Exemple de point fixe elliptique non topologiquement stable en dimension 4. (English) Zbl 0535.58015

Douady, Raphaël; Le Calvez, Patrice

C. R. Acad. Sci., Paris, Sér. I 296, 895-898 (1983).

The authors introduce a technical condition (T) on diffeomorphisms of \(R^ 4\). Then they show that all these diffeomorphisms have an orbit with the origin in its closure. Now regard \(R^ 4\) as a direct product of the two symplectic manifolds \(R^ 2\). The authors construct a symplectic diffeomorphism of \(R^ 4\) satisfying property (T) and having the origin as elliptic fixed point, thereby showing the existence of a symplectic diffeomorphism of \(R^ 4\) with a nondegenerate elliptic fixed point and an orbit containing this fixed point in its closure.

Reviewer: T.Ratiu

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58C30	Fixed-point theorems on manifolds
57R50	Differential topological aspects of diffeomorphisms

Keywords:

Birkhoff invariants; stable and unstable manifolds; symplectic diffeomorphism; elliptic fixed point