Disjoint hypercyclic linear fractional composition operators
Let \(X\) be a topological vector space and, for \(2\leq N\), let \(X^N\) be endowed with the product topology. The operators \(T_1,\dots,T_N\) are \textit{disjoint hypercyclic} (or \textit{d-hypercyclic} for short) if there is an \(x\in X\) such that \((x,\dots,x)\) is hypercyclic for \(\bigoplus_{i=1}^NT_i\) acting on \(X^N\). There is an analogous definition for \textit{d-supercyclicity.} These concepts were introduced simultaneously and independently by \textit{J. Bès} and \textit{A. Peris} [J. Math. Anal. Appl. 336, No. 1, 297--315 (2007; Zbl 1129.47007)] and \textit{L. Bernal-Gónzalez} [Stud. Math. 182, No. 2, 113--131 (2007; Zbl 1134.47006)]. \textit{P. S. Bourdon} and \textit{J. H. Shapiro} [Mem. Am. Math. Soc. 596 (1997; Zbl 0996.47032)] started the study of cyclic properties of composition operators \(C_{\varphi}\) on the Hardy space \(H^2(\mathbb D)\). (Here, \(\mathbb D\) is the open unit disc and \(\varphi\) is an analytic self-map of \(\mathbb D.\)) They and other authors continued their study in different settings; in particular, \textit{E. Gallardo-Gutiérrez} and \textit{A. Montes-Rodríguez} [Mem. Am. Math. Soc. 791 (2004; Zbl 1054.47008)] determined cyclic properties of linear fractional composition operators on the weighted Hardy spaces \(S_{\nu}\) with \(\nu \in \mathbb R\). For \(\nu=-\frac{1}{2}\), \(\nu=0\), and \(\nu=\frac{1}{2}\), the spaces \(S_{\nu}\) correspond, respectively, to the classical Bergman, Hardy and Dirichlet spaces. Below is one of the main theorems obtained by the authors, but first one more concept is needed. The operators \(T_1,\dots,T_N\) with \(2\leq N\) are \textit{d-mixing} if for every nonempty open subsets \(V_0, V_1,\dots, V_N\) of \(X\) there exists \(m \in \mathbb N\) such that \(\emptyset \neq V_0\cap T_1^{-j}(V_1)\cap \dots \cap T_N^{-j}(V_N)\) for all \(j \geq m\). Theorem. Let \(\nu <\frac{1}{2}\) and let \(\varphi_1,\dots, \varphi_N\) be distinct linear fractional maps of \(\mathbb D\) into itself. The following are equivalent: (a) The operators \(C_{\varphi_1},\dots, C_{\varphi_N}\) are d-supercyclic on \(S_{\nu}\). (b) The operators \(C_{\varphi_1},\dots, C_{\varphi_N}\) are d-mixing on \(S_{\nu}\). (c) The symbols \(\varphi_1,\dots, \varphi_N\) satisfy \hskip .2in (i) each \(\varphi_j\) is either a parabolic automorphism or a hyperbolic map without fixed points in \(\mathbb D\), and \hskip .2in (ii) there are no two symbols \(\varphi_l,\varphi_j\) having a common fixed point \(\alpha\) such that \(\varphi_l^{\prime}(\alpha)=\varphi_j^{\prime}(\alpha)<1\).
scientific article; zbMATH DE number 5909863
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