Hausdorff dimension of quadratic Julia sets and Siegel-disk scaling in the unbounded-type regime

For the quadratic polynomial $P_\alpha(z)=e^{2\pi i\alpha}z+z^2$ with irrational rotation number $\alpha$, McMullen (Acta Math. 180, 1998) proved that if $\alpha$ is of bounded type then the Julia set $J(P_\alpha)$ has Hausdorff dimension strictly less than two, the boundary dynamics is universal and uniformly twisting, and for quadratic irrational $\alpha$ (eventually periodic continued fraction) the Siegel disk boundary is exactly self-similar about the critical point. He closed by noting that under fast growth of the continued-fraction digits there is no Siegel disk and the dimension is two, and that 'the regime of unbounded type still presents many mysteries'. Later work resolved the two ends of the unbounded regime: for almost every $\alpha$ the Julia set is locally connected of zero area and the Siegel disk is a Jordan domain (Petersen-Zakeri 2004); quadratic Julia sets with a Siegel disk and positive area exist, necessarily at unbounded type (Buff-Cheritat 2012); and every non-Herman $\alpha$ of sufficiently high type gives Hausdorff dimension two (Cheraghi-DeZotti-Yang, 2026 preprint). What remains open is the middle: for unbounded-type Brjuno rotation numbers that are Herman numbers - a class containing the unbounded-type members of the full-measure Petersen-Zakeri class, which still form a set of full Lebesgue measure - no value of the Hausdorff dimension of $J(P_\alpha)$ is known in either direction, the rotation numbers giving positive area are uncharacterized, and no scaling self-similarity of the Siegel disk boundary is known or excluded outside the eventually periodic case. The problem asks for the determination of $\dim_H J(P_\alpha)$ in this regime, for the positive-area characterization, and for the scaling behaviour of the Siegel disk boundary.

Background

Polynomial dynamics studies the iteration of a map; for a polynomial $f$, the filled Julia set $K(f)$ is the set of points whose forward orbits remain bounded, and the Julia set $J(f)=\partial K(f)$ is the compact fractal locus on which the dynamics is chaotic. This problem concerns the quadratic family

$$P_\alpha(z)=e^{2\pi i\alpha}z+z^2,\qquad \alpha\in\mathbb{R}\setminus\mathbb{Q},$$

which has a fixed point at the origin of multiplier of modulus one, called an irrationally indifferent fixed point. Write the continued fraction expansion of the rotation number as $\alpha=[a_1,a_2,a_3,\dots]$ with rational convergents $p_n/q_n=[a_1,\dots,a_n]$. The Siegel disk $\Delta_\alpha$ is the maximal domain containing the origin on which $P_\alpha$ is conformally conjugate to the rigid rotation $z\mapsto e^{2\pi i\alpha}z$. By a theorem of Siegel for Diophantine rotation numbers (numbers that rational approximation can reach only at a definite polynomial rate), extended via the Brjuno condition - an explicit summability condition on the denominators $q_n$ - and proved optimal for quadratics by Yoccoz, the Siegel disk exists exactly when $\alpha$ is a Brjuno number, a condition satisfied by Lebesgue-almost every irrational; otherwise the origin is a Cremer point and no linearization exists. The type of $\alpha$ measures the growth of its continued-fraction digits: $\alpha$ is of bounded type if $\sup_n a_n<\infty$ and of unbounded type otherwise; the high-type sub-regime consists of numbers all of whose digits are uniformly large, and fast growth means $a_n\to\infty$, which typically violates Brjuno so that no Siegel disk exists at all.

The geometric quantities of interest are the Hausdorff dimension $\dim_H$, which for a plane set takes values up to $2$ and equals $2$ whenever the set has positive Lebesgue area; the Lebesgue area of $J(P_\alpha)$ itself; and the scaling geometry of the Siegel disk boundary $\partial\Delta_\alpha$. The postcritical set $\Lambda(P_\alpha)$ - the closure of the forward orbit of the unique critical point of $P_\alpha$ - is a dynamically distinguished subset of $K(P_\alpha)$ that contains the boundary of the Siegel disk and much of the Julia set. A quasicircle is a Jordan curve that is the image of a round circle under a quasiconformal homeomorphism (a homeomorphism with bounded infinitesimal distortion); quasicircles have uniformly controlled geometry, while at unbounded type the Siegel disk boundary is known to lose this regularity and to admit only trans-quasiconformal control, i.e. control by David homeomorphisms, whose distortion may exceed every uniform bound on sets of small but positive measure.

For bounded-type rotation numbers the picture is complete: $\partial\Delta_\alpha$ is a quasicircle through the critical point (by quasiconformal surgery on circle maps - surgery here means cutting a holomorphic dynamical system apart and reassembling the pieces into a new one; published analytic account: Światek 1998), $J(P_\alpha)$ is locally connected (every point admits arbitrarily small connected neighborhoods) and has zero area (Petersen 1996), and McMullen (1998) proved the fine-geometric results this problem builds on: $\dim_H J(P_\alpha)<2$; universality of the boundary dynamics - after conformal change of scale the dynamics near the boundary depends only on the combinatorial type, not on $\alpha$; uniform twisting - at every scale near every point of the Julia set, some iterate of the map contains a degree-two branched covering of controlled geometry; and, when $\alpha$ is a quadratic irrational, exact self-similarity of $\partial\Delta_\alpha$ about the critical point: there is a conformal pinching symmetry with constant derivative contraction that repeats all boundary features at geometric scales, and the rescaled first-return maps converge exponentially to a fixed point of renormalization (renormalization here means the study of first-return maps under rescaling, whose convergence expresses self-similarity). In the opposite direction, Shishikura proved that the boundary of the Mandelbrot set - the connectedness locus of the family $z\mapsto z^2+c$ - has Hausdorff dimension two, that $\dim_H J(P_\alpha)=2$ for a residual (topologically generic) set of rotation numbers with no known arithmetic characterization, and, as McMullen records citing him, that when the digits $a_n$ grow sufficiently fast there is no Siegel disk, $J(P_\alpha)$ is not locally connected, and $\dim_H J(P_\alpha)=2$. McMullen closed his account: 'The regime of unbounded type still presents many mysteries.'

