Rigorous classification of real two-petal parabolic parameters of $z^2+c$ by the Hausdorff-dimension threshold $4/3$

For the real quadratic family $f_c(z)=z^2+c$, let $d(c)$ denote the Hausdorff dimension of the Julia set $J_c$. Jaksztas and Zinsmeister (2020) proved a trichotomy for the behavior of $d'(c)$ at any real parabolic parameter $c_0$ whose parabolic cycle has multiplier $-1$ (two petals): if $d(c_0)<4/3$ then $d'(c)\to-\infty$ at a power-law rate, if $d(c_0)=4/3$ the divergence is logarithmic, and if $d(c_0)>4/3$ the derivative has a finite limit. Which regime applies at a given parameter, however, has never been determined rigorously: at $c_0=-3/4$ every sharp result since 2011 rests on the numerically verified but unproven assumption $d(-3/4)<4/3$, and below the Feigenbaum parameter nothing is known. The problem is to assign, with rigorous proof, the regime (sign of $d(c_0)-4/3$) for every real two-petal parabolic parameter — in particular to decide $d(-3/4)$ versus $4/3$ — thereby making the known derivative asymptotics unconditional. Dudko, Gorbovickis and Tucker (2023) resolved the supercritical side on the interval $(c_{\mathrm{Feig}},-3/4)$ by rigorous computer-assisted lower bounds, sharpening the open core to $c_0=-3/4$ itself and to all two-petal parameters below the Feigenbaum point.

Background

For a polynomial $f$, the filled-in Julia set is the set of points with bounded forward orbits, and the Julia set $J(f)$ is its boundary; for $f_c(z)=z^2+c$ the connectedness locus in the $c$-plane is the Mandelbrot set $\mathcal M$. The Hausdorff dimension $d(c)=\dim_H(J_c)$ is a classical observable of this family: by Ruelle's theory it is real-analytic on hyperbolic components (regions where the map is uniformly expanding on its Julia set) and on the exterior of $\mathcal M$, while Shishikura proved that the boundary $\partial\mathcal M$ itself has Hausdorff dimension $2$ and Zdunik proved that the Julia set of a parabolic parameter always has Hausdorff dimension strictly greater than $1$. A parameter $c_0$ is parabolic when $f_{c_0}$ has a periodic point whose multiplier (the derivative along the cycle) is a root of unity; the number of attracting petals of the local Leau–Fatou flower (the flower-shaped local configuration at a parabolic point, whose attracting petals are the sectors along which iterates converge to the parabolic cycle) equals the order of that root of unity. On the real axis the multiplier is real, so parabolic parameters have multiplier $+1$ (one petal) or $-1$ (two petals); the two-petal parameters are exactly the period-doubling bifurcation parameters ($c_0=-3/4$, $-5/4$, $\approx-1.3680989$, …, accumulating at the Feigenbaum parameter $c_{\mathrm{Feig}}\approx-1.401155$, the limit of this period-doubling cascade, and occurring in every hyperbolic window of $[-2,c_{\mathrm{Feig}})$). The threshold $4/3$ is structural: a classical dichotomy in parabolic thermodynamic formalism (recalled in Jaksztas–Zinsmeister 2020) states that for a polynomial with a single parabolic cycle of $p$ petals, a finite invariant measure equivalent to the $s$-conformal measure (the canonical reference measure on the Julia set, which rescales locally by the Jacobian factor $|f'|^s$) exists exactly when the Julia-set dimension exceeds $2p/(p+1)$, which for $p=2$ is $4/3$. Studying $d(c)$ near parabolic parameters requires thermodynamic formalism for families that degenerate as the parabolic parameter is approached, since the spectral gap of the transfer operator (the linear operator acting on functions over the Julia set whose spectral data locate the dimension as the zero of a pressure function) collapses near the parabolic orbit. Jaksztas (2011, 2014) obtained one-sided derivative asymptotics at $c_0=-3/4$ under the numerically verified assumption $d(-3/4)<4/3$, and Jaksztas–Zinsmeister (2020) generalized this to a trichotomy (their Theorem 1.1) valid at every real two-petal parameter: if $d(c_0)<4/3$ then $d'(c)\cdot|c_0-c|^{-(3/2)d(c_0)+2}$ tends to finite negative constants from each side; if $d(c_0)=4/3$ then $d'(c)/(-\log|c_0-c|)$ tends to a negative constant; if $d(c_0)>4/3$ then $d'(c)$ has a finite limit, so $d$ is differentiable at $c_0$. Thus the differentiability structure of the dimension function at each two-petal parameter is completely characterized once the sign of $d(c_0)-4/3$ is known — and that sign is exactly what is missing. Dudko–Gorbovickis–Tucker (2023) developed a rigorous computer-assisted method for lower bounds on the hyperbolic dimension (the supremum of dimensions of invariant uniformly hyperbolic subsets, always at most the Hausdorff dimension) and proved that every real parabolic parameter in $(c_{\mathrm{Feig}},-3/4)$ has dimension $>4/3$; combined with the trichotomy this verified the Jaksztas–Zinsmeister conjecture that $c\mapsto d(c)$ is continuously differentiable on that interval. But the method gives lower bounds only: no rigorous upper bound is known that could certify $d(-3/4)<4/3$, and no information exists for two-petal parameters below $c_{\mathrm{Feig}}$, where dimensions approach $1$ near the tip $c=-2$.

