Compositional frequent hypercyclicity on weighted Dirichlet spaces
Let \(\mathbb D\) be the open unit disc of \(\mathbb C\) and let LFT(\(\mathbb D\)) be the linear fractional transformations \(\psi(z)=\frac{az+b}{cz+d}\) such that \(\psi(\mathbb D) \subset \mathbb D\). Thus, in particular, such a \(\psi\) is a holomorphic self-map. \textit{P.\,S.\thinspace Bourdon} and \textit{J.\,H.\thinspace Shapiro} [``Cyclic phenomena for composition operators'' (Mem.\ Am.\ Math.\ Soc.\ 596) (1997; Zbl 0996.47032)] were the first to study for which holomorphic self-maps \(\psi\) the composition operators \(C_{\psi}\) acting on the Hardy space \(H^2(\mathbb D)\) are either cyclic or hypercyclic. They relied on a ``linear fractional'' classification of maps. The weighted Dirichlet spaces \(S_{\nu}\) are of the form \(\{f \in H(\mathbb D):f(z)=\sum_{n=0}^{\infty} a_nz^n,\;\sum_{n=0}^{\infty} |a_n|^2(n+1)^{2\nu}<\infty\}\), where \(\nu \in \mathbb R\). Classical cases are the Hardy space, the Bergman space and the Dirichlet space. A result by \textit{P.\,R.\thinspace Hurst} [Arch.\ Math.\ 68, No.\,6, 503--513 (1997; Zbl 0902.47030)] says that the composition operator \(C_{\psi}: S_{\nu} \to S_{\nu}\) is bounded whenever \(\nu \in \mathbb R\) and \(\psi \in\) LFT(\(\mathbb D\)). The authors prove that, in many cases, the scalar multiple \(\lambda C_{\psi}\) of an operator as above is frequently hypercyclic if and only if is hypercyclic. This occurs for all triples \((\nu,\lambda,\psi)\) except perhaps for those such that \(\nu \in [1/4,1/2)\), \(|\lambda|=1\), and \(\psi\) is a parabolic automorphism. The proof of their result is based on the work of \textit{E.\,A.\thinspace Gallardo-Gutiérrez} and \textit{A.\,Montes-Rodríguez} [``The role of the spectrum in the cyclic behavior of composition operators'' (Mem.\ Am.\ Math.\ Soc.\ 791) (2004; Zbl 1054.47008)] who characterized those \(\lambda C_{\psi}\) which are hypercyclic. Another important ingredient in the proof is the Frequent Hypercyclicity Criterion of \textit{F.\,Bayart} and \textit{S.\,Grivaux} [Trans.\ Am.\ Math.\ Soc.\ 358, No.\,11, 5083--5117 (2006; Zbl 1115.47005)].
scientific article; zbMATH DE number 5689247
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