  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.
