Connected components of attractors and other stable sets (Q1356813)

The action of a continuous map \(f\) on a locally connected metric space, especially the action induced by \(f\) on the space of connected components of an attractor \(A\) or a stable set \(Y\) of \(f\), is studied. The main result: if the compact invariant set \(A\) is an attractor in the sense of Conley, then \(A\) has a finite number of connected components, and these are permuted by \(f\); the permutation is cyclic if and only if the attractor \(A\) is indecomposable. This is related to some results of \textit{J. Buescu} and \textit{I. Stewart} [Ergodic Theory Dyn. Syst. 15, No. 2, 271-290 (1995; Zbl 0848.54027)] and \textit{I. Melbourne}, \textit{M. Dellnitz} and \textit{M. Golubitsky} [Arch. Ration. Mech. Anal. 123, No. 1, 75-98 (1993; Zbl 0805.58043)].