  The attracting set and the inverse limit set are important objects associated
to a self-map on a set. We call \emph{stable set} of the self-map the
projection of the inverse limit set. It is included in the attracting set, but
is not equal in the general case. Here we determine whether or not the equality
holds in several particular cases, among which are the case of a dense range
continuous function on an Hilbert space, and the case of a substitution over
left infinite words.
