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Iteration theory of holomorphic maps on taut manifolds. (English) Zbl 0747.32002
Research and Lecture Notes in Mathematics. Commenda di Rende (Italy): Mediterranean Press, xvii, 416 p. (1989).
The study of the family $${\mathcal F}=\{f^ k,k=1,2,\ldots\}$$ of iterates of a holomorphic map $$f:M\to M$$ from a Riemann surface $$M$$ into itself has two distinct aspects. On the one hand, there are the questions originating with the classical work of Fatou and Julia on describing the set of points where the family is normal, and their complements, and which have caught a great deal of attention in recent years thanks to the stunning graphic images revealed with the help of modern technology. On the other hand, there are interesting complex geometric questions about the family $${\mathcal F}$$ in case it is normal on all of $$M$$, for example, when $$M$$ is the open unit disc $$\Delta$$. The author’s book deals with the latter case, and it presents us a well organized and attractive tour of the basic general theory, and an in depth study of several results, both in the one variable context, and in the appropriate setting in several complex variables.
The first part of the book deals with the one dimensional theory. The natural spaces $$M$$ considered here are the hyperbolic Riemann surfaces, i.e., those whose universal covering is the unit disc. The author builds up the general theory around the invariant version of the Schwarz Lemma and the Poincaré metric, and he discusses in detail the theorems of Wolff-Denjoy on the asymptotic behavior of the sequence $${\mathcal F}$$ of iterates, of Shields on fixed points of $${\mathcal F}$$, and of Julia-Wolff- Carathéodory on angular derivatives. This part of the book is essentially self-contained, and it could be used very well for a course in geometric function theory from a modern point of view.
The second, and major part of the book deals with the multidimensional theory. The general theory has its roots in classical results of Carathéodory and H. Cartan on holomorphic self maps of bounded domains in $$\mathbb{C}^ n$$. The author first transfers these results to their natural intrinsic setting, which involves the Kobayashi metric (a generalization of the Poincaré metric), the concept of hyperbolic manifold, and the closely related concept of taut manifold, introduced by H. Wu in 1967, and which appears in the title of the book: a complex manifold $$M$$ is taut, if the family of holomorphic maps from $$M$$ into $$M$$ is normal. The remaining sections of the book present higher dimensional versions of the three theorems mentioned above, first in the simpler setting of the unit ball, and then on convex or strictly convex domains, as appropriate.
The author provides us with an excellent comprehensive exposition on a branch of complex analysis which has been very actively investigated during the past two decades. The book collects systematically numerous results scattered in the literature, thus making them easily accessible to newcomers to the field. Several major results from multidimensional complex analysis are introduced without proof, but precise references are given. There are long introductory sections, so that the reader always knows what is going to happen, and why, and there are extensive notes which trace the origin of most results. This book complements very well recent texts in multidimensional complex analysis, and I recommend it highly to students and researchers in complex analysis.
Reviewer: R.M.Range (Albany)

##### MSC:
 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable 32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable