Renormalization of the Hutchinson 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.