• holomorphic dynamics
• small divisors
• normal forms
• linearization
• resonances
• sistemi dinamici
• piccoli divisori
• linearizzazione
• normalizzazione
• oggetti locali
• convergenza
• divergenza

In this thesis, one may find a survey of the results about normalization of holomorphic vector fields obtained by A. D. Bryuno and clearly reviewed by J. Martinet. Also we extend many facts to the case of holomorphic maps.

The existence of transformations that reduce an holomorphic vector field in a neighborhood of a singular point (or an holomorphic diffeomorphism in a neighborhood of a fixed point) to a simple form, a normal form, was initiated by Poincaré in his doctoral thesis and has subsequently been investigated by many authors. Brjuno's papers contain a definitive study of the existence of formal transformations and their convergence or divergence in the case of vector fields.

H. Rüssmann proved a similar theorem for diffeomorphisms in the linear case. We start the study of a possible Bruno-Rüssmann theorem in the resonant case.