Non-hyperbolic Iterated Function Systems: semifractals and the chaos game

TL;DR

This paper explores the dynamics of non-hyperbolic iterated function systems on compact spaces, introducing target sets as semifractals, and establishes conditions for local and global attractors, highlighting new classes of semifractals.

Contribution

It introduces the concept of target sets for non-hyperbolic IFSs, providing conditions for attractors and demonstrating the existence of new semifractal classes.

Findings

Target sets can be nonempty in non-regular IFSs.

Conditions for local and global attractors are established.

Random orbits tend to stable target sets.

Abstract

We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do for regular IFSs. We study sufficient conditions which guarantee that the closure of the target set is a local attractor for the IFS. As a corollary, we establish necessary and sufficient conditions for the IFS having a global attractor. We give an example of a non-regular IFS whose target set is nonempty, showing that our approach gives rise to a "new class" of semifractals. Finally, we show that random orbits generated by IFSs draws target sets that are "stable".