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Normal form of holomorphic dynamical systems. (English) Zbl 1146.37033
Craig, Walter (ed.), Hamiltonian dynamical systems and applications. Proceedings of the NATO Advanced Study Institute on Hamiltonian dynamical systems and applications, Montreal, Canada, June, 18-29, 2007. Berlin: Springer (ISBN 978-1-4020-6963-5/pbk; 978-1-4020-6962-8/hbk). NATO Science for Peace and Security Series B: Physics and Biophysics, 249-284 (2008).
The paper under review contains the notes of the author’s series of lectures on normal forms of holomorphic dynamical systems (mainly vector fields). After an introductory chapter where examples and motivations are given, the author treats resonances and PoincarĂ©-Dulac normal forms, Hamiltonian vector fields, Diophantine conditions and linearization. Then, starting from theorems of Bryuno and of Vey, he introduces his own notion of singular complete integrability for a family of vector fields.
Very roughly speaking, a family of vector fields is Diophantine if a mixed Bryuno-type condition is satisfied and it is formally complete integrable if there exists a formal diffeomorphism tangent to the identity which conjugates the family to a linear family of vector fields, modulo the ideal of formal first integrals. Then the main theorem of the author is that any Diophantine formally complete integrable system of holomorphic vector fields is holomorphically normalizable.
Many applications of such a theorem (such as recovering Bryuno and Vey’s theorems) are given. Finally, the author presents a rather detailed proof of the theorem.
For the entire collection see [Zbl 1131.37002].

37F75 Dynamical aspects of holomorphic foliations and vector fields
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
32S65 Singularities of holomorphic vector fields and foliations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics