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arXiv · 0908.0021

Iteration theory of $L$-index and Multiplicity of brake orbits

Chungen Liu·Duanzhi Zhang
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Abstract

In this paper, we first establish the Bott-type iteration formulas and some abstract precise iteration formulas of the Maslov-type index theory associated with a Lagrangian subspace for symplectic paths. As an application, we prove that there exist at least $[\frac{n}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $Σ$ in $\mathbb{R}^{2n}$ satisfying the reversible condition $NΣ=Σ$, furthermore, if all brake orbits on this hypersurface are nondegenerate, then there are at least $n$ geometrically distinct brake orbits on it. As a consequence, we show that there exist at least $[\frac{n}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $\mathbb{R}^{n}$, furthermore, if all brake orbits in this domain are nondegenerate, then there are at least $n$ geometrically distinct brake orbits in it. In the symmetric case, we give a positive answer to the Seifert conjecture of 1948 under a generic condition.

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Chungen Liu, Duanzhi Zhang. 2011-06-02. Iteration theory of $L$-index and Multiplicity of brake orbits. https://arxiv.org/abs/0908.0021

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