哈密顿微分同胚的不变集与谱刚性
Invariant sets and spectral rigidity of Hamiltonian diffeomorphisms
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中文总结 AI 辅助
该研究解决了Polterovich2002年关于二维哈密顿微分同胚Hofer度量刚性的问题，在高维自治哈密顿流等情形取得进展，相关结果适用于Viterbo谱度量等，运用定量拉格朗日Floer理论完成论证。
中文摘要 AI 辅助
我们解决了Polterovich在2002年提出的关于二维哈密顿微分同胚在Hofer度量下刚性的著名问题：亏格至少为1的曲面上的每个非平凡哈密顿微分同胚，其所有正次迭代在Hofer度量下都与恒等映射分离。在高维情形下，我们在自治哈密顿流的问题上取得进展：证明了辛双曲流形的Hofer非回归性，并完全刻画了单调 toric 流形上矩多面体函数的Hofer回归性。这些结果同样适用于Viterbo谱度量，解决了γ-刚性猜想的若干情形，因此在许多情况下也适用于C⁰度量。我们的方法基于辛可见不变集支配辛非回归现象的理念。在二维情形下，我们利用低维动力学提供的不变环带，并将C⁰辛拓扑方法应用于近似不变的拉格朗日子流形。对于辛双曲流形，不变集由自治哈密顿量的亚水平集和超水平集提供；而在 toric 情形下，它们包含拉格朗日环面纤维。在整个论证中，我们运用了定量拉格朗日Floer理论及其与经典动力学概念的关系。
英文摘要
We solve a well-known question of Polterovich from 2002 regarding the rigidity of Hamiltonian diffeomorphisms in Hofer's metric in dimension two: every non-trivial Hamiltonian diffeomorphism of a surface of genus at least one has all positive iterations separated from the identity in Hofer's metric. In higher dimensions, we make progress on the case of autonomous Hamiltonian flows: we prove Hofer non-recurrence for symplectically hyperbolic manifolds, and completely characterize Hofer recurrence for functions of the moment polytope on monotone toric manifolds. These results apply equally well to Viterbo's spectral metric, settling cases of the $γ$-rigidity conjecture, and therefore in many cases also to the $C^0$-metric. Our approach relies on the philosophy that symplectically visible invariant sets govern symplectic non-recurrence phenomena. In dimension two, we use invariant annuli provided by low-dimensional dynamics, and apply methods of $C^0$ symplectic topology to approximately invariant Lagrangian submanifolds. For symplectically hyperbolic manifolds, the invariant sets are provided by sublevel and superlevel sets of autonomous Hamiltonians, while in the toric case, they comprise Lagrangian torus fibers. Across the arguments, we use quantitative Lagrangian Floer theory and its relation to notions of classical dynamics.
