Non-degeneracy of closed orbits for generic potentials
[Non dégénérescence des orbites périodiques pour un potentiel générique]
Bernard, Patrick1
1PSL Research University, Université Paris-Dauphine, CEREMADE (UMR CNRS 7534), 75775 Paris Cedex 16, France
Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 363-393

We prove that Mañé generic convex Hamiltonians have only non-degenerate periodic orbits on a given energy level. This result was stated, but not proved, in the literature.

On démontre qu’un hamiltonien convexe générique au sens de Mañé n’a que des orbites périodiques non dégénérées sur un niveau d’énergie donné. Ce résultat a déjà été énoncé, mais pas démontré, dans la littérature.

Reçu le :
Accepté le :
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DOI : 10.5802/jep.255
Classification : 70H12
Keywords: Hamiltonian systems, periodic orbits, generic properties, classical systems
Mots-clés : Systèmes hamiltoniens, orbites périodiques, propriétés génériques, systèmes classiques

Bernard, Patrick  1

1 PSL Research University, Université Paris-Dauphine, CEREMADE (UMR CNRS 7534), 75775 Paris Cedex 16, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Bernard, Patrick. Non-degeneracy of closed orbits for generic potentials. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 363-393. doi: 10.5802/jep.255

[1] Abraham, Ralph Transversality in manifolds of mappings, Bull. Amer. Math. Soc., Volume 69 (1963), pp. 470-474 | DOI | Zbl | MR

[2] Abraham, Ralph Bumpy metrics, Global Analysis (Berkeley, Calif., 1968) (Proc. Sympos. Pure Math.), Volume XIV, American Mathematical Society, Providence, RI, 1970, pp. 1-3 | Zbl | MR

[3] Anosov, D. V. Generic properties of closed geodesics, Izv. Akad. Nauk Armjan. SSR Ser. Mat., Volume 46 (1982) no. 4, pp. 675-709 English transl.: Math. USSR-Izv. 21 (1983), no. 1, p. 1–29 | MR | Zbl

[4] Aslani, Shahriar; Bernard, Patrick Normal form near orbit segments of convex Hamiltonian systems, Internat. Math. Res. Notices (2022) no. 11, pp. 8196-8208 | DOI | Zbl | MR

[5] Aslani, Shahriar; Bernard, Patrick Bumpy metric theorem in the sense of Mañe for non-convex Hamiltonians, 2022 | arXiv

[6] Bolotin, S. V.; Kozlov, V. V. Libration in systems with many degrees of freedom, J. Appl. Math. Mech., Volume 42 (1978) no. 2, pp. 256-261 | DOI | Zbl

[7] Contreras, Gonzalo Geodesic flows with positive topological entropy, twist maps and hyperbolicity, Ann. of Math. (2), Volume 172 (2010) no. 2, pp. 761-808 | DOI | Zbl | MR

[8] Devaney, Robert L. Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., Volume 218 (1976), pp. 89-113 | DOI | MR | Zbl

[9] de Gosson, Maurice Symplectic geometry and quantum mechanics, Operator Theory: Advances and App., 166, Birkhäuser Verlag, Basel, 2006 | DOI | MR

[10] Kozlov, V. V. The principle of least action and periodic solutions in problems of classical mechanics, J. Appl. Math. Mech., Volume 40 (1976), pp. 363-370 | DOI | Zbl

[11] Kupka, Ivan Contribution à la théorie des champs génériques, Contrib. Differential Equations, Volume 2 (1963), pp. 457-484 | Zbl | MR

[12] Lazrag, Ayadi; Rifford, Ludovic; Ruggiero, Rafael O. Franks’ lemma for C 2 -Mañé perturbations of Riemannian metrics and applications to persistence, J. Modern Dyn., Volume 10 (2016), pp. 379-411 | DOI | Zbl | MR

[13] Oliveira, Elismar R. Generic properties of Lagrangians on surfaces: the Kupka-Smale theorem, Discrete Contin. Dynam. Systems, Volume 21 (2008) no. 2, pp. 551-569 | DOI | MR | Zbl

[14] Rifford, Ludovic; Ruggiero, Rafael O. Generic properties of closed orbits of Hamiltonian flows from Mañé’s viewpoint, Internat. Math. Res. Notices (2012) no. 22, pp. 5246-5265 | DOI | Zbl | MR

[15] Robinson, R. Clark Generic properties of conservative systems, Amer. J. Math., Volume 92 (1970), pp. 562-603 | DOI | Zbl

[16] Robinson, R. Clark Generic properties of conservative systems. II, Amer. J. Math., Volume 92 (1970), pp. 897-906 | DOI | MR | Zbl

[17] Smale, S. Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), Volume 17 (1963), pp. 97-116 | MR | Numdam | Zbl

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