Renormalization of the Hutchinson Operator

Yann Demichel (1)

1 MODAL'X - Modélisation aléatoire de Paris X

One of the easiest and common ways of generating fractal sets in $\R^D$ is as attractors of iterated function systems. The classic theory requires that the functions of such systems are contractive. 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 sequence $(H_n(K_0))_n$ with each set being a priori rescaled by its distance from $0$. We state several results for the convergence of these orbits and give a geometrical description of the corresponding attractors. We link these new sequences to the classic ones, in particular for the linear case, which provides another point of view about the classical theory. We illustrate our results with several various examples. Finally, we discuss some possible generalizations.

Mots clés: Hutchinson operator; Iterated function system; Attractor; Fractal sets

Domaines: Systèmes dynamiques [math.DS]

Yann Demichel. Renormalization of the Hutchinson Operator. 2014. ⟨hal-00990204v1⟩