Universality of the Basilica

TL;DR

The paper proves a universality phenomenon in conformal dynamics by showing that the fat Basilica Julia set $J(z^2-\frac{3}{4})$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial and to Bers boundary limit sets, thereby establishing a new link between Julia sets and Kleinian limit sets. The authors develop a unified fragmented-dynamics framework, introducing Basilica $\mathfrak{Q}$-maps and Basilica Bowen-Series maps to model disparate systems (polynomials, Kleinian groups, Schwarz reflections) within a common puzzle/Markov structure, enabling quasiconformal and David-type conjugacies. Consequences include conformal removability of geometrically finite Bers boundary limit sets, quasiconformal uniformization to round Basilicas for polynomial Fat Basilicas, and the existence of infinitely many non-commensurable uniformly quasi-symmetric surface subgroups arising from Basilica dynamics. The work extends to cuspidal Basilicas and cubic polynomials and culminates in a David-hierarchy view, where the archbasilica $J(z^2-1)$ dominates the hierarchy; collectively, these results reveal a broad, robust universality class for Basilica-type fractals across rational maps, Kleinian groups, and Schwarz reflection dynamics.

Abstract

We establish universality of the fat Basilica Julia set $J(z^2-\frac34)$ in conformal dynamics in the following sense: $J(z^2-\frac34)$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial as well as to the limit set of any geometrically finite closed surface Bers boundary group. We thus obtain the first example of a connected rational Julia set, not homeomorphic to the circle or the sphere, that is quasiconformally equivalent to a Kleinian limit set. It follows that any geometrically finite Bers boundary limit set is conformally removable. Other consequences of this universality result include quasi-symmetric uniformization of polynomial fat Basilicas by round Basilicas, and the existence of infinitely many non-commensurable uniformly quasi-symmetric surface subgroups of the Basilica quasi-symmetry group. We apply our techniques to cuspidal Basilica Julia sets arising from Schwarz reflections and cubic polynomials, yielding further universality classes. We also show that the standard Basilica Julia set $J(z^2-1)$ is the archbasilica in the David hierarchy.

Figures (11)

• Figure 1.1: A fat Basilica Julia set (for the polynomial $Q(z) = z^2-\frac{3}{4}$) on the left and a Basilica limit set of a genus $2$ closed surface group on the right. These two sets are quasiconformally homeomorphic.
• Figure 1.2: A Basilica limit set with persistently parabolic elements. Note that as we zoom in at a persistently parabolic point, the limit set converges to a line in the Hausdorff topology. On the other hand, at any non-contact point of the fat Basilica limit set in Figure \ref{['fig:Basilica']}, we can choose a specific sequence of zoom-ins so that the limit set converges to a closed set that is not topologically a line. This shows that these two Basilica limit sets are quasiconformally different (see Proposition \ref{['prop:qsinv']} for more details).
• Figure 1.3: Left: The cuspidal Basilica Julia set for the polynomial $R(z)$ is shown. The large black bounded Fatou component has the parabolic fixed point $1$ and the simple critical point $1-\frac{1}{4a}$ on its boundary. Right: The limit set of a Schwarz reflection in $\Sigma_3^*$ arising from the rational map $f(z)=z+\frac{2}{3z}-\frac{1}{3z^3}$ is displayed. These two sets are quasiconformally homeomorphic.
• Figure 1.4: Two cubic Basilica Julia sets on the boundary of the main hyperbolic component $\mathcal{H}$ of Milnor's $\mathop{\mathrm{Per}}\nolimits_1(0)$ curve. The left polynomial $z^3+2iz^2$ has a parabolic fixed point and hence a critical point in the parabolic basin. The right polynomial $z^3+\frac{3}{2}z^2$ has a strictly pre-periodic Julia critical point as a contact point, and its Julia set is quasiconformally equivalent to the standard Basilica $J(Q_{\mathop{\mathrm{pcf}}\nolimits})$. Any geometrically finite polynomial on $\partial\mathcal{H}$ has a Basilica Julia set, and is quasiconformal to one of these two models.
• Figure 1.5: A geometrically finite Basilica Julia set on $\partial \mathcal{H}$ and its magnification (zoom) on the right. It follows from Theorem \ref{['thm:cubicclass']} that it is quasiconformally equivalent to $J(z^3+2iz^2)$ in Figure \ref{['fig:cubicBasilica']} left. The quasiconformal homeomorphism sends the small cauliflower Fatou component in the center of the zoomed figure to the most prominent cauliflower Fatou component in Figure \ref{['fig:cubicBasilica']} left. This illustrates the flexibility of the quasiconformal map obtained in Theorem \ref{['thm:cubicclass']}.

Theorems & Definitions (83)

• Theorem A
• Theorem B
• Corollary 1.1: Quasiconformal Universality
• Remark 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem C
• Conjecture 1.7