Dynamics of composition operators on weighted Bergman spaces (Q904190)

The aim of the paper under review is to study multiples of composition operators associated to linear fractional maps on weighted Bergman spaces, from the point of view of linear dynamics. Let \(\alpha >-1\), \(1<p<+\infty \), and \(\varphi \) a linear fractional map on the unit disk \(\mathbb{D}\). The composition operator \(C_{\varphi }:f\longmapsto f\circ \varphi \) is always bounded on the Bergman space \(A_{\alpha }^{p}\). The author characterizes the frequent hypercyclicity of multiples \(\lambda C_{\varphi }\) of \(C_{\varphi }\), when \(\varphi\) is either a parabolic automorphism, a hyperbolic automorphism, or a hyperbolic non-automorphism of \(\mathbb{D}\). In all these cases, \(\lambda C_{\varphi }\) is frequently hypercyclic on \(A_{\alpha }^{p}\) if and only if it is hypercyclic, if and only if it is mixing. The proofs rely on classical criteria, such as the Frequent Hypercyclicity Criterion. Disjoint hypercyclicity of finite tuples of composition operators on \(A_{\alpha }^{p}\) associated to linear fractional maps is also studied.