×

zbMATH — the first resource for mathematics

Stabilité ou instabilité des points fixes elliptiques. (Stability or instability of elliptic fixed points). (French) Zbl 0656.58020
This long and thorough paper shows that J. Moser’s result on topological stability of an elliptic fixed point of a smooth area preserving plane diffeomorphism has no generalization to symplectic diffeomorphisms of higher dimensional symplectic manifolds. Here, topological stability cannot be read off the jet of infinite order of the diffeomorphism at the fixed point. Under mild nondegeneracy conditions, in dimensions \(\geq 4\), a symplectic diffeomorphism enjoying a totally elliptic fixed point can be modified to one for which the fixed point is topologically stable, as well as to one for which the fixed point is topologically unstable, without changing the jet of infinite order at the fixed point. Moreover, the modifications can be found in any \(C^{\infty}\) neighborhood. (Cf. II.2, II.3). Analogue results hold if an invariant torus, provided by the theorem of Kolmogorov-Arnol’d-Moser, replaces the fixed point, or if Hamiltonian systems are considered. It is pointed out that this paper disproves announcements by V. N. Tkhai.
Reviewer: D.Erle

MSC:
37C75 Stability theory for smooth dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57R50 Differential topological aspects of diffeomorphisms
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] V. I. ARNOL’D , Instability of Dynamical Systems with Many Degrees of Freedom (Dokl. Akad. Nauk. U.S.S.R., vol. 156, n^\circ 1, 1964 , p. 9-12). MR 29 #329 | Zbl 0135.42602 · Zbl 0135.42602
[2] V. I. ARNOL’D , Méthodes mathématiques de la mécanique classique (MIR, Moscou, 1974 ). Zbl 0385.70001 · Zbl 0385.70001
[3] V. I. ARNOL’D et A. AVEZ , Problèmes ergodiques de la mécanique classique , Gauthier-Villars, Paris, 1967 . MR 35 #334 | Zbl 0149.21704 · Zbl 0149.21704
[4] G. D. BIRKHOFF , Dynamical Systems (A.M.S. Colloq. Pub., vol. 9, 1927 ). JFM 53.0732.01 · JFM 53.0732.01
[5] J. B. BOST , Tores invariants des systèmes dynamiques hamiltoniens (Séminaire Bourbaki, n^\circ 639, février 1985 , Astérisque, vol. 133-134, 1986 ). Numdam | Zbl 0602.58021 · Zbl 0602.58021
[6] A. DELSHAMS-VALDÈS , Por qué la difusion de Arnol’d aparece genéricamente en los sistemas hamiltonianos con màs de dos grados de liberdad , Tesa, facultad de matematicas, Universidad de Barcelona, Espagne, 1984 .
[7] R. DOUADY , Une démonstration directe de l’équivalence des théorèmes de tores invariants pour difféomorphismes et champs de vecteurs (C.R. Acad. Sci. Paris, t. 295, 1982 , p. 201-204). MR 84b:58040 | Zbl 0502.58013 · Zbl 0502.58013
[8] R. DOUADY et P. LE CALVEZ , Exemple de point fixe elliptique non topologiquement stable en dimension 4 (C.R. Acad. Sci. Paris, t. 296, 1983 , p. 895-898). MR 85d:58027 | Zbl 0535.58015 · Zbl 0535.58015
[9] V. GUILLEMIN et S. STERNBERG , Geometric asymptotics (A.M.S. Survey, vol. 14, Amer. Math. Soc., Providence, R.I., 1977 ). MR 58 #24404 | Zbl 0364.53011 · Zbl 0364.53011
[10] J. MOSER , On Invariant Curves of Area-preserving Mappings of an Annulus (Nachr. Akad. Wiss. Göttingen, M.P.K., 1962 , p. 1-20). MR 26 #5255 | Zbl 0107.29301 · Zbl 0107.29301
[11] N. N. NEKHOROSHEV , An Exponential Estimate of the Time of Stability of Nearly-integrable Hamiltonian Systems (Russian Math. Surveys, vol. 32, n^\circ 6, 1977 , p. 1-65). MR 58 #18570 | Zbl 0389.70028 · Zbl 0389.70028
[12] J. PALIS , On Morse-Smale Dynamical Systems (Topology, vol. 8, 1969 , p. 385-405). MR 39 #7620 | Zbl 0189.23902 · Zbl 0189.23902
[13] H. POINCARÉ , Les méthodes nouvelles de la mécanique céleste , Gauthier-Villars, Paris, 1892 - 1899 . JFM 24.1130.01 · JFM 24.1130.01
[14] M. SHUB , Stabilité globale des systèmes dynamiques (Astérisque, vol. 56, S.M.F., 1978 ). MR 80c:58015 | Zbl 0396.58014 · Zbl 0396.58014
[15] C. L. SIEGEL et J. MOSER , Lectures on Celestial Mechanics , Springer, Grundlehren Bd., vol. 187, 1971 . MR 58 #19464 | Zbl 0312.70017 · Zbl 0312.70017
[16] V. N. TKHAI , The Stability of Multidimensional Hamiltonian Systems (PMM U.S.S.R., vol. 49, t. 3, 1985 , p. 273-281). MR 87k:58096 | Zbl 0596.70022 · Zbl 0596.70022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.