Hypercyclic Composition Operators on Hv0‐spaces

We consider analytic self‐maps φ on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {D}$\end{document} and prove that the composition operator Cφ acting on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{v}^0$\end{document} is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {D}$\end{document} which yield non‐hypercyclic composition operators Cφ on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{v}^0$\end{document}.