Example of a non-topologically stable elliptic fixed point in dimension 4. (Exemple de point fixe elliptique non topologiquement stable en dimension 4.) (English) Zbl 0535.58015

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.


37J12 Fixed points and periodic points of finite-dimensional Hamiltonian and Lagrangian systems
58C30 Fixed-point theorems on manifolds
57R50 Differential topological aspects of diffeomorphisms