The Hutchinson-Barnsley theory for generalized iterated function systems by means of infinite iterated function systems

TL;DR

This paper links generalized iterated function systems (GIFS) to infinite iterated function systems (IIFS), showing that GIFS attractors and measures can be approximated as limits of IIFS attractors and measures using specific metrics.

Contribution

It introduces a new approach to analyze GIFS as limits of IIFS, extending the Hutchinson-Barnsley theory to a broader class of systems.

Findings

GIFS attractors are limits of IIFS attractors in Hausdorff-Pompeiu metric.

Hutchinson measures for GIFS are limits of IIFS measures in Monge-Kantorovich metric.

Provides a unified framework connecting GIFS and IIFS theories.

Abstract

The study of generalized iterated function systems (GIFS) was introduced by Mihail and Miculescu in 2008. We provide a new approach to study those systems as the limit of the Hutchinson-Barnsley setting for infinite iterated function systems (IIFS) which has been developed by many authors in the last years. We show that any attractor of a contractive generalized iterated function system is the limit with respect to Hausdorff-Pompeiu metric of attractors of contractive infinite iterated function systems. We also prove that any Hutchinson measure for a contractive generalized iterated function system with probabilities is the limit with respect to the Monge-Kantorovich metric of the Hutchinson measures for contractive infinite iterated function systems with probabilities.