Dynamics of weighted composition operators (Q403683)

In the paper under review, the author is concerned with the dynamics of the weighted composition operator \(C_{\omega,\varphi}: H(\Omega)\to H(\Omega)\) given by \(C_{\omega,\varphi}(f)(z)=\omega(z)(f\circ \varphi)(z)\), for \(z\in \Omega\), where \(H(\Omega)\) denotes the space of holomorphic functions on a simply connected domain \(\Omega\) of the complex field, endowed with the compact open topology, \(\varphi\) is a holomorphic self map of \(\Omega\), and \(\omega\in H(\Omega)\). In particular, the author shows that any such operator \(C_{\omega,\varphi}\) is weakly supercyclic if and only if it is topologically mixing and, in the case that the weight is bounded, if and only if the operator has a hypercyclic subspace. The author also provides conditions on the symbol in order that the operator \(C_{\omega,\varphi}\) is Devaney-chaotic and has a frequently hypercyclic subspace.