Spiders' webs and locally connected Julia sets of transcendental entire functions

TL;DR

This paper establishes a connection between locally connected Julia sets and spider's web structures in transcendental entire functions, revealing new insights into their topological properties.

Contribution

It proves that locally connected Julia sets are spider's webs and that spider's web Julia sets are locally connected at dense buried points, expanding understanding of Julia set topology.

Findings

Locally connected Julia sets are spider's webs.

Spider's web Julia sets are locally connected at dense buried points.

The residual Julia set can also form a spider's web.

Abstract

We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.