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

Abstract

Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFSs) 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.