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Normal form of holomorphic dynamical systems

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41 pages
Normal form of holomorphic dynamical systems Laurent Stolovitch Abstract This article represents the expanded notes of my lectures at the ASI “Hamiltonian Dynamical Systems and applications”. We shall present various re- cent results about normal forms of germs of holomorphic vector fields at a fixed point in Cn. We shall explain how relevant it is for geometric as well as for dy- namical purpose. We shall first give some examples and counter-examples about holomorphic conjugacy. Then, we shall state and prove a main result concerning the holomorphic conjugacy of a commutative family of germs of holomorphic vector fields. For this, we shall explain the role of diophantine condition and the notion of singular complete integrability. 1 Definitions and examples Let us consider the pendulum with normalized constants : ? + sin? = 0 (1) We would like to understand the behavior of the motion for the small oscillations of the pendulum, that is to say when ? is small. We are tempted to say that sin? is well approximated by ? and then we would like to consider the much simpler equation (?) ? +? = 0 instead of (1). If we set ?1 = ? and ?2 = ?˙ , equation (?) can be written as Laurent Stolovitch CNRS UMR 5580, Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France, e-mail: stolo@picard.

  • hamiltonian dynamical

  • nilpotent matrix

  • holomorphic conjugacy

  • dy- namical purpose

  • end point

  • vector fields

  • ∑q?nn

  • ∑q?nn fqxq

  • formal power


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Normal form of holomorphic dynamical systems
Laurent Stolovitch
AbstractThis article represents the expanded notes of my lectures at the ASI “Hamiltonian Dynamical Systems and applications”. We shall present various re-cent results about normal forms of germs of holomorphic vector fields at a fixed point inCn. We shall explain how relevant it is for geometric as well as for dy-namical purpose. We shall first give some examples and counter-examples about holomorphic conjugacy. Then, we shall state and prove a main result concerning the holomorphic conjugacy of a commutative family of germs of holomorphic vector fields. For this, we shall explain the role of diophantine condition and the notion of singular complete integrability.
1 Definitions and examples
Let us consider the pendulum with normalized constants :
¨ θ+sinθ=0
(1)
We would like to understand the behavior of the motion for the small oscillations of the pendulum, that is to say whenθis small. We are tempted to say that sinθ is well approximated byθand then we would like to consider the much simpler ¨ ˙ equation()θ+θ=0 instead of(1). If we setθ1=θandθ2=θ, equation()can be written as
Laurent Stolovitch CNRS UMR 5580, Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France, e-mail: stolo@picard.ups-tlse.fr
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