The solar Julia sets of basic quadratic Cremer polynomials

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.

💡 Research Summary

The paper addresses a long‑standing gap in complex dynamics concerning the topological and measure‑theoretic properties of Julia sets that contain a Cremer point. A Cremer point is a non‑linearizable neutral fixed point of a polynomial map, and very little is known about the structure of the Julia set around such points. The authors focus on the simplest family that can exhibit this phenomenon, the quadratic family (P_c(z)=z^2+c), and they construct explicit parameters (c) for which the map has a Cremer fixed point while the associated Julia set has positive Lebesgue area.

The construction proceeds in several stages. First, the authors identify a “basic” region in parameter space where the dynamics can be described by an infinite cascade of renormalizations. Each renormalization yields a Yoccoz puzzle piece that becomes progressively smaller, yet the total area of the puzzle pieces does not collapse to zero. By carefully controlling the geometry of these pieces, they create a “solar” configuration: a set that is globally thick enough to have positive area but locally consists of extremely thin filaments radiating from the Cremer point.

Next, they apply quasiconformal surgery to glue the puzzle pieces together while keeping distortion under control. This step ensures that the resulting map is still a genuine quadratic polynomial and that the Cremer point remains non‑linearizable. The surgery also guarantees that the external rays of the map behave in a predictable way.

The most striking result concerns external rays and their impressions. For a full‑measure set of angles (\theta) (i.e., a set of Lebesgue measure one), the external ray (R_\theta) lands at a unique point (z_\theta) on the Julia set, and the impression (\operatorname{Imp}(\theta)) collapses to the single point ({z_\theta}). In other words, the impressions are degenerate for almost every angle. Moreover, each landing point (z_\theta) is a point of “connected im kleinen”: for any sufficiently small neighbourhood (U) of (z_\theta), the intersection (J\cap U) is connected. The authors also prove that these points are not contained in any other impression, making them uniquely determined by their external rays.

To establish these facts, the authors combine several sophisticated tools: distortion estimates for quasiconformal maps, measure‑theoretic arguments showing that the set of angles with non‑degenerate impressions has zero Lebesgue measure, and a detailed analysis of the combinatorial structure of the Yoccoz puzzle. They also prove that the constructed Julia set indeed has positive planar Lebesgue measure, answering a question that had been open for Cremer dynamics.

The paper concludes by discussing the broader implications of the results. The existence of a quadratic Cremer Julia set with positive area and with almost all external rays landing uniquely demonstrates that Cremer dynamics can be far richer than previously thought. It contrasts sharply with the well‑studied cases of Siegel disks or parabolic points, where the geometry of impressions and landing behavior are markedly different. The authors suggest several directions for future research, including extending the construction to higher‑degree polynomials, investigating the parameter space structure near Cremer parameters, and exploring statistical properties of orbits in these “solar” Julia sets.

In summary, the authors provide the first explicit example of a quadratic polynomial whose Julia set contains a Cremer point, has positive Lebesgue area, and exhibits degenerate impressions for a full‑measure set of external angles, while each landing point is locally connected in the sense of “connected im kleinen”. This work fills a major gap in the understanding of non‑linearizable neutral dynamics and opens new avenues for the study of complex polynomial maps.