The largest set of rotation numbers for the Buff–Chéritat perturbative area-preservation property in quadratic dynamics

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 $\alpha$ , consider the quadratic polynomial $P_\alpha(z)=e^{2\pi i\alpha}z+z^2$ , whose fixed point $0$ has multiplier $e^{2\pi i\alpha}$ of modulus one. The filled-in Julia set $K_\alpha$ is the set of points with bounded forward orbits, and the Julia set $J_\alpha=\partial K_\alpha$ 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 $\alpha$ expressed through its continued-fraction expansion $\alpha=[a_0;a_1,a_2,\dots]$ with convergents $p_n/q_n$ . A classical theorem of Yoccoz states that $P_\alpha$ is locally linearizable at $0$ (i.e. holomorphically conjugate to a rotation near $0$ ) if and only if $\alpha$ is a Brjuno number, meaning $\sum_n \log q_{n+1}/q_n<+\infty$ ; when linearizable, the maximal linearization domain is called a Siegel disk, and when not, $0$ is called a Cremer point. If the partial quotients $a_n$ are bounded, $\alpha$ 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 $J_\alpha$ has Hausdorff dimension strictly less than 2, hence area zero — although $K_\alpha$ itself has positive area since it contains the Siegel disk. Conversely, Petersen and Zakeri proved that for Lebesgue-almost every $\alpha$ 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 $S$ of bounded-type irrationals such that for every $\alpha\in S$ and every $\varepsilon>0$ one can find $\alpha'\in S$ with $|\alpha'-\alpha|<\varepsilon$ , such that $P_{\alpha'}$ has a periodic orbit in the disk $D(0,\varepsilon)$ centered at 0 (a small cycle, excluding 0 itself), and such that $\operatorname{area}(K_{\alpha'})\ge(1-\varepsilon)\operatorname{area}(K_\alpha)$ . 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 $\alpha\mapsto\operatorname{area}(K_\alpha)$ . The proof restricts $S$ to bounded-type numbers with all continued-fraction entries at least some large $N$ (the set $S_N$ ) and inserts entries $A_n$ subject to a growth condition $\limsup_n \sqrt[q_n]{\log A_n}\le 1$ (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 $S$ 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 $P_\alpha(z)=e^{2\pi i\alpha}z+z^2$ for irrational $\alpha$ , with $K_\alpha$ its filled-in Julia set. Say that a set $S$ of irrationals has the Buff–Chéritat preservation property if for every $\alpha\in S$ and every $\varepsilon>0$ there exists $\alpha'\in S$ such that $|\alpha'-\alpha|<\varepsilon$ , $P_{\alpha'}$ has a periodic orbit in $D(0,\varepsilon)\setminus\{0\}$ , and $\operatorname{area}(K_{\alpha'})\ge(1-\varepsilon)\operatorname{area}(K_\alpha)$ . Buff and Chéritat proved this for a nonempty class of bounded-type irrationals (contained in $S_N$ , bounded-type numbers whose continued-fraction entries are all at least $N$ , with growth restrictions on the inserted entries), and asked: what is the largest set $S$ for which the property holds? Determine it, i.e. characterize the maximal admissible set $S$ 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 $S$ equal to the set of all bounded-type irrationals, i.e. that all entry lower bounds and all growth conditions on the inserted entries $(A_n)$ 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 $S$ 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 $S$ .

References

Xavier Buff and Arnaud Chéritat, Quadratic Julia sets with positive area, preprint (2006, revised 2008), arXiv:math/0605514, https://arxiv.org/abs/math/0605514
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
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
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
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
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
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)
Dzmitry Dudko and Mikhail Lyubich, Uniform a priori bounds for neutral renormalization, preprint (2022), arXiv:2210.09280, https://arxiv.org/abs/2210.09280
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
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)
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
Hongyu Qu, On postcritical sets of quadratic polynomials with a neutral fixed point, preprint (2024), arXiv:2407.14476, https://arxiv.org/abs/2407.14476
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