On weakly hyperbolic iterated functions systems

TL;DR

This paper extends the theory of weakly hyperbolic iterated function systems to more general compact spaces and parameter spaces, establishing existence of attractors, their ergodic properties, and methods to visualize them.

Contribution

It generalizes Edalat's weakly hyperbolic IFS framework to complete spaces with compact parameter spaces, proving attractor existence and ergodicity, and adapts the chaos game for this setting.

Findings

Existence of topological and measure-theoretic attractors.

Measure-theoretic attractor is ergodic.

Extension of chaos game to compact parameter spaces.

Abstract

We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint, and prove that the measure theoretical attractor is ergodic. We also define weakly hyperbolic iterated functions systems for complete spaces and compact parameter space, and prove that this definition extends the one given by Edalat. Furthermore, we study the question of existence of the attractors in this setting. Finally, we prove a version of the results of Barnsley and Vince about drawing the attractor (also called the chaos game), for the case of compact parameter space.