Attractor of Cantor Type with Positive Measure

TL;DR

This paper constructs a specific iterated function system with increasing contractions that produces an attractor of positive Lebesgue measure, challenging typical measure-zero attractors.

Contribution

It introduces a novel iterated function system with disjoint images and positive measure attractor, expanding understanding of measure properties in fractal constructions.

Findings

Constructed an IFS with disjoint images and positive measure attractor

Demonstrated the existence of attractors with positive Lebesgue measure

Challenged the common notion that attractors of IFSs have measure zero

Abstract

We construct an iterated function system consisting of strictly increasing contractions $f,g\colon [0,1]\to [0,1]$ with $f([0,1])\cap g([0,1])=\emptyset$ and such that its attractor has positive Lebesgue measure.