Uniformly perfect and hereditarily non uniformly perfect analytic and conformal non-autonomous attractor sets

TL;DR

This paper investigates conditions under which non-autonomous analytic iterated function systems in the complex plane have either uniformly perfect or hereditarily non uniformly perfect attractor sets, with applications to Julia sets and generalizations of autonomous IFS results.

Contribution

It provides new criteria for uniform perfectness and non-uniform perfectness in non-autonomous IFS, along with examples and analysis of their relationships to autonomous systems.

Findings

Conditions for uniformly perfect attractors established

Examples illustrating main theorems and generalizations

Applications to non-autonomous Julia sets included

Abstract

Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFS's) in the complex plane have uniformly perfect attractor sets, while other conditions imply the attractor is pointwise thin, and thus hereditarily non uniformly perfect. Examples are given to illustrate the main theorems, as well as to indicate how they generalize other results. Examples are also given to illustrate how possible generalizations of corresponding results for autonomous IFS's do not hold in general in this more flexible setting. Further, applications to non-autonomous Julia sets are given. Lastly, since our definition of NIFS is in some ways more general than others found in the literature, a careful analysis is given to show when certain familiar relationships still hold, along with detailed examples showing when other relationships do not hold.