Subharmonic solutions of Hamiltonian equations on tori

• 成果类型: Article
• 署名作者: Hingston, Nancy
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.529
• 发表日期: 2009
• 页码: 529-560
• 关键词: fixed-point theorem; CLOSED GEODESICS; morse-theory; index

摘要

Let the torus T-2n be equipped with the standard symplectic structure and a periodic Hamiltonian H is an element of C-3(S-1 x T-2n, R). We look for periodic orbits of the Hamiltonian flow (u) over dot(t) = J del H(t, (t)). A subharmonic solution is a periodic orbit with minimal period an integral multiple in of the period of H, with in > 1. We prove that if the Hamiltonian flow has only finitely many orbits with the same period as H, then there are subharmonic solutions with arbitrarily high minimal period. Thus there are always infinitely many distinct periodic orbits. The results proved here were proved in the nondegenerate case by Conley and Zehnder and in the case n = 1 by Le Calvez.