Finest Morse decompositions for semigroup actions on fiber bundles (Q616197)

Let \(S\) be a semigroup acting on a topological space \(M\), and let \(\mathcal{F}\) be a family of subsets of \(S\). The \(\omega\)-limit set of a subset \(X\subset M\) for \(\mathcal{F}\) is defined as \[ \omega(X,\mathcal{F}):=\bigcap_{A\in\mathcal{F}}\mathrm{clos}(AX). \] This generalizes the classical concept of an \(\omega\)-limit set for semiflows where \(S=\mathbb{R}^{+}_{0}\) and \(\mathcal{F}=\{(a,\infty):a\geq 0\}\). The authors extend some concepts from dynamical systems and Conley index theory to the setting above, in particular attractor-repeller pairs, Morse decompositions, and chain-transitivity.