Complete Integrability of Diffeomorphisms and Their Local Normal Forms
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Complete Integrability of Diffeomorphisms and Their Local Normal Forms Kai Jiang1,2 · Laurent Stolovitch2 A la mémoire de Walter Received: 22 April 2020 / Revised: 25 May 2020 / Accepted: 30 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of Kn (K = C or R). We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp. a smooth) transformation if the initial diffeomorphisms are analytic (resp. smooth). Keywords Local Dynamical Systems · Fixed point · Normal form · Complete integrability · Hyperbolicity Mathematics Subject Classification 37C05 · 37C79 · 37C25 · 37D99 · 37F44 · 58K50
1 Introduction When studying dynamical systems with continuous time (i.e. systems of differential equations) or discrete time (i.e. diffeomorphisms), special solutions, such as fixed points also called equilibrium points, attract a lot of attention. In particular, one needs to understand the behavior of nearby solutions. This usually requires some deep analysis involving normal forms [2], which are models supposed to capture the very nature of the dynamics to which the initial dynamical system is conjugate. When considering analytic or smooth dynamical systems, one needs extra assumptions in order to really obtain dynamical and geometrical
This work has been supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference Number ANR-15-IDEX-01 and by ANR project BEKAM with the reference number ANR-15-CE40-0001-03.
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Laurent Stolovitch [email protected] Kai Jiang [email protected]
1
Beijing International Center for Mathematical Research, Peking University, Beijing, China
2
Laboratoire J.A. Dieudonné, UMR CNRS 7351, Université Côte d’Azur, Nice, France
123
Journal of Dynamics and Differential Equations
information on the initial dynamical system via its normal form. These assumptions can sometimes be understood as having a lot of symmetries. This led to the concept of integrability. In the framework of differential equations or vector fields, a first attempt to define such a notion for Hamiltonian systems is due to Liouville [18]. This led, much later, to the now classic Liouville–Mineur–Arnold theorem [1] which provides action-angle coordinates by a canonical transformation. For a general concept of action-angle coordinates we refer to [31]. Vey studied in the groundbreaking work [24], a family of n Poisson commuting analytic Hamiltonian functions in a neighborhood of a common critical point. Under a generic condition on their Hessians, he proved that the family can be simultaneously transformed into a (Birkhoff) normal form. This system of Hamiltonians has to be understood as “completely integrable system”. Later, Eliasson, Ito,
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