.29326v1

For the sequence of Misiurewicz parameters defined by the stated critical-orbit relation, Theorem 2 gives the full normalized asymptotic rather than only a liminf. The orbit relation fixes the long central return and makes its length the asymptotic parameter. The pressure expansion includes the finite postcritical contribution, and that contribution is lower order after the displayed normalization. Thus the constant agrees with Theorem 1 without interchanging the index limit and a separate parameter limit.

Proposition 1.1Correct

Pressure-root comparison for induced repellers

Pages 4–5 · Proposition 1.1 · arXiv:2607.29326v1

The proposition brackets hyperbolic dimension by zeros of the truncated and full induced pressure functions. Monotonicity in the dimension variable and bounded distortion justify passing from partition sums to Hausdorff dimension. The hypotheses on the parameter neighbourhood and truncation are those available later, so both Theorems 1 and 2 may invoke the proposition with a common normalization.

02Proofs3 reported findingsCorrect

Uniform tail control and passage to the tip. The inducing scheme isolates the long excursion near the endpoint fixed point and represents the dimension by the zero of an induced pressure. Distortion estimates compare each branch derivative with its model value uniformly in the tip parameter, and the summable tail permits truncation before the parameter limit. Proposition 1.1 converts these pressure estimates into dimension bounds. The finite truncations give repellers for the lower inequality; the full tail estimate gives the upper inequality. Sending the truncation level to infinity only after the uniform parameter estimate recovers the displayed square-root constant.

Pressure and induced-repeller analysisCorrect and complete

Uniform tail control and passage to the tip

Pages 6–24 · Sections 2–6 · arXiv:2607.29326v1

The inducing scheme isolates the long excursion near the endpoint fixed point and represents the dimension by the zero of an induced pressure. Distortion estimates compare each branch derivative with its model value uniformly in the tip parameter, and the summable tail permits truncation before the parameter limit. Proposition 1.1 converts these pressure estimates into dimension bounds. The finite truncations give repellers for the lower inequality; the full tail estimate gives the upper inequality. Sending the truncation level to infinity only after the uniform parameter estimate recovers the displayed square-root constant.

Sections 8–10Correct and complete

Uniform branch tails and the Misiurewicz specialization

Pages 17–23 · Sections 8–10 · arXiv:2607.29326v1

The branch domains are paired across nearby parameters and their inverse derivatives are bounded by a summable model sequence. This gives a uniform error when the induced transfer operator is truncated. At a Misiurewicz parameter the critical itinerary closes at the declared time, so the exceptional branch is finite and its contribution can be computed directly. Proposition 10.1 records the resulting pressure expansion with an error uniform in the two variables used in the final limit; substituting it into Proposition 1.1 yields Theorem 2.

Final proof of Theorem 1Correct and complete

Matching upper and lower normalized bounds

Page 24 · proof of Theorem 1 · arXiv:2607.29326v1

For an arbitrary sequence tending to the tip, the full induced pressure estimate yields the normalized upper bound. Conversely, the Misiurewicz parameters from Theorem 2 form an admissible sequence tending to the tip and their finite repellers realize the same constant, giving the lower envelope. These are the two inequalities required for the liminf statement; no assertion about convergence along every parameter sequence is used.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.29326v1
Authors listed: Neil Dobbs, Jacek Graczyk, Nicolae Mihalache
Audit date: August 18, 2026

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Precise asymptotics at the tip of the Mandelbrot set

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Square-root lower-envelope asymptotic at the Mandelbrot tip

Misiurewicz-parameter asymptotic

Pressure-root comparison for induced repellers

Uniform tail control and passage to the tip

Uniform branch tails and the Misiurewicz specialization

Matching upper and lower normalized bounds

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