Subsequent work has mapped the two ends of the unbounded regime while leaving its middle open. Petersen and Zakeri proved that for every $\alpha$ with $\log a_n=O(\sqrt n)$ - a class of full Lebesgue measure containing many unbounded-type numbers, hence almost every irrational - $J(P_\alpha)$ is locally connected and has zero area, and $\Delta_\alpha$ is a Jordan domain whose boundary contains the critical point. Buff and Cheritat constructed quadratic polynomials with a Siegel disk whose Julia set has positive area, therefore dimension two; since bounded type forces dimension strictly below two, such examples are necessarily of unbounded type. Shishikura and Yang proved that for rotation numbers of sufficiently high type the Siegel disk is a Jordan domain whose boundary contains the critical point exactly when $\alpha$ is a Herman number - a rotation number for which every orientation-preserving analytic circle diffeomorphism of that rotation number is analytically conjugate to a rigid rotation; the Herman class contains all Diophantine numbers and hence all bounded-type and all Petersen-Zakeri-type numbers, and is properly contained in the Brjuno class. Cheraghi proved a topological trichotomy for the postcritical set at high type: a Jordan curve in the Herman case, a one-sided hairy Jordan curve (a Jordan curve with attached hairs) in the Brjuno non-Herman case, or a Cantor bouquet - a compact union of uncountably many arcs, each accumulated from both sides by further arcs - in the non-Brjuno case. Most recently, Cheraghi, DeZotti and Yang proved that for every non-Herman $\alpha$ of sufficiently high type the postcritical set and hence the Julia set have Hausdorff dimension two, and state explicitly that it remains open for which $\alpha$ the Julia set has positive area and that the unbounded regime remains mysterious for general polynomials.

What is still missing is the middle: for unbounded-type Brjuno rotation numbers that are Herman - a class that includes the unbounded-type members of the full-measure Petersen-Zakeri class, which still form a set of full Lebesgue measure - no value of $\dim_H J(P_\alpha)$ is known in either direction, no example with dimension below two is known, dimension two is not proved for any arithmetic subclass of them, the positive-area locus is uncharacterized, and nothing is known about scaling self-similarity of the Siegel disk boundary once the continued fraction is not eventually periodic.

Problem Statement

Let $P_\alpha(z)=e^{2\pi i\alpha}z+z^2$ with $\alpha$ an irrational number of unbounded type ($\sup_n a_n=\infty$) satisfying the Brjuno condition, so that the Siegel disk $\Delta_\alpha$ exists. This is precisely the regime left open between McMullen's bounded-type theorems and the fast-growth regime where no Siegel disk exists. Determine the fine geometry of $J(P_\alpha)$ and $\partial\Delta_\alpha$ in this regime:

(1) Dimension of the Julia set. Determine the set of unbounded-type Brjuno rotation numbers $\alpha$ for which $\dim_H J(P_\alpha)=2$, equivalently those for which $\dim_H J(P_\alpha)<2$; that is, whether some unbounded-type Brjuno $\alpha$ has $\dim_H J(P_\alpha)<2$, or whether $\dim_H J(P_\alpha)=2$ for every $\alpha$ in an arithmetic class that contains Lebesgue-almost every irrational. The Herman-type unbounded numbers, which include the full-measure set of unbounded-type Petersen-Zakeri numbers $\{\alpha:\log a_n=O(\sqrt n),\ \sup_n a_n=\infty\}$, are the central open case; for bounded type $\dim_H J(P_\alpha)<2$ is known, and for the non-Herman high-type regime $\dim_H J(P_\alpha)=2$ is known. In addition, as a second required part of this sub-question, characterize the rotation numbers $\alpha$ for which $J(P_\alpha)$ has positive Lebesgue area (known to occur for some $\alpha$ of sufficiently high type, and to fail for bounded type and for Lebesgue-almost every $\alpha$).

(2) Scaling of the Siegel disk boundary. Outside the eventually periodic (quadratic irrational) case, the scaling self-similarity of $\partial\Delta_\alpha$ refers to convergence of the rescaled first-return maps about the critical point to a nontrivial fixed point of renormalization. For unbounded-type Brjuno $\alpha$, the question is whether such a renormalization limit exists - whether at least one unbounded-type Brjuno $\alpha$ has a Siegel disk boundary admitting such a limit, or whether no such limit can occur for any unbounded-type Brjuno $\alpha$.

Sub-questions (1) and (2) are independent and both are posed, though each may be resolved on its own; the positive-area characterization is part of sub-question (1). The common objective is the fate, outside the bounded-type window, of the geometric rigidity McMullen established inside it - dimension strictly below two and self-similar renormalization limits.

References

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
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