AI 中文总结
本文从几何方法研究刘维尔域上的辛动力学，采用管构造构建高维刘维尔域，建立动力学稳定性，探讨拓扑熵关系、辛容量及度量范畴化，还研究了管的过滤辛同调与哈密顿弗勒同调的关系。
AI 中文摘要
这是一系列从几何方法研究刘维尔域$(W,\lambda)$上辛动力学的论文中的第一篇。我们采用厄舍的管构造从$(W,\lambda;H)$构建一个高维刘维尔域，其中$H$是$(W,\lambda)$上的哈密顿函数，可能是非自治的。我们建立了关于哈密顿量上新的汤普森型伪度量的动力学稳定性，其下限由巴拿赫 - 马祖尔型距离界定。我们还探讨了哈密顿动力学及其诱导的瑞布动力学的拓扑熵之间的关系，研究了管构造下的辛容量，特别是格罗莫夫宽度，并通过广义管构造（称为隧道构造）对汤普森型度量进行了范畴化。最后，我们研究了所得管的过滤辛同调与给定基$(W,\lambda)$上输入哈密顿量及其迭代的哈密顿弗勒同调之间 的关系。
英文摘要
This is the first in a series of papers studying symplectic dynamics on Liouville domains $(W, λ)$ from a geometric approach. We employ Usher's tube construction to build a higher-dimensional Liouville domain from $(W,λ; H)$, where $H$ is a Hamiltonian function on $(W, λ)$, possibly non-autonomous. We establish dynamical stabilities with respect to a new Thompson-type pseudo-metric on Hamiltonians, bounded below by Banach--Mazur type distances. We also explore relations between the topological entropies of the Hamiltonian dynamics and its induced Reeb dynamics, examine symplectic capacities, especially the Gromov width, under the tube construction, and give a categorification of the Thompson-type metric via a generalized tube construction (called the tunnel construction). Finally, we investigate the relation between the filtered symplectic homology of the resulting tube and the Hamiltonian Floer homologies of the input Hamiltonians, as well as their iterates, on the given base $(W, λ)$.
Comments70 pages, 3 figures
