The construction by Buff and Chéritat of quadratic Julia sets with positive Lebesgue measure rests on a perturbative property (their Proposition 1): there is a nonempty set S of bounded-type irrational rotation numbers such that every member admits arbitrarily close perturbations inside S for which the polynomial acquires a periodic orbit arbitrarily close to the indifferent fixed point while the filled-in Julia set loses an arbitrarily small fraction of its area. The authors asked what the largest such set S is, conjectured it might be all bounded-type irrationals, and flagged the continued-fraction growth hypotheses used in their proof as probably unnecessary. Later work removed the growth condition (Qiao–Qu 2024) and the type restrictions on postcritical-set control (Qu 2024), and Qiao–Qu (2026) gave explicit arithmetic conditions implying positive area together with the far stronger conjecture that every quadratic polynomial with a Cremer point, or with a Siegel disk whose boundary avoids the critical point, has a positive-area Julia set. Determining the largest admissible S — in particular whether it contains all bounded-type irrationals — remains open.

Background

For an irrational number α\alpha, consider the quadratic polynomial Pα(z)=e2πiαz+z2P_\alpha(z)=e^{2\pi i\alpha}z+z^2, whose fixed point 00 has multiplier e2πiαe^{2\pi i\alpha} of modulus one. The filled-in Julia set KαK_\alpha is the set of points with bounded forward orbits, and the Julia set Jα=∂KαJ_\alpha=\partial K_\alpha is the locus of chaotic dynamics. Whether a polynomial Julia set can have positive Lebesgue measure (area) was a long-standing question dating back to Fatou. The answer depends on the arithmetic of α\alpha expressed through its continued-fraction expansion α=[a0;a1,a2,… ]\alpha=[a_0;a_1,a_2,\dots] with convergents pn/qnp_n/q_n. A classical theorem of Yoccoz states that PαP_\alpha is locally linearizable at 00 (i.e. holomorphically conjugate to a rotation near 00) if and only if α\alpha is a Brjuno number, meaning ∑nlog⁡qn+1/qn<+∞\sum_n \log q_{n+1}/q_n<+\infty; when linearizable, the maximal linearization domain is called a Siegel disk, and when not, 00 is called a Cremer point. If the partial quotients ana_n are bounded, α\alpha is said to be of bounded type; then the critical point lies on the Siegel disk boundary (which is a quasicircle), and a theorem of McMullen gives that JαJ_\alpha has Hausdorff dimension strictly less than 2, hence area zero — although KαK_\alpha itself has positive area since it contains the Siegel disk. Conversely, Petersen and Zakeri proved that for Lebesgue-almost every α\alpha the Julia set has area zero. Buff and Chéritat proved (arXiv:math/0605514; Ann. of Math. 2012) that positive-area Julia sets do exist for quadratic polynomials in three cases: with a Cremer fixed point, with a Siegel disk, and with infinitely many satellite renormalizations (infinitely renormalizable maps whose renormalization periods are generated by a common indifferent cycle).

Their engine is Proposition 1: there exists a nonempty set SS of bounded-type irrationals such that for every α∈S\alpha\in S and every ε>0\varepsilon>0 one can find α′∈S\alpha'\in S with ∣α′−α∣<ε|\alpha'-\alpha|<\varepsilon, such that Pα′P_{\alpha'} has a periodic orbit in the disk D(0,ε)D(0,\varepsilon) centered at 0 (a small cycle, excluding 0 itself), and such that area⁡(Kα′)≥(1−ε)area⁡(Kα)\operatorname{area}(K_{\alpha'})\ge(1-\varepsilon)\operatorname{area}(K_\alpha). Iterating such perturbations produces a limiting parameter with small cycles of arbitrarily high order (hence a Cremer point, whose Julia set equals its filled-in Julia set) whose area survives by Douady's upper semicontinuity of α↦area⁡(Kα)\alpha\mapsto\operatorname{area}(K_\alpha). The proof restricts SS to bounded-type numbers with all continued-fraction entries at least some large NN (the set SNS_N) and inserts entries AnA_n subject to a growth condition lim sup⁡nlog⁡Anqn≤1\limsup_n \sqrt[q_n]{\log A_n}\le 1 (Proposition 6). The restrictions enter through three pillars: parabolic explosion, the technique (due to Chéritat) describing how cycles of controlled multiplier emerge when a parameter with a rational rotation number is perturbed; the shape control of perturbed Siegel disks, which keeps a definite fraction of the disk's area in each open subset (Proposition 6); and the control of the postcritical set (the forward orbit of the critical point), available at the time only for rotation numbers of bounded high type. The authors state they do not know the largest SS for which Proposition 1 holds, that it might be the set of all bounded-type irrationals, and that they believe the growth condition on the inserted entries is not necessary. Since the union of any family of sets satisfying the Proposition-1 property again satisfies it, the largest such set is well defined as the union of all admissible classes, so the question is well posed.

