Multivalued maps, selections and dynamical systems
Let \(X\) be a locally compact space and \((x,t)\mapsto \varphi(x,t)\) a continuous flow on \(X\). Let \(\psi=J^{+}\) be the multivalued map with \(D(\psi)\)=\(\{x\in X\); \(\emptyset \neq J^{+}(x)\)=compact\(\}\) and \(\Psi\) its associated map (from \(2^{D(\psi)}\) to \(2^X\)): \(\Psi(K)=J^{+}(K)\). Theorem. Let \(K\subseteq X\) be compact and invariant. Then (i) \(K\) is stable iff \(\Psi(k)=k\), (ii) \(K\) is a global attractor iff \(\Psi(\text{cl}\{k\})=k\), (iii) \(K\) is an attractor iff \(\Psi(\text{cl} \{k\} \cap W)=k\), for some neighborhood \(W\) of \(k\) in \(2^X\). If, in addition, \(X\) is compact and \(K\) is an attractor then (iv) \(\text{int} (K)= \emptyset\) iff \(\Psi(k^*)=x\) (where \(k^*\) is the dual repeller of \(k\)). Further aspects involving the attractors in \(\mathbb R^n\) are discussed.
scientific article; zbMATH DE number 5275711
scientific article