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Continuous semigroups of holomorphic self-maps of the unit disc. (English) Zbl 1441.30001

Springer Monographs in Mathematics. Cham: Springer (ISBN 978-3-030-36781-7/hbk; 978-3-030-36782-4/ebook). xxvii, 566 p. (2020).
Research on continuous one-parameter semigroups of holomorphic self-maps of the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) has a long history. Recently, applications of this theory appeared in mathematics and other sciences. The present book deals with dynamical properties of semigroups, the analytical concept of infinitesimal operators and the geometry of Koenigs functions. The authors introduce a new point of view based on dynamical aspects of semigroups involving the hyperbolic distance and Gromov’s hyperbolicity theory.
The book is divided into two parts consisting of 18 chapters. Part I contains preliminary results stated with necessary proofs. Chapter 1 provides basic tools such as the hyperbolic metric and Julia’s lemma on the horocycle image under a holomorphic self-map of \(\mathbb D\), the theorems of Lindelöf and Fatou on angular limits, boundary limits in connection with the Julia-Wolff-Carathéodory and Denjoy-Wolff theorems. Chapter 2 is devoted to holomorphic functions \(p\) in \(\mathbb D\), \(\text{Re}\,p(z)>0\), \(z\in\mathbb D\), their Herglotz representation, growth estimates, finite contact points and boundary behavior. Univalent functions appear in Chapter 3 together with boundary behavior properties, the area and distortion theorems and the Carathéodory kernel convergence theorem. Carathéodory’s prime ends theory and the Carathéodory topology and extension theorems are presented in Chapter 4. In Chapter 5, the authors examine the notion of hyperbolic metric in simply connected domains and the notion of geodesics as distance minimizing length curves. They obtain useful estimates of hyperbolic metric and distance. The authors introduce Gromov’s quasi-geodesics in Chapter 6 and prove that close to every quasi-geodesic there is a geodesic. Some localization results for hyperbolic distance are proved as well. The first part is ended by Chapter 7 where the harmonic measures in \(\mathbb D\) and in simply connected domains are studied carefully along with Bloch’s functions and diameter distortion theorems.
The core of the book is Part II. The first main result in Chapter 8 is the continuous Denjoy-Wolff theorem which says that, for a semigroup \((\phi_t)\) in \(\mathbb D\) that is not a group of hyperbolic rotations, there exists a unique Denjoy-Wolff point \(\tau\), \(|\tau|\leq1\), such that \(\lim_{t\to\infty}\phi_t(z)=\tau\) for all \(z\in\mathbb D\). Next, in Chapter 9, the authors define a universal holomorphic model giving an essentially unique Koenigs function \(h:\mathbb D\to\mathbb C\) of the semigroup \((\phi_t)\). Geometric and analytic characteristics of \(h\) for \((\phi_t)\) with a prescribed Denjoy-Wolff point \(\tau\) are fully described. The characteristic feature of \((\phi_t)\) is the infinitesimal generator \(G\) determined in Chapter 10 so that \[\frac{\partial\phi_t(z)}{\partial t}=G(\phi_t(z)),\;\;\;t\geq0,\;\;\;z\in\mathbb D.\] The Berkson-Porta formula establishes that \(G(z)=(z-\tau)(\overline\tau z-1)p(z)\), where \(p\) is holomorphic in \(\mathbb D\), \(\text{Re}\,p(z)>0\), and \(|\tau|\leq1\). There are other representations of \(G\) including formulas due to Abate and Aharonov-Elin-Reich-Shoikhet. The semigroup equation and the model functional equation can be extended up to the boundary as it is shown in Chapters 11 and 12. This leads to appearance of boundary regular fixed points which are the repelling fixed points of a semigroup and correspond to boundary regular critical points of infinitesimal generators. Repelling fixed points are associated with petals, that is, simply connected domains in \(\mathbb D\) which are completely invariant for the semigroup. In Chapter 13, petals and repelling fixed points are characterized via the geometry of the Koenigs mapping. In Chapter 14, the authors study contact arcs, namely, open arcs on \(\partial\mathbb D\) whose non-tangential image under the semigroup is on \(\partial\mathbb D\). Special boundary points which give rise to regular poles for infinitesimal generators are in the focus of Chapter 15. The total, orthogonal and tangential speeds are used in Chapter 16 to estimate the rate of convergence for orbits of non-elliptic semigroups to the Denjoy-Wolff point. The slope problem discussed in Chapter 17 is concerned with the landing of trajectories of a semigroup at the Denjoy-Wolff point. In particular, it is shown that the convergence of the trajectories of a non-elliptic semigroup to the Denjoy-Wolff point is non-tangential if and only if the image \(\Omega\) of the Koenigs map possesses certain symmetry properties with respect to any vertical line through \(\Omega\). In the final Chapter 18, the authors consider topological invariants of semigroups.
The book can serve as a reference source for mathematicians working in the holomorphic mappings area. Also. it is intended for beginners with a background in real and complex analysis. All the important results are proved in detail. Every chapter is ended with a publication history and interdependent connections.

MSC:

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C20 Conformal mappings of special domains
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
22A15 Structure of topological semigroups
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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