Attractors of iterated function systems with uncountably many maps and infinite sums of Cantor sets

Abstract

We study the topological properties of attractors of Iterated function systems (IFS) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation IFS: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non–singular, hyperbolic, IFS (of first generation), or it possesses a particular dissection property, the attractor of the second generation IFS consists of the union of a finite number of closed intervals. We also prove a theorem that generalizes this result to certain infinite sums of compact sets, in the sense of Minkowski and under the Hausdorff metric.