AI 中文总结
本文针对耗散的共形辛哈密顿流提出两种可积性定义，证明无共轭点的Tonelli哈密顿流自动满足Hopf可积性，并给出渐近Maslov指标结果。
AI 中文摘要
本文旨在研究共形辛哈密顿流在可积性视角下的动力学。由于共形辛哈密顿流的动力学是耗散的，并且与其保守对应物有根本不同，我们首先提出几个更适合该问题的可积性概念。我们将提出两个可积性概念：$C^1$-可积性和Hopf可积性，它们依赖于全局吸引子的存在及其形状。然后，我们的主要定理关注于Tonelli哈密顿量，其共形辛流没有共轭点。我们证明这样的流自动是Hopf可积的。证明是几何性的，研究了垂直子空间在流作用下的长时间演化。它还利用了（折扣）弱KAM理论。我们还建立了关于可积共形辛哈密顿流的渐近Maslov指标的若干结果。最后，我们描述了一些例子，以说明辛哈密顿流与共形辛哈密顿流之间的差异，并说明我们可积性定义的相关性。
英文摘要
The goal of this paper is to study the dynamics of conformally symplectic Hamiltonian flows under the light of integrability. As the dynamics of conformally symplectic Hamiltonian flows are dissipative and differ fundamentally from their conservative counterpart we start by proposing several notions of integrability that are better suited to the problem. We will propose two notions of integrability: $C^1$-integrability and Hopf integrability, that depend on the existence of a global attractor and on its shape. Then our main theorem focuses on Tonelli Hamiltonians whose conformally symplectic flows do not have conjugate points. We prove that such flows are automatically Hopf integrable. The proof is geometric and studies the long time evolution of vertical subspaces under the flow. It also makes use of (discounted) weak KAM theory. We also establish several results about the asymptotic Maslov index for integrable conformally symplectic Hamiltonian flows. Finally, we describe some examples to illustrate differences between symplectic and conformally symplectic Hamiltonian flows and to illustrate the pertinence of our definitions of integrability.
Comments29 pages, 3 figures
