Stable set of self map

TL;DR

This paper investigates the properties of the stable set of a self-map, comparing it to the attracting set and inverse limit set, and identifies conditions under which these sets coincide in specific mathematical contexts.

Contribution

It characterizes when the stable set equals the attracting set for certain classes of self-maps, including continuous functions on Hilbert spaces and substitutions over infinite words.

Findings

The stable set is included in the attracting set but not always equal.

Equality holds for dense range continuous functions on Hilbert spaces.

Equality also holds for substitutions over left infinite words.

Abstract

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.