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Rescaling methods in complex analysis. (Méthodes de changement d’échelles en analyse complexe.) (French) Zbl 1123.37019
The author discusses several rescaling methods in complex analysis and geometry and applies them to get elementary proofs of some classical results. The Bloch principle plays an important role in this approach and yields to a somewhat unified point of view.
The first chapter, “Renormalizations of holomorphic disks”, is centered on the usual Bloch principle. This principle explains the defect of normality of the family of holomorphic disks in a compact variety by the presence of bounded integer curves situated in the closure of this family. These curves are produced by the Zalcman renormalization process. The author applies them to some fundamental properties of the Julia ensembles of rational fractions and to certain questions of hyperbolicity in the Kobayashi sense. This approach considerably simplifies the classic demonstrations.
The second chapter, “Dilatation of coordinates”, is dedicated to the Pinchuk method of coordinate dilatation. It concerns the study of biholomorphisms by blowing up their domains (source or goal) in order to create a bounded biholomorphism between a simple model domains (for example, an automorphism of a nonbounded realization of the Euclidean ball). The author stands strictly in the pseudoconvex case to simplify but he adopts an easy presentation to transpose to the weakly pseudoconvex case of finite type. The author starts with studying the normalization problem inherent in this method; it is very naturally linked with the Bloch principle. This method is applied to the study of the Kobayashi metric compartment, to the Wong-Rosay theorem and to the Fefferman theorem on the differentiable extension of biholomorphisms.
In the third chapter, “Normalization and linearization along orbits”, the normalization of invertible germs (theorems of Poincaré and Dulac) is treated . It has there nothing new, it is only the proof that the author gives simpler than those found in the literature. The author is interested then in the linearization along orbits of a holomorphic endomorphism of the projective space. In a certain way it generalizes the linearization around a stationary point; the orbit now plays the role of the point. The phenomenon is in statistical nature and makes fully intervene the measure of balance of the endomorphism and its Lyapunov exponents. One recovers the problems of resonance that this time expresses itself at the level of the Lyapounov exponents. The reader will find there “historical” elements and some landmarks permitting to move in the literature on the topic.

MSC:
37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
37F25 Renormalization of holomorphic dynamical systems
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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