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

Zhang, Duanzhi; Liu, Chungen

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 3, pp. 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 $N=\mathrm{diag}(-I_n,I_n)$. 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 $n=4\text{and}5$, 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.

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

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