Universal holomorphic maps with slow growth. I: An algorithm. (Q6583548)

A linear operator \(T \colon E \to E\) of a topological vector space \(E\) is called \textit{hypercyclic} if there exists a \textit{hypercyclic vector} \(f \in V\) for \(T\), i.e., its forward orbit \(\{ f, \, T(f), \, T^2(f), \, \dots \}\) is dense in \(E\).\N\NGiven \(a \in \mathbb{C}^n\), we denote the corresponding \textit{translation operator} by \(T_a \colon \mathcal{O}(\mathbb{C}^n) \to \mathcal{O}(\mathbb{C}^n)\) which is defined as \(f(z) \mapsto f(z + a)\).\N\NThe study of hypercyclic operators was initiated by \textit{G. D. Birkhoff} [C. R. Acad. Sci., Paris 189, 473--475 (1929; JFM 55.0192.07)] who considered the case of translation operators on \(\mathcal{O}(\mathbb{C})\). On can also ask for vectors that are hypercylclic w.r.t.\ several operators, and one can impose further growth conditions (first established by [\textit{S. M. Duyos Ruiz}, Sov. Math., Dokl. 27, 9--13 (1983; Zbl 0549.30017); translation from Dokl. Akad. Nauk SSSR 268, 18--22 (1983)] in one complex variable). For a detailed account of this area of mathematics which is now sometimes referred to as Linear Chaos, see [\textit{K.-G. Grosse-Erdmann} and \textit{A. Peris Manguillot}, Linear chaos. Berlin: Springer (2011; Zbl 1246.47004)].\N\NThis article studies the interesting question of combining hypercyclicity and growth rate of the map in higher dimensions.\N\NTheorem A. Given a growth function \(\phi \colon \mathbb{R}_+ \to \mathbb{R}_+\), \(\phi(r) = \sum_{j=0}^{\infty} C_j r^j\) with \(C_j > 0\) and \(\lim_{j \to \infty} C_{j+1}/C_j = 0\), and given countably many directions \(I = \{ \Theta_k \}_{k \in \mathbb{N}}\) in the unit sphere \(S^ {2n-1} \subset \mathbb{C}^n\), there exists a holomorphic map \(F \colon \mathbb{C}^n \to \mathbb{C}^m\) with slow growth, i.e.,\N\[\N\forall z \in \mathbb{C}^n \quad \| F(z) \| \leq \phi(\|z\|)\N\]\Nand for every \(k \in \mathbb{N}\) there exists \(a_k \in \mathbb{R}_+ \cdot \theta_k\) such that \(F\) is a hypercyclic vector for the translation operator \(T_{a_k}\).\N\NThe results of Theorem D and Theorem E indicate that the result can't be sharpened much, at least not in the case \(n=m=1\): The set of hypercyclic directions \(I\) must have Hausdorff measure zero; A mapping \(F\) can be constructed where \(I\) is uncountable, e.g., a Cantor set.\N\NAt the heart of the proof of Theorem A is a constructive algorithm that is developed in this article. In one variable, it gives a new proof of the results of [\textit{S. M. Duyos Ruiz}, Sov. Math., Dokl. 27, 9--13 (1983; Zbl 0549.30017); translation from Dokl. Akad. Nauk SSSR 268, 18--22 (1983)], but the method can be extended to several variables.\N\NThe article contains also a section that explains very clearly the relevant ideas of [\textit{G. D. Birkhoff}, C. R. Acad. Sci., Paris 189, 473--475 (1929; JFM 55.0192.07); \textit{S. M. Duyos Ruiz}, Sov. Math., Dokl. 27, 9--13 (1983; Zbl 0549.30017); translation from Dokl. Akad. Nauk SSSR 268, 18--22 (1983); \textit{N. Sibony} (private communication, January (2021); \textit{Z. Chen} et al., J. Geom. Anal. 33, No. 9, Paper No. 308, 12 p. (2023; Zbl 1522.32006)].