Problem Statement

Let Pα(z)=e2πiαz+z2P_\alpha(z)=e^{2\pi i\alpha}z+z^2 for irrational α\alpha, with KαK_\alpha its filled-in Julia set. Say that a set SS of irrationals has the Buff–Chéritat preservation property if for every α∈S\alpha\in S and every ε>0\varepsilon>0 there exists α′∈S\alpha'\in S such that ∣α′−α∣<ε|\alpha'-\alpha|<\varepsilon, Pα′P_{\alpha'} has a periodic orbit in D(0,ε)∖{0}D(0,\varepsilon)\setminus\{0\}, and area⁡(Kα′)≥(1−ε)area⁡(Kα)\operatorname{area}(K_{\alpha'})\ge(1-\varepsilon)\operatorname{area}(K_\alpha). Buff and Chéritat proved this for a nonempty class of bounded-type irrationals (contained in SNS_N, bounded-type numbers whose continued-fraction entries are all at least NN, with growth restrictions on the inserted entries), and asked: what is the largest set SS for which the property holds? Determine it, i.e. characterize the maximal admissible set SS exactly. The two directions below are jointly required for such a determination, and each is independently refutable; resolving either one alone is a partial result that leaves the other open. Concretely: (a) prove or disprove the authors' conjecture that the property holds for SS equal to the set of all bounded-type irrationals, i.e. that all entry lower bounds and all growth conditions on the inserted entries (An)(A_n) can be removed (a disproof must exhibit a bounded-type rotation number at which the property fails, and settles only the bounded-type component, leaving the extent beyond bounded type open); and (b) decide how far the largest admissible SS extends beyond bounded type — whether it contains all Brjuno numbers, or all irrationals — or prove that the property fails beyond bounded type for a specified arithmetically-defined class of rotation numbers, since exhibiting isolated failing rotation numbers does not determine the largest SS.

References

  1. Xavier Buff and Arnaud Chéritat, Quadratic Julia sets with positive area, preprint (2006, revised 2008), arXiv:math/0605514, https://arxiv.org/abs/math/0605514
  2. Xavier Buff and Arnaud Chéritat, Quadratic Julia sets with positive area, Annals of Mathematics (2) 176 (2012), no. 2, 673–746, DOI 10.4007/annals.2012.176.2.1, https://doi.org/10.4007/annals.2012.176.2.1
  3. Curtis T. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Mathematica 180 (1998), no. 2, 247–292, DOI 10.1007/BF02392901, https://doi.org/10.1007/BF02392901
  4. Carsten Lunde Petersen and Saeed Zakeri, On the Julia set of a typical quadratic polynomial with a Siegel disk, Annals of Mathematics (2) 159 (2004), no. 1, 1–52, DOI 10.4007/annals.2004.159.1, https://doi.org/10.4007/annals.2004.159.1
  5. Arnaud Chéritat, The hunt for Julia sets with positive measure, in Complex Dynamics: Families and Friends, A K Peters, Wellesley, MA, 2009, 539–599, DOI 10.1201/b10617-19, https://doi.org/10.1201/b10617-19
  6. Davoud Cheraghi, Typical orbits of quadratic polynomials with a neutral fixed point: non-Brjuno type, Annales scientifiques de l'École normale supérieure (4) 52 (2019), no. 1, 59–138, DOI 10.24033/asens.2384, https://doi.org/10.24033/asens.2384
  7. Artur Avila and Mikhail Lyubich, Lebesgue measure of Feigenbaum Julia sets, Annals of Mathematics (2) 195 (2022), no. 1, 1–88, DOI 10.4007/annals.2022.195.1.1, https://doi.org/10.4007/annals.2022.195.1.1 (preprint arXiv:1504.02986, https://arxiv.org/abs/1504.02986)
  8. Dzmitry Dudko and Mikhail Lyubich, Uniform a priori bounds for neutral renormalization, preprint (2022), arXiv:2210.09280, https://arxiv.org/abs/2210.09280
  9. Jianyong Qiao and Hongyu Qu, Area of Julia sets of non-renormalizable cubic polynomials, preprint (2020), arXiv:2004.08088, https://arxiv.org/abs/2004.08088
  10. Jianyong Qiao and Hongyu Qu, On the area rate of perturbed Siegel disks, Nonlinearity 37 (2024), no. 12, 125022, DOI 10.1088/1361-6544/ad8c0e, https://doi.org/10.1088/1361-6544/ad8c0e (preprint arXiv:2111.10802, https://arxiv.org/abs/2111.10802)
  11. Hongyu Qu, Jianyong Qiao, and Guangyuan Zhang, On quadratic Siegel disks with a class of unbounded type rotation numbers, preprint (2023), arXiv:2307.10851, https://arxiv.org/abs/2307.10851
  12. Hongyu Qu, On postcritical sets of quadratic polynomials with a neutral fixed point, preprint (2024), arXiv:2407.14476, https://arxiv.org/abs/2407.14476
  13. Jianyong Qiao and Hongyu Qu, Arithmetic conditions for Julia sets to have positive area, preprint (2026), arXiv:2608.24316, https://arxiv.org/abs/2608.24316