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

In this paper, we prove that there exist at least $\left[\frac{n+1}{2}\right]+1$ geometrically distinct brake orbits on every ${C}^{2}$ compact convex symmetric hypersurface Σ in ${𝐑}^{2n}$ for $n⩾2$ satisfying the reversible condition $N\Sigma =\Sigma$ with $N=\mathrm{diag}\left(-{I}_{n},{I}_{n}\right)$. As a consequence, we show that there exist at least $\left[\frac{n+1}{2}\right]+1$ 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 $n=4\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}5$, 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.

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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