Universality of the Basilica

TL;DR

This paper proves a universality property of the fat Basilica Julia set, showing it is quasiconformally equivalent to various other complex structures, revealing deep connections in conformal and Kleinian group dynamics.

Contribution

It establishes the first example of a connected rational Julia set quasiconformally equivalent to a Kleinian limit set, expanding understanding of Julia set universality.

Findings

Fat Basilica Julia set is quasiconformally equivalent to any polynomial's Julia set.

Fat Basilica Julia set is quasiconformally equivalent to limit sets of geometrically finite Bers boundary groups.

Standard Basilica Julia set is identified as the archbasilica in the David hierarchy.

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.