Split square and split carpet as examples of non-metrizable IFS attractors

TL;DR

This paper introduces two non-metrizable compact spaces, the split square and split carpet, demonstrating that they can serve as attractors for iterated function systems, expanding the understanding of IFS attractors beyond metrizable spaces.

Contribution

The paper constructs two novel non-metrizable attractors, the split square and split carpet, using products of split intervals, and proves their attractor properties.

Findings

Both the split square and split carpet are non-metrizable compact spaces.

These spaces can be realized as attractors of iterated function systems.

The construction uses products of split intervals to achieve non-metrizability.

Abstract

We define two non-metrizable compact spaces and show that they are attractors of iterated function systems. Both of them, the split square and the split carpet, are constructed using the product of split intervals.