Problem Statement

For the real quadratic family $f_c(z)=z^2+c$ with Julia set $J_c$ and dimension $d(c)=\dim_H(J_c)$, determine with rigorous proof, for every real parameter $c_0$ at which $f_{c_0}$ has a parabolic periodic cycle with multiplier $-1$ (a two-petal parabolic parameter, i.e. a period-doubling bifurcation parameter of the real axis), whether $d(c_0)<4/3$, $d(c_0)=4/3$, or $d(c_0)>4/3$; in particular, decide rigorously whether $d(-3/4)<4/3$, replacing the numerically verified assumption on which all sharp derivative asymptotics at $-3/4$ (Jaksztas 2011, 2014; Jaksztas–Zinsmeister 2020; Jaksztas 2025) currently rest. This determination also settles whether the borderline value $4/3$ is attained at any two-petal parameter. Via the known trichotomy (Jaksztas–Zinsmeister 2020, Theorem 1.1), this is equivalent to determining, unconditionally, the differentiability structure of the dimension function $c\mapsto d(c)$ at every real two-petal parabolic parameter: a power-law $d'(c)\to-\infty$ when $d(c_0)<4/3$, a logarithmic divergence when $d(c_0)=4/3$, and a finite derivative when $d(c_0)>4/3$. The already-certified supercritical regime on $(c_{\mathrm{Feig}},-3/4)$ (Dudko–Gorbovickis–Tucker 2023) may be assumed; the open instances are $c_0=-3/4$ and all two-petal parameters in $[-2,c_{\mathrm{Feig}})$.

References

1. Ludwik Jaksztas and Michel Zinsmeister, "On the derivative of the Hausdorff dimension of the Julia sets for z^2 + c, c ∈ R at parabolic parameters with two petals", Advances in Mathematics 363 (2020), article 106981. DOI: 10.1016/j.aim.2020.106981; arXiv:1712.03102, https://arxiv.org/abs/1712.03102
2. Ludwik Jaksztas, "On the derivative of the Hausdorff dimension of the quadratic Julia sets", Transactions of the American Mathematical Society 363 (2011), 5251–5291. DOI: 10.1090/s0002-9947-2011-05208-6, https://doi.org/10.1090/s0002-9947-2011-05208-6
3. Ludwik Jaksztas, "On the left side derivative of the Hausdorff dimension of the Julia sets for z^2 + c at c = −3/4", Israel Journal of Mathematics 199 (2014), 1–43. DOI: 10.1007/s11856-014-0001-y, https://doi.org/10.1007/s11856-014-0001-y
4. Ludwik Jaksztas, "On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4", Nonlinearity 33 (2020), 5919–5960. DOI: 10.1088/1361-6544/ab9a1a; arXiv:1905.09126, https://arxiv.org/abs/1905.09126
5. Artem Dudko, Igors Gorbovickis, and Warwick Tucker, "Lower bounds on the Hausdorff dimension of some Julia sets", Nonlinearity 36 (2023), 2867–2893. DOI: 10.1088/1361-6544/acc71b; arXiv:2204.07880, https://arxiv.org/abs/2204.07880
6. Ludwik Jaksztas, "On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at −3/4", Mathematische Annalen 391 (2025), 5427–5483 (online December 2024). DOI: 10.1007/s00208-024-03060-y, https://doi.org/10.1007/s00208-024-03060-y
7. Ludwik Jaksztas, "On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at −2", Advances in Mathematics 433 (2023), article 109297. DOI: 10.1016/j.aim.2023.109297; arXiv:2301.04519, https://arxiv.org/abs/2301.04519
8. Neil Dobbs, Jacek Graczyk, and Nicolae Mihalache, "Hausdorff Dimension of Julia Sets in the Logistic Family", Communications in Mathematical Physics 399 (2022), 673–716. DOI: 10.1007/s00220-022-04568-7; arXiv:2007.07661, https://arxiv.org/abs/2007.07661
9. Neil Dobbs, Jacek Graczyk, and Nicolae Mihalache, "Precise asymptotics at the tip of the Mandelbrot set", preprint (2026). arXiv:2607.29326, https://arxiv.org/abs/2607.29326