Strongly-Fibred Iterated Function Systems and the Barnsley--Vince triangle

TL;DR

This paper reviews the theory of semiattractors for non-contractive IFSs, applies it to a Barnsley--Vince example, and characterizes the semiattractor using invariant measures and specific criteria.

Contribution

It introduces criteria for semiattractor existence and strong-fibredness in non-contractive IFSs, with a detailed analysis of the Barnsley--Vince system.

Findings

Established existence of semiattractors for the example

Identified an invariant measure for the Barnsley--Vince IFS

Characterized the semiattractor as strongly-fibred

Abstract

We review the theory of semiattractors associated with non-contractive Iterated Function Systems (IFSs) and demonstrate its applications on a concrete example. In particular, we present criteria for the existence of semiattractors due to Lasota and Myjak. We also discuss the Kieninger criterion which allows us to characterise when a semiattractor is strongly-fibred. Finally, we consider a specific example of a non-contractive IFS introduced by Barnsley and Vince. We find an invariant measure for this system which allows us to describe its semiattractor. The difficulty in analysing this IFS stems from the fact that it is neither eventually contractive nor contractive on average.