Multiple brake orbits on compact convex symmetric reversible hypersurfaces in 𝐑 2n
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, p. 531-554
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In this paper, we prove that there exist at least [n+1 2]+1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in 𝐑 2n for n2 satisfying the reversible condition NΣ=Σ with N= diag (-I n ,I n ). As a consequence, we show that there exist at least [n+1 2]+1 geometrically distinct brake orbits in every bounded convex symmetric domain in 𝐑 n with n2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n=3. As an application, for n=4and5, 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.
DOI : https://doi.org/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},
     publisher = {Elsevier},
     volume = {31},
     number = {3},
     year = {2014},
     pages = {531-554},
     doi = {10.1016/j.anihpc.2013.03.010},
     zbl = {1300.52006},
     language = {en},
     url = {http://www.numdam.org/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, Volume 31 (2014) no. 3, pp. 531-554. doi : 10.1016/j.anihpc.2013.03.010. http://www.numdam.org/item/AIHPC_2014__31_3_531_0/

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