Topological classification of scattered IFS-attractors

TL;DR

This paper investigates which countable compact spaces can serve as attractors for iterated function systems, providing examples and classifications based on their topological height and scattered properties.

Contribution

It introduces new criteria for when countable compact spaces are IFS-attractors, including constructions for spaces of specific heights and limitations for limit height spaces.

Findings

A convergent sequence in the real line is not an IFS-attractor.

Countable compact spaces of height δ+1 can be embedded as IFS-attractors.

Scattered compact metric spaces of limit height are not IFS-attractors.

Abstract

We study countable compact spaces as potential attractors of iterated function systems. We give an example of a convergent sequence in the real line which is not an IFS-attractor and for each countable ordinal $\delta$ we show that a countable compact space of height $\delta+1$ can be embedded in the real line so that it becomes the attractor of an IFS. On the other hand, we show that a scattered compact metric space of limit height is never an IFS-attractor.