Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $ {\mathbf{R}}^{2n}$

Zhang, Duanzhi; Liu, Chungen

Multiple brake orbits on compact convex symmetric reversible hypersurfaces in

𝐑^{2 n}

Zhang, Duanzhi ; Liu, Chungen

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 531-554 / Harvested from Numdam

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Résumé

In this paper, we prove that there exist at least $[\frac{n + 1}{2}] + 1$ geometrically distinct brake orbits on every $C^{2}$ compact convex symmetric hypersurface Σ in $𝐑^{2 n}$ for $n ⩾ 2$ satisfying the reversible condition $N Σ = Σ$ with $N = diag (- I_{n}, I_{n})$ . As a consequence, we show that there exist at least $[\frac{n + 1}{2}] + 1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $𝐑^{n}$ with $n ⩾ 2$ which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for $n = 3$ . As an application, for $n = 4 and 5$ , we prove that if there are exactly n geometrically distinct closed characteristics on Σ, then all of them are symmetric brake orbits after suitable time translation.

Publié le : 2014-01-01

Zbl 1300.52006

DOI : https://doi.org/10.1016/j.anihpc.2013.03.010
Classification: 58E05, 70H05, 34C25

@article{AIHPC_2014__31_3_531_0,
     author = {Zhang, Duanzhi and Liu, Chungen},
     title = {Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $ {\mathbf{R}}^{2n}$
      },
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {531-554},
     doi = {10.1016/j.anihpc.2013.03.010},
     zbl = {1300.52006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_3_531_0}
}

Zhang, Duanzhi; Liu, Chungen. Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $ {\mathbf{R}}^{2n}$
      . Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 531-554. doi : 10.1016/j.anihpc.2013.03.010. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_3_531_0/

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