On \(\omega\)-limit sets and attraction of non-autonomous discrete dynamical systems (Q2898893)

The authors study fundamental properties of \(\omega\)-limit sets and discuss the relationship between \(\omega\)-limit sets and attraction of non-autonomous discrete dynamical systems.NEWLINENEWLINE Some of the main results of the paper are:NEWLINENEWLINELet \((X,f_{1,\infty})\) be a non-autonomous discrete dynamical system and let \(A, B, F, K \subseteq X\) and \(K\) be a compact set. Then the following properties hold: \begin{itemize}\item[1)] if \(A \subseteq B\) and \(F\) attracts \(B\), then \(F\) attracts also \(A\); \item[2)] if \(F\) attracts \(A\) and \(B\), then \(F\) attracts \(A \cap B\); \item[3)] if \(F\) attracts \(A\) and \(B\), then \(F\) attracts \(A \cup B\); \item[4)] if \(X\) is a regular space and \(K\) attracts \(A\), then \(K\) attracts also \(\overline A\).NEWLINENEWLINELet \((X,f_{1,\infty})\) be a non-autonomous discrete dynamical system, where \(X\) is a regular topological space. Let \(E \subseteq X\) and \(K\) be a compact set of \(X\) such that \(K\) attracts \(E\). Then the following properties hold:NEWLINENEWLINE\(\omega (E,f_{1,\infty})\) is a nonempty compact set;NEWLINENEWLINEif \(F\subseteq X\) is a closed set, then then \(F\) attracts \(E\) if and only if \(\omega (E,f_{1,\infty}) \subseteq F\);NEWLINENEWLINE\(\omega (E,f_{1,\infty})\) attracts \(E\).\end{itemize}