A Thompson group for the basilica

TL;DR

This paper introduces a Thompson-like group acting on the basilica Julia set, using novel arc pair diagrams to demonstrate finite generation and virtual simplicity of the group.

Contribution

It defines a new group of homeomorphisms for the basilica Julia set and develops arc pair diagrams as a tool for analyzing its properties.

Findings

The group is finitely generated.

The group is virtually simple.

Arc pair diagrams are effective for group analysis.

Abstract

We describe a Thompson-like group of homeomorphisms of the basilica Julia set. Each element of this group acts as a piecewise-linear homeomorphism of the unit circle that preserves the invariant lamination for the basilica. We develop an analogue of tree pair diagrams for this group which we call arc pair diagrams, and we use these diagrams to prove that the group is finitely generated. We also prove that the group is virtually simple.