The persistence of \(\omega\)-limit sets defined on compact spaces (Q2338702)

Let \(X\) be a compact metric space, \({\mathcal K}\) stand for the class of all closed subsets of \(X\), \({\mathcal K}^*\) denote the family of closed subsets of \({\mathcal K}\), and \(\underline{S}(f)\) indicate the collection of \(p\)-stable periodic points of \(f\in C(X,X)\). Also, let \(I\) denote a (closed) unit interval of \(\mathbb R\) and \(M\) be a \(n\)-manifold. The authors' main results are the following: { Theorem 1.} The map \({\mathcal L}:C(I,I)\to {\mathcal K}^*\) is continuous if and only if \(\text{cl}(\underline{S}(f))={\mathcal L}(f)={\mathcal L}^*(f)\). { Theorem 2.} The map \({\mathcal L}:C(I,I)\to {\mathcal K}^*\) is continuous on a residual subset of \(C(I,I)\). { Theorem 3.} The map \(\overline{{\mathcal L}}:C(M,M)\to {\mathcal K}^*\) is upper semicontinuous at \(f\), if and only if \(\text{cl}({\mathcal L}(f))={\mathcal L}^*(f)\). { Theorem 4.}The map \(\overline{{\mathcal L}}:C(M,M)\to {\mathcal K}^*\) is continuous on a residual subset of \(C(M,M)\). { Theorem 5.} If \(f\in C(X,X)\) and \(\text{cl}(\underline{S}(f))={\mathcal L}^*(f)\), then \(\overline{{\mathcal L}}:C(X,X)\to {\mathcal K}^*\) is continuous at \(f\). Some comments involving certain related statements in the area are also given.