Counterexamples for IFS-attractors

TL;DR

This paper explores the properties of attractors in iterated function systems (IFS), providing counterexamples that highlight the necessity of various definitions within fractal theory related to IFS and weak IFS.

Contribution

It introduces counterexamples demonstrating that different definitions of IFS-attractors and related fractals are essential and not interchangeable.

Findings

Counterexamples show the non-equivalence of IFS definitions.

Highlights the importance of specific conditions in IFS attractor theory.

Clarifies distinctions between Banach and topological fractals.

Abstract

In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.