On local aspects of sensitivity in topological dynamics (Q738480)

Let \((X,f)\) be a topological dynamical system. Denote by \(P(f)\) the set of all periodic points and by \({\mathcal K}(X)\) the class of all compact subsets in \(X\). Further, let \(A\) be a (nonempty) subset of \(X\). The following are the main results in this paper: Theorem 1. Suppose that \(A\) is an infinite transitive set of \((X,f)\) and \(P(f)\) is dense in \(A\). Then, \(A\) is an s-set of \((X,f)\). Theorem 2. Suppose that \(A\) is closed and \(P(f)\) is dense in \(A\). Then, \(P(\overline{f})\) is dense in \({\mathcal K}(A)\). Theorem 3. Suppose that \(A\) is closed and \({\mathcal K}(A)\) is a sensitive set of \(({\mathcal K}(X),\overline{f})\). Then, \(A\) is an s-set of \((X,f)\). Finally, an extension of these facts to set-valued discrete dynamical systems is being discussed.