Dynamics of composition operators with holomorphic symbol (Q380435)

In this interesting paper, the relation of various dynamical properties of composition operators on the Fréchet space of holomorphic functions on a simply connected plane domain is investigated. For a simply connected plane domain \(\Omega\subseteq\mathbb{C}\), the holomorphic functions \(H(\Omega)\) endowed with the compact open topology is a Fréchet space on which, for a holomorphic \(\varphi:\Omega\rightarrow\Omega\), the corresponding composition operator \(C_\varphi:H(\Omega)\rightarrow H(\Omega)\), \(f\mapsto f\circ\varphi\), is continuous. In recent years, the dynamical behaviour of such composition operators on \(H(\Omega)\) has become a subject of intense study. In the paper under review, among other things, it is shown that, under the above hypotheses, \(C_\varphi\) is weakly supercyclic if and only if, for each non-constant polynomial \(P\), the operator \(P(C_\varphi)\) satisfies the so-called Frequent Hypercyclicity Criterion (and is thus frequently hypercyclic, mixing, and chaotic) if and only if \(\varphi\) is univalent and has no fixed point in \(\Omega\). By a result of Shapiro, the later is equivalent to \(C_\varphi\) being hypercyclic. Apart from \textit{A. Bonilla} and \textit{K. G. Grosse-Erdmann}'s variant [Ergodic Theory Dyn. Syst. 27, No. 2, 383--404 (2007; Zbl 1119.47011); Monatsh. Math. 168, No. 3--4, 305--320 (2012; Zbl 1290.47007)] of the Frequent Hypercyclicity Criterion originally due to \textit{F. Bayart} and \textit{S. Grivaux} [Trans. Am. Math. Soc. 358, No. 11, 5083--5117 (2006; Zbl 1115.47005)], one of the main tools is the following form of what the author calls Shapiro's Comparison Principle: For a simply connected domain \(\Omega\) and a univalent, holomorphic \(\varphi:\Omega\rightarrow\Omega\) without fixed points, there is a non-trivial translation \(\varphi_a(z)=z+a\), a simply connected domain \(\Omega_a\) satisfying \(\varphi_a(\Omega_a)\subseteq \Omega_a\), and a biholomorphic map \(\sigma:\Omega\rightarrow\Omega_a\) with \(\sigma\circ\varphi=\varphi_a\circ\sigma\). So, in particular, \(C_\varphi\) and \(C_{\varphi_a}\) are conjugate via \(C_\sigma:H(\Omega_a)\rightarrow H(\Omega)\), \(f\mapsto f\circ\sigma\).