Uniformly perfect analytic and conformal attractor sets

TL;DR

This paper establishes conditions under which attractor sets of analytic iterated function systems in the complex plane are uniformly perfect, highlighting cases where this property holds or fails.

Contribution

It provides new criteria for uniform perfectness of attractor sets in analytic IFSs and presents an example illustrating the limits of these conditions.

Findings

Attractor sets of finitely generated conformal IFSs with two or more points are uniformly perfect.

An example of a finitely generated analytic attractor set that is not uniformly perfect is constructed.

Conditions for uniform perfectness depend on the structure and points contained in the attractor set.

Abstract

Conditions are given which imply that analytic iterated function systems (IFS's) in the complex plane have uniformly perfect attractor sets. In particular, it is shown that the attractor set of a finitely generated conformal IFS is uniformly perfect when it contains two or more points. Also, an example of a finitely generated analytic attractor set which is not uniformly perfect is given.