Fat Basilica Julia Set and QC Universality

• The paper establishes that the fat Basilica Julia set is a universal QC model for both polynomial Julia sets and Bers-boundary Kleinian limit sets.
• Fat Basilica Julia set is a connected fractal defined by Q(z)=z²-3/4, characterized by non-empty interior and tangentially contacting quasi-disks.
• The quasiconformal universality theorem and QC surgery techniques offer new methods for uniformizing complex dynamical systems and classifying fractal geometries.

The fat Basilica Julia set, denoted $J(Q)$ for $Q(z) = z^2 - \frac{3}{4}$, is a fundamental object in conformal dynamics, serving as a universal model for certain classes of polynomial Julia sets and Kleinian group limit sets. Unlike the classical (thin) basilica $J(z^2-1)$, $J(Q)$ has a connected structure with non-empty interior (hence “fat”) arising from a degree-2 polynomial with a parabolic fixed point at $-\frac{1}{2}$. The fat Basilica is both combinatorially and topologically distinguished by the geometry of its bounded Fatou components and their pairwise tangential contacts, a configuration that becomes the basis of a new universality theorem in the quasiconformal (QC) classification of dynamical systems.

1. Defining the Fat Basilica and Comparison with the Classical Basilica

The fat Basilica Julia set $J(Q)$, for $Q(z) = z^2 - \frac{3}{4}$, is defined as the boundary of the set of points whose forward iterates under $Q$ diverge to infinity: $$J(Q) = \partial \bigl\{z : Q^n(z)\to\infty\bigr\} .$$ This set is connected and possesses non-empty interior in $\mathbb{C}$. Every bounded Fatou component (domain of normality) of $Q$ is a Jordan quasi-disk, and any two bounded components, if they are in contact, intersect tangentially—meaning that the two boundary arcs share a tangent and meet on one side only.

A compact set $K \subset \mathbb{C}$ is defined as a fat Basilica if: (a) $K$ is homeomorphic to $J(Q)$; (b) every bounded component of $\mathbb{C} \setminus K$ is a quasi-disk; (c) any touching bounded components do so tangentially.

In contrast, the classical basilica $J_{\mathrm{pc}} = J(z^2-1)$ (the "rabbit" at parameter $-1$) is a dendrite-like Julia set with precisely two bounded Fatou components forming a period-2 superattracting cycle. Its complement’s bounded components are not “fat,” as the Julia set itself has empty interior and fails the quasi-disk property.

2. Quasiconformal Universality Theorem

The principal universality result asserts a rigidity in the QC classification of fat Basilica Julia sets. Let $J_{3/4} := J(z^2-\frac{3}{4})$, let $\mathrm{Poly}_{\mathrm{fatB}}$ denote the set of polynomials with fat Basilica Julia sets, and let $\mathrm{Ber}_\mathrm{B}$ be the set of geometrically finite Bers-boundary Kleinian groups whose limit set is a (necessarily fat) basilica. The universality theorem states: $$\forall\,P\in\mathrm{Poly}_{\mathrm{fatB}},\quad J(P)\;\;\text{is QC‐equivalent to}\;\;J_{3/4},$$

$$\forall\,G\in\mathrm{Ber}_\mathrm{B},\quad \Lambda(G)\;\;\text{is QC‐equivalent to}\;\;J_{3/4}.$$

Thus, the QC class $[J_{3/4}]$ encompasses all fat Basilica Julia sets of complex polynomials and all limit sets of geometrically finite Bers-boundary surface groups, providing the first explicit example of a connected rational Julia set (not homeomorphic to a circle or sphere) that is QC-equivalent to a Kleinian limit set (Luo et al., 20 Jan 2026).

3. Proof Strategy and Key Constructions

The proof proceeds in three conceptual stages:

I. Fragmented Dynamics and Markov Maps:

To every relevant dynamical system—either a polynomial or a Bers-boundary group—one associates a piecewise-conformal Markov map acting on its limit or Julia set. For polynomials, these are constructed from inverse branches; for Kleinian groups, from group elements. This combinatorial model captures the adjacency and dynamical correspondence of Fatou components and their pointwise contacts.

II. Modeling via the Fat Basilica Polynomial:

One constructs a parallel Markov map directly on $J(Q)$. The Markov partition is designed such that, near tangency points between Fatou components, its expansion mimics that of the target system. The approach uses internal/external rays and pinched neighborhoods to define “puzzle pieces” and combinatorial conjugacies.

III. QC Surgery and Extension:

Combinatorial conjugacies between Markov maps on the Julia/limit sets are shown to be quasisymmetric (or David-symmetric) on the boundary circles of the Fatou pieces. Using Beurling–Ahlfors or David extension theorems, these local conjugacies are extended to global quasiconformal or David homeomorphisms of $\mathbb{C}$, then dynamically pulled back to fill the Fatou components.

Key intermediate structures include:

• Basilica Bowen–Series maps: Piecewise-Möbius Markov maps constructed for Bers-boundary groups.
• Expansion and distortion estimates via Markov maps on unions of circles (Propositions 3.4 and 3.5).
• “Puzzle blow-up” lemmas controlling quasi-symmetric distortion (Lemmas 4.5, 4.7).

4. Consequences and Novel Classifications

Several rigidity and uniformization phenomena arise from universality:

• Conformal Removability: As $J_{3/4}$ is conformally removable, any QC image—including any Bers-boundary limit set $\Lambda(G)$—is also removable, implying all geometrically finite Bers-boundary limit sets are conformally removable (Corollary 1.8).
• QC Uniformization: Every fat Basilica Julia set can be QC-mapped to a “round” basilica, where the bounded Fatou components become Euclidean circles (Corollary 1.9).
• Uniform QC Surface Subgroups: The QC automorphism group of $J_{3/4}$ contains infinitely many non-commensurable surface subgroups with uniformly bounded QC distortion, constructed via Bers-surgery. This stands in contrast to the more rigid scenario for Sierpinski carpets and similar sets.

5. Role of the Classical Basilica and the David Hierarchy

The standard basilica $J(z^2-1)$ occupies a distinct structural position as the “archbasilica” in the so-called David map hierarchy. While $J(z^2-1)$ lacks “fat” bounded complementary domains and is dendritic, its position in the hierarchy is characterized by the David-symmetry (measurable conformal structure that is less rigid than QC but more controlled than arbitrary homeomorphisms).

6. Extensions and Further Universality Classes

The techniques extend to:

• Cuspidal basilica Julia sets emerging from Schwarz reflection dynamics and cubic polynomials, yielding additional universality classes with analogous fatness properties.
• Limit sets of finite Bers-boundary Kleinian groups, supporting the broader application of QC and David-symmetry methodology to newly constructed fractal geometries within complex dynamics (Luo et al., 20 Jan 2026).

7. Significance Within Conformal Dynamics and Kleinian Theory

The identification of $J(z^2-\frac{3}{4})$ as a universal QC model for both fat Basilica Julia sets and Bers-boundary limit sets bridges a major gap between the theory of iterated rational maps and Kleinian group theory. By establishing explicit combinatorial, geometric, and analytic correspondences, this work yields the first known connected Julia set model—beyond the circle and sphere—admitting such a universal QC class. This establishes new directions for the classification of dynamical and geometric structures in holomorphic dynamics, complex analysis, and low-dimensional topology (Luo et al., 20 Jan 2026).