Canonical models for holomorphic iteration (Q2790592)

In this very interesting and well-written paper, the authors propose a categorial construction of intertwining models for any holomorphic self-map which is univalent on an absorbing domain, without to assume any regularity condition at the Denjoy-Wolff point. In particular, their results apply to univalent self-maps.NEWLINENEWLINELet \(X\) be a complex manifold of dimension \(q\), and let \(f\in \text{Hol}(X,X)\) be a univalent mapping. The authors define a semi-model for \(f\) as a triple \((\Omega,h,\psi)\), where the base space \(\Omega\) is a complex manifold of dimension \(k\in {\mathbb N}\), \(\psi\in \text{Aut}(\Omega)\), and \(h\in \text{Hol}(X,\Omega)\), with \(h\circ f=\psi\circ h\) and \(\bigcup_{m\in {\mathbb N}}\psi^{-m}(h(X))=\Omega\). The semi-model is said to be a model if \(h\) is univalent. One of the main results of this paper is Theorem 1.1, in which the authors prove that models exist, are essentially unique, and satisfy a universal property. More precisely, they prove the following strong result:NEWLINENEWLINETheorem 1.1: Let \(X\) be a complex manifolds and let \(f\in\text{Hol}(X,X)\) be a univalent map. Then there exists a model \((\Omega,h,\psi)\) for \(f\). Moreover, if \((\Lambda,l,\phi)\) is any semi-model for \(f\), then there exists a surjective holomorphic map \(g:\Omega\to\Lambda\) such that \(l=g\circ h\) and \(g\circ \psi=\phi\circ g\). In particular, if \((\Lambda,l,\phi)\) is a model (i.e., \(l\) is univalent), then \(g\) is a biholomorphism.NEWLINENEWLINEIn connection with this fundamental result, recall that in several complex variables there is no uniformization theorem, and thus even if \(X={\mathbb B}^q\), the complex structure of \(\Omega\) and its automorphism group may be quite complicated.NEWLINENEWLINEIn Section 4 the authors show that if \(X={\mathbb B}^q\), one can single out a special semi-model whose base space is a possible lower-dimensional ball \({\mathbb B}^k\) and which still provides interesting information about the map \(f\). In the same section, the authors introduce the canonical Kobayashi hyperbolic semi-model, while in the fifth section the authors focus on the model problem for univalent self-maps of the unit ball \({\mathbb B}^q\). They describe the canonical Kobayashi hyperbolic semi-model of a hyperbolic univalent map, and obtain another main result, which is presented in Theorem 1.2 of this paper.NEWLINENEWLINEIn the last section, the authors apply their main results obtained in this paper to study semigroups of holomorphic maps. In particular, in the case of the unit ball they solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups. The paper also contains many suggestive and useful examples.