A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to linear dynamics (Q2859492)

In the paper under review, the author proves a Birkhoff type transitivity theorem for continuous maps acting on non-separable completely metrizable spaces. Precisely, he shows that, if \(T\) is a topologically transitive continuous map acting on a completely metrizable space \(X\) and \(x\in X\), then there exists a dense \(G_\delta\) subset \(D\) of \(X\) such that every point \(z\in D\) is recurrent and \(x\in\overline{O(w,T)}\) for every \(w\in D\). Moreover, the author gives some applications of such a result for dynamics of bounded linear operators acting on complex Fréchet spaces. In particular, he proves that any positive power and any unimodular multiple of a topologically transitive linear operator is topologically transitive, thus extending similar results of S. I. Ansari, resp. F. Léon-Saavedra and V. Müller for hypercyclic operators.