凸哈密顿系统的轨道段附近的Normal form
Normal form near orbit segments of convex Hamiltonian systems
摘要：
在余切丛上的哈密顿系统的研究中，通过增加势（仅依赖于基点的函数）来扰动哈密尔顿系统是很自然的。这导致了Ma{}{é} genericity的定义：一个性质是generic的，如果给定一个哈密顿量H，使得H + u满足该性质的势集u是generic的。这一概念主要用于在p中是凸的哈密尔顿算子的上下文中，在这个意义上，$\partial$ 2 pp H在每个点上都是正定的。我们也将把我们的研究限制在这种情况下。小附加势对哈密顿量的扰动与接近于1的正因子对哈密顿量的扰动之间有密切的关系。事实上，哈密顿量H + u和H/（1 - u）具有相同的一级能量面，因此它们在该能量面上的动力学是彼此的重新参数化，这就是莫佩屠斯原理。当H在纤维中是齐次的（对应于芬斯勒度量）或甚至纤维二次的（对应于黎曼度量）时，这一点尤其重要。在这些情况下，哈密顿量的势的扰动对应于度规的共形扰动，直到参数化。广泛研究的一个方面是了解在多大程度上返回地图相关的周期轨道可以扰动通过添加一个小的潜力。这类问题在很大程度上取决于提出这些问题的背景。一些研究最多的情况下，在增加的顺序的困难，扰动一般向量场，扰动的哈密顿系统类内的哈密顿系统，扰动的黎曼度量类内的黎曼度量，马{}{é}扰动凸哈密尔顿。例如，众所周知，每个向量场可以被扰动为仅具有双曲周期轨道的向量场，这是Kupka-Smale定理的一部分，参见[5，13]。在哈密尔顿向量场的上下文中没有这样的结果，但是每个哈密尔顿算子都可以被扰动为仅具有非退化周期轨道（包括迭代轨道）的哈密尔顿算子，参见[11，12]。同样的结果在黎曼度量的上下文中也是正确的：每个黎曼度量都可以被扰动到一个只有非退化闭测地线的黎曼度量，这就是颠簸度量定理，参见[4，2，1]。这个问题是最近才调查的背景下，马{}{é}扰动凸哈密顿，见[9，10]。文[10]证明了同样的结果：若H是凸Hamilton，a是H的正则值，则存在任意小的位势u使得H + u在能量a处的所有周期轨道（包括迭代周期轨道）都是非退化的。[10]中给出的证明实际上与关于黎曼度量扰动的论文中给出的证明相当类似。在所有这些证明中，
英文摘要：
In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Ma{ñ}{é} genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that $\partial$ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians H + u and H/(1 -- u) have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when H is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be perturbed by adding a small potential. This kind of question depend strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, Ma{ñ}{é} perturbations of convex Hamiltonians. It is for example well-known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such result in the context of Hamiltonian vector fields, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated only much more recently in the context of Ma{ñ}{é} perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : If H is a convex Hamiltonian and a is a regular value of H, then there exist arbitrarily small potentials u such that all periodic orbits (including iterated ones) of H + u at energy a are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work
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