Renormalization of the Hutchinson Operator

TL;DR

This paper introduces a renormalized version of the Hutchinson operator for affine iterated function systems, allowing for the generation of fractal sets without the strict contractivity condition, and analyzes their convergence and geometric properties.

Contribution

It proposes a new renormalized operator $H_ ho$ that relaxes contractivity requirements, enabling the construction and analysis of more general fractal attractors.

Findings

Convergence results for the renormalized Hutchinson orbits.

Geometrical descriptions of the limit sets.

Method to construct eigensets for the operator.

Abstract

One of the easiest and common ways of generating fractal sets in ${\mathbb R}^D$ is as attractors of affine iterated function systems (IFS). The classic theory of IFS's requires that they are made with contractive functions. In this paper, we relax this hypothesis considering a new operator $H_\rho$ obtained by renormalizing the usual Hutchinson operator $H$. Namely, the $H_\rho$-orbit of a given compact set $K_0$ is built from the original sequence $\big(H^n(K_0)\big)_n$ by rescaling each set by its distance from $0$. We state several results for the convergence of these orbits and give a geometrical description of the corresponding limit sets. In particular, it provides a way to construct some eigensets for $H$. Our strategy to tackle the problem is to link these new sequences to some classic ones but it will depend on whether the IFS is strictly linear or not. We illustrate the different results with various detailed examples. Finally, we discuss some possible generalizations.