Lagrangian 集合的完成及其在动力学中的应用——基于C. Viterbo的讲座
The completion of the set of Lagrangians and applications to dynamics -- Based on lectures by C. Viterbo
AI总结：
本文介绍Lagrangian集合的完成方法及其在共形辛动力学中的应用，扩展了Birkhoff吸引子的概念，并提出了多个开放性问题。
摘要：
这些讲座的目标是介绍一种将辛流形的Lagrangian子流形集合按照谱度量进行完成的方法，该度量最初由V. Humilière提出，最近又被C. Viterbo重新审视。我们建立了这种完成的一些基本性质，特别是通过γ-支撑的概念，该概念是对Humilière原始概念的细化。然后，我们展示了这些概念在共形辛动力学中的应用，扩展了G.D. Birkhoff、M. Charpentier以及更近期P. Le Calvez定义和研究的Birkhoff吸引子的概念。最后，我们简要提到了Humilière完成的其他应用，并强调了许多开放性问题。这些是基于C. Viterbo在意大利Cetraro（CS）举行的CIME学校“辛动力学与拓扑”上以相同标题进行的讲座所撰写的笔记，时间从2025年6月16日至20日。
英文摘要：
The goal of these lectures is to introduce the completion of the set of Lagrangian submanifolds of a symplectic manifold with respect to the spectral metric first introduced by V. Humilière and recently revisited by C. Viterbo. We establish a number of basic properties of this completion, in particular through the notion of $γ$-support, which we develop as a refinement of Humilière's original concept. We then present an application of these notions to conformally symplectic dynamics, generalizing the notion of Birkhoff attractor as defined and studied by G.D. Birkhoff, M. Charpentier, and more recently P. Le Calvez. Finally, we briefly mention several other applications of the Humilière completion and highlight many open questions. These are notes elaborated from the lectures with the same title given by C. Viterbo at the CIME School ''Symplectic Dynamics and Topology'' held in Cetraro (CS), Italy, from 16th to 20th June 2025.
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