Attractors of Iterated Function Systems with uncountably many maps

TL;DR

This paper investigates the topological structure of attractors in second generation iterated function systems on the real line, showing conditions under which they form finitely many closed intervals.

Contribution

It introduces the concept of second generation I.F.S. with uncountably many maps and characterizes their attractors under specific conditions.

Findings

Attractors can be finitely many closed intervals under certain conditions.

Uncountably many maps lead to complex attractor structures.

Dissection properties influence the topological nature of attractors.

Abstract

We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: 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, I.F.S. (of first generation), or it possesses a particular dissection property, the attractor of the second generation I.F.S. consists of finitely many closed intervals.