The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems
The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems
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DOI：
10.1016/j.jfa.2009.01.001
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发表时间：
2008-06
期刊：
影响因子：
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通讯作者：
Guangcun Lu
中科院分区：
文献类型：
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作者：
Guangcun Lu
最近由Franks和Handel[J.Franks，M.Handel，[J.Franks，M.Handel]证明的Conley猜想，哈密尔顿曲面微分同态的周期点，Geom.白杨。7(2003)713-756](对于正亏格的曲面)，Hingston[N.Hingston，Tori，Ann上的哈密尔顿方程的次调和解。Conley猜想，Arxiv：Math.SG/0610956v1(对于闭的辛非球面流形)证明了C3-光滑紧致流形M的余切丛上的C1-哈密顿系统，时间为1周期C2-光滑哈密顿H：R×T*M→R，它是强凸的且在纤维上二次增长.也就是说，我们证明了这样的哈密顿系统有无穷多个可压缩的积分周期解序列，使得它们中的任何一个都不能通过迭代从其他人那里获得。如果对任意(t，q，p)−R×T*M，H也满足H(−t，q，∈p)=H(t，q，p)，则证明了该哈密顿系统的时间-1-映射(如果存在)在T*M的零点上有无穷多个周期点.如果M是C5-光滑的，且DIMM>；1，H是C4类的，且与时间t无关，则对任意τ>；相应的系统存在τ的整数倍周期的可压缩周期解的无穷序列，使得它们中的任何一个都不能通过迭代或旋转从其他人那里获得。这些结果是通过证明H，L：R×TM→R的芬切尔变换的拉格朗日系统的类似结果而得到的，该系统被证明是强凸的，并且在速度上仍有二次增长。
In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713–756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C1-Hamiltonian systems on the cotangent bundle of a C3-smooth compact manifold M without boundary, of a time 1-periodic C2-smooth Hamiltonian H:R×T*M→R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t,q,−p)=H(t,q,p) for any (t,q,p)∈R×T*M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T*M. If M is C5-smooth and dimM>1, H is of C4class and independent of time t, then for any τ>0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L:R×TM→R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.