Iteration theory of L-index and multiplicity of brake orbits

作者：Liu, Chungen^*; Zhang, Duanzhi

来源：Journal of Differential Equations, 2014, 257(4): 1194-1245.

DOI：10.1016/j.jde.2014.05.006

• Brake orbit
• Maslov-type index
• Bott-type iteration formula
• Convex symmetric domain

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 [n/2] + 1 geometrically distinct brake orbits on every C-2 compact convex symmetric hypersurface Sigma in R-2n satisfying the reversible condition N Sigma = Sigma. 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 [1/2] + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in 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.

• 出版日期2014-8-15
• 单位南开大学