Porosity and Hausdorff dimension of Julia sets of quadratic rational maps with a Siegel disk and a parabolic fixed point beyond bounded type

For a quadratic rational map carrying both a Siegel disk and a parabolic fixed point of multiplier 1, Zhou and Liao (2014) proved that the Julia set is shallow (uniformly porous) and hence has Hausdorff dimension strictly less than 2, under the hypothesis that the Siegel rotation number is an irrational of bounded type. The problem asks whether this conclusion survives without the bounded-type hypothesis: for the same class of maps, with a Brjuno rotation number of unbounded type, decide whether the Julia set is still shallow, or at least of Hausdorff dimension below 2, or whether such a Julia set can have dimension 2. The nearest results — positive-area quadratic polynomial Julia sets for suitable unbounded-type rotation numbers (Buff and Chéritat) and almost-every-rotation-number area-zero results (Petersen and Zakeri) — concern polynomials without parabolic points and do not settle the rational case.

Background

A rational map is a map of the form $f(z)=p(z)/q(z)$ with coprime complex polynomials; its degree is the maximum of the degrees of $p$ and $q$. Iterating a rational map of degree at least two partitions the Riemann sphere into the Fatou set, the open set where the family of iterates is normal (equitable, stable behavior), and its complement, the Julia set $J(f)$, the locus of chaotic behavior; for the maps considered here $J(f)$ is a fractal subset of the sphere. An indifferent fixed point $z_0$ has multiplier $f'(z_0)$ of modulus 1; when the multiplier is $e^{2\pi i\theta}$ with $\theta$ irrational, the point is linearizable if some neighborhood of $z_0$ is conformally conjugate to the irrational rotation $w\mapsto e^{2\pi i\theta}w$. The maximal such neighborhood is a Siegel disk, a Fatou component on which the map acts as an irrational rotation, and $\theta$ is its rotation number. The Siegel–Brjuno theorem guarantees linearizability whenever $\theta$ belongs to the Brjuno class, a number-theoretic condition defined by summability of the logarithms of the denominators of the convergents of $\theta$, and Yoccoz proved the converse for nondegenerate quadratic germs: for such germs a non-Brjuno rotation number forces a Cremer point, a non-linearizable indifferent fixed point, instead of a Siegel disk. Writing the continued fraction expansion $\theta=[a_1,a_2,\dots]$, the rotation number is of bounded type when the partial quotients $a_n$ are uniformly bounded and of unbounded type otherwise; bounded-type numbers are Brjuno, while almost every irrational is Brjuno of unbounded type. A parabolic fixed point is one whose multiplier is a root of unity; here the multiplier is 1, the point lies on the Julia set, and its attracting petals form part of the Fatou set. A planar set is called porous (shallow in the terminology used by Zhou and Liao) if there is a constant $c>0$ such that every sufficiently small disk centered at a point of the set contains a subdisk of radius scaled by $c$ that avoids the set; a classical result in fractal geometry says that uniformly porous planar sets have Hausdorff dimension strictly less than 2, so porosity is a standard route to dimension upper bounds for Julia sets.

The dimension of Siegel and parabolic Julia sets has a clear lineage, all of it under bounded-type hypotheses. McMullen (1998) proved that for the quadratic polynomial $e^{2\pi i\theta}z+z^2$ with $\theta$ of bounded type, the Julia set is porous and $\dim_H J<2$, via self-similar renormalization of the Siegel disk. Geyer (1999) proved porosity of parabolic Julia sets for geometrically finite rational maps (maps whose postcritical set meets the Julia set in finitely many points). Shen (2006) extended McMullen's conclusion to the polynomials $e^{2\pi i\theta}z(1+z/d)^d$ of arbitrary degree $d\geq 2$, and Shen and Wu (2010) to quadratic rational maps with two Siegel disks both of bounded type. A key structural tool is the theorem of Zhang (2011) that every bounded-type Siegel disk of every rational map is a quasidisk, that is, the image of a round disk under a quasiconformal homeomorphism of the plane (a homeomorphism with bounded infinitesimal distortion). Zhou and Liao (2014) combined the two mechanisms, proving that a quadratic rational map with a Siegel disk of bounded-type rotation number and a parabolic fixed point of multiplier 1 has a shallow, hence dimension-smaller-than-two, Julia set. Their result is genuinely non-hyperbolic: in this quadratic rational setting a critical point must accumulate on the Siegel disk boundary, so the map is not geometrically finite and Geyer's parabolic porosity alone does not suffice; the bounded-type hypothesis enters through the quasidisk property and self-similarity of the Siegel boundary.

The unbounded-type side is where the neighboring theory becomes delicate, and so far only for polynomials. Petersen and Zakeri (2004) proved that for a full-measure set of rotation numbers (satisfying $\log a_n=O(\sqrt{n})$, which includes many unbounded-type numbers) the quadratic Siegel polynomial has a locally connected Julia set of zero area, though without a dimension-below-2 conclusion. In the opposite direction, Buff and Chéritat (2012) constructed quadratic polynomials with Julia sets of positive area — hence Hausdorff dimension exactly 2 — in the presence of a Cremer point, of a Siegel disk with a suitable unbounded-type rotation number (for instance of high type, meaning the partial quotients are eventually large), or of infinitely many satellite renormalizations (nested smaller copies of the quadratic dynamics around a periodic point); Qiao and Qu (2024) subsequently removed an auxiliary condition from the Siegel-disk case, enlarging the known positive-area class. These constructions are polynomial-specific: no positive-area example is known for a rational map with a parabolic point and a Siegel disk, and no technique is known that proves $\dim_H J<2$ for such maps once the rotation number is of unbounded type. The problem below sits exactly in this gap, inside the broader and largely open problem of the area and dimension of Julia sets of rational maps with recurrent critical points, where even the Feigenbaum case — infinitely renormalizable quadratic-like maps with universal scaling constants — was settled (zero area) only recently by Avila and Lyubich (2022), and only for measure, not dimension.

