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Instability of resonant totally elliptic points of symplectic maps in dimension 4. (English) Zbl 1156.37313
Loday-Richaud, Michèle (ed.), Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. Volume en l’honneur de Jean-Pierre Ramis. Paris: Société Mathématique de France (ISBN 2-85629-168-6/pbk). Astérisque 297, 79-116 (2004).
Summary: A well known Moser stability theorem states that a generic elliptic fixed point of an area-preserving mapping is Lyapunov stable. We investigate the question of Lyapunov stability for 4-dimensional resonant totally elliptic fixed points of symplectic maps. We show that generically a convex, resonant, totally elliptic point of a symplectic map is Lyapunov unstable. The proof heavily relies on a proof of J. Mather of existence of Arnold diffusion for convex Hamiltonians in 2.5 degrees of freedom. The latter proof is announced in [J. N. Mather, J. Math. Sci., New York 124, No. 5, 5275–5289 (2004; Zbl 1069.37044)], but still unpublished.
For the entire collection see [Zbl 1061.00005].

MSC:
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J50 Action-minimizing orbits and measures (MSC2010)
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