标题： 保角辛系统的几何、拓扑和动力学性质，正常双曲不变流形以及散射映射 显示英文标题

标题： Geometric, topological and dynamical properties of conformally symplectic systems, normally hyperbolic invariant manifolds, and scattering maps

摘要： 共形辛微分同胚$f:M \rightarrow M$将流形$M$上的辛形式$\omega$转换为自身的倍数，$f^* \omega = \eta \omega$。我们假设$\omega$是有界的，因为否则一些结果可能不成立。我们表明流形的拓扑性质、映射的动力学性质和不变流形的几何性质之间存在深刻的相互作用。我们表明，当辛形式不是恰当的时候，可能的共形因子$\eta$与流形的拓扑性质有关。对于某些流形，共形因子被限制为代数数。我们还发现了动力学性质（向量的增长率与$\eta$之间的关系）与辛性质之间的关系。正常双曲不变流形（NHIM）及其（不稳定）流形是组织长期动力行为的重要标志。我们证明，当且仅当速率满足某些配对规则，并且当速率和共形因子满足某些（自然）不等式时，NHIM 是辛的。到NHIM的同宿跃迁通过散射映射进行定量描述。这些映射给出了未来渐近的轨迹作为过去渐近的轨迹的函数。我们证明，即使动力学是耗散的，散射映射也是辛的。我们还表明，如果辛形式是恰当的，那么即使动力学不是恰当的，散射映射也是恰当的。我们在共形辛设置中给出了散射映射的变分解释。 我们还证明了当$\omega$为预辛时，NHIMs 和散射图的类似性质仍然成立。 在具有多种速率的动力系统中 （例如，接近多个共振的准可积系统），预辛几何自然出现。 显示英文摘要

摘要： Conformally symplectic diffeomorphisms $f:M \rightarrow M$ transform a symplectic form $\omega$ on a manifold $M$ into a multiple of itself, $f^* \omega = \eta \omega$. We assume $\omega$ is bounded, as some of the results may fail otherwise. We show that there are deep interactions between the topological properties of the manifold, the dynamical properties of the map, and the geometry of invariant manifolds. We show that, when the symplectic form is not exact, the possible conformal factors $\eta$ are related to topological properties of the manifold. For some manifolds the conformal factors are restricted to be algebraic numbers. We also find relations between dynamical properties (relations between growth rate of vectors and $\eta$) and symplectic properties. Normally hyperbolic invariant manifolds (NHIM) and their (un)stable manifolds are important landmarks that organize long-term dynamical behaviour. We prove that a NHIM is symplectic if and only if the rates satisfy certain pairing rules and if and only if the rates and the conformal factor satisfy certain (natural) inequalities. Homoclinic excursions to NHIMs are quantitatively described by scattering maps. These maps give the trajectory asymptotic in the future as a function of the trajectory asymptotic in the past. We prove that the scattering maps are symplectic even if the dynamics is dissipative. We also show that if the symplectic form is exact, then the scattering maps are exact, even if the dynamics is not exact. We give a variational interpretation of scattering maps in the conformally symplectic setting. We also show that similar properties of NHIMs and scattering maps hold in the case when $\omega$ is presymplectic. In dynamical systems with many rates (e.g., quasi-integrable systems near multiple resonances), pre-symplectic geometries appear naturally.