The stable set of a self-map

The attracting set and the inverse limit set are important objects associated with a self-map on a set. By stable set of the self-map we mean 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 the equality holds in several particular cases, among which are the case of a dense range continuous function on a Hilbert space, and the case of a substitution over right-infinite words.