Asymptotically stable sets and the stability of \(\omega\)-limit sets (Q5916181)

Let \(f\) be a continuous interval map. A non empty closed set \(A\) is said to be stable if for each open set \(U\) containing \(A\) there is another open set \(V\) containing \(A\) such that any orbit starting from a point \(x\in V\) is included in \(U\). If in addition there exists an open set \(U_0\) containing \(A\) such that the omega limit set \(\omega (x,f)\subset A\) for any \(x\in U_0\), the set \(A\) is said to be asymptotically stable. Let \(Q(x,f)\) be the intersection of all the asymptotically stable sets of \(f\) which contain \(\omega (x,f)\). The paper surveys some results concerning the set \(Q(x,f)\) and the maps \(Q_f:[0,1]\rightarrow K\) as \(Q_f(x)=Q(x,f)\) and \(Q:[0,1]\times C \rightarrow K\), as \(Q((x,f))=Q(x,f)\), where \(C\) is the set of continuous interval maps and \(K\) is the family of sets of the form \(Q(x,f)\), \(x\in [0,1]\).