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

Zhang, Duanzhi; Liu, Chungen

doi:10.1016/j.anihpc.2013.03.010

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) no. 3, pp. 531-554

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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.

Zbl

DOI : 10.1016/j.anihpc.2013.03.010

Classification : 58E05, 70H05, 34C25
Keywords: Brake orbit, Maslov-type index, Seifert conjecture, Convex symmetric

@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},
     pages = {531--554},
     year = {2014},
     publisher = {Elsevier},
     volume = {31},
     number = {3},
     doi = {10.1016/j.anihpc.2013.03.010},
     zbl = {1300.52006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.03.010/}
}

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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) no. 3, pp. 531-554. doi: 10.1016/j.anihpc.2013.03.010

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