On the Attractor of One-Dimensional Infinite Iterated Function Systems

TL;DR

This paper investigates the attractor of infinite one-dimensional iterated function systems, proposing a conjecture that, in certain cases, the attractor comprises finitely many non-overlapping intervals, supported by numerical and partial theoretical evidence.

Contribution

It introduces a conjecture about the structure of attractors in second generation infinite IFS and provides numerical methods and partial proofs to support it.

Findings

Numerical techniques suggest the attractor may be finitely many intervals.

A partial rigorous proof supports the conjecture in specific cases.

The study advances understanding of infinite IFS attractors.

Abstract

We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.