Problem Statement

The question below concerns the exact family treated by Zhou and Liao: the geometric complexity of Julia sets of quadratic rational maps in the combined Siegel–parabolic regime once the bounded-type hypothesis is removed.

Let $f$ be a quadratic rational map that has a Siegel disk $\Delta$ of rotation number $\theta$ and a parabolic fixed point of multiplier 1. Zhou and Liao proved that if $\theta$ is an irrational of bounded type, then $J(f)$ is shallow and in particular $\dim_H J(f)<2$. Is the same true when $\theta$ is a Brjuno irrational of unbounded type — that is, is $J(f)$ shallow, or at least of Hausdorff dimension strictly less than 2 — or does there exist such a map with $\dim_H J(f)=2$? (The admissible class is nonempty for every Brjuno $\theta$: the quadratic rational map $f(z)=z+\frac{z^{2}}{1+\lambda z}$ with $\lambda=\frac{e^{2\pi i\theta}}{1-e^{2\pi i\theta}}$ has a parabolic fixed point of multiplier $1$ at $0$ and a fixed point of multiplier $e^{2\pi i\theta}$ at $\infty$, which by the Siegel–Brjuno theorem is the center of a Siegel disk of rotation number $\theta$. The question is therefore posed precisely over the Brjuno class minus the bounded-type numbers.)

References

1. Jia Zhou, Liang-Wen Liao: Hausdorff dimension of quadratic rational Julia sets. Acta Mathematica Sinica, English Series 30(2), 331–342 (2014). DOI: 10.1007/s10114-013-1640-3. https://doi.org/10.1007/s10114-013-1640-3
2. Curtis T. McMullen: Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Mathematica 180, 247–292 (1998). DOI: 10.1007/BF02392901. https://doi.org/10.1007/BF02392901
3. Lukas Geyer: Porosity of parabolic Julia sets. Complex Variables, Theory and Application 39, 191–198 (1999). DOI: 10.1080/17476939908815191. https://doi.org/10.1080/17476939908815191
4. Liang Shen: Hausdorff dimension of Julia set of a polynomial with a Siegel disk. Science in China Series A: Mathematics 49, 1284–1296 (2006). DOI: 10.1007/s11425-006-1284-1. https://doi.org/10.1007/s11425-006-1284-1
5. Liang Shen, Sheng Jian Wu: A note on a quadratic rational map with two Siegel disks. Acta Mathematica Sinica, English Series 26, 1393–1402 (2010). DOI: 10.1007/s10114-010-6611-3. https://doi.org/10.1007/s10114-010-6611-3
6. Gaofei Zhang: All bounded type Siegel disks of rational maps are quasi-disks. Inventiones mathematicae 185, 421–466 (2011). DOI: 10.1007/s00222-011-0312-0. https://doi.org/10.1007/s00222-011-0312-0 (arXiv:0810.2733)
7. Carsten L. Petersen, Saeed Zakeri: On the Julia set of a typical quadratic polynomial with a Siegel disk. Annals of Mathematics (2) 159, 1–52 (2004). DOI: 10.4007/annals.2004.159.1. https://doi.org/10.4007/annals.2004.159.1
8. Xavier Buff, Arnaud Chéritat: Quadratic Julia sets with positive area. Annals of Mathematics (2) 176, 673–746 (2012). DOI: 10.4007/annals.2012.176.2.1. https://doi.org/10.4007/annals.2012.176.2.1 (arXiv:math/0605514)
9. Jianyong Qiao, Hongyu Qu: On the area rate of perturbed Siegel disks. Nonlinearity 37, article 125022 (2024). DOI: 10.1088/1361-6544/ad8c0e. https://doi.org/10.1088/1361-6544/ad8c0e (arXiv:2111.10802)
10. Mitsuhiro Shishikura, Fei Yang: The high type quadratic Siegel disks are Jordan domains. Journal of the European Mathematical Society 27(11), 4501–4562 (2025; published online 2024). DOI: 10.4171/JEMS/1481. https://doi.org/10.4171/JEMS/1481 (arXiv:1608.04106)
11. Yuming Fu, Fei Yang, Gaofei Zhang: Quadratic rational maps with a 2-cycle of Siegel disks. The Journal of Geometric Analysis 32(10), article 244 (2022). DOI: 10.1007/s12220-022-00989-x. https://doi.org/10.1007/s12220-022-00989-x (arXiv:2012.10818)
12. Artur Avila, Mikhail Lyubich: Lebesgue measure of Feigenbaum Julia sets. Annals of Mathematics (2) 195(1), 1–88 (2022). DOI: 10.4007/annals.2022.195.1.1. https://doi.org/10.4007/annals.2022.195.1.1 (arXiv:1504.02986)