Fractals as Sculptures

TL;DR

This paper demonstrates that attractors of iterated function systems in boundedly compact metric spaces can be characterized as intersections of decreasing sequences of compact, invariant subspaces, providing a new perspective on their structure.

Contribution

It introduces a novel method to represent IFS attractors as intersections of decreasing compact invariant subspaces in boundedly compact metric spaces.

Findings

Attractors can be expressed as intersections of decreasing compact invariant subspaces.

Provides a new characterization of IFS attractors in boundedly compact metric spaces.

Enhances understanding of the structure of fractal attractors.

Abstract

The attractors of iterated function systems are usually obtained as the Hausdorff limit of any non-empty compact subset under iteration. In this note we show that an iterated function system on a boundedly compact metric space has compact, invariant subspaces so that the attractor of the IFS can also be expressed as the intersection of a sequence of decreasing compact spaces.