arXiv 1602.044622016-02-18math.SG

关于辛动力学

On Symplectic Dynamics

Stéphane Tchuiaga

摘要：

本文继续对闭辛流形的Banyaga拓扑进行了基础性研究[3]。我们写这篇文章的目的是提供几个辛类似的研究中发现的哈密顿动力学的一些结果。特别地，在不要求辛位移能为正的情况下，指出了BanyagaHofer度量的$L^\infty$版本对研究辛映射序列的$C^0-$极限的辛性质的影响.这个结果是在Hofer-Zehnder [8]中证明的一个结果的辛模拟（对于$\mathbb{R}^{2n}$上的紧支撑的Hamilton同态），然后在Oh-Müller [10]中重新表述为一般的Hamilton同态。进一步，我们将Polterovich [11]中引入的Hamilton路的正则化方法推广到辛合痕上，并利用它证明了定义在辛同胚群的单位连通区上的BanyagaHofer-like范数的两种形式之间的等价性.这一结果在[2]中公布。证明了BanyagaHofer-like几何的唯一性，并给出了Polterovich [11]中一个结果的辛模拟.最后，我们详细讨论了Oh-Müller [10]中某些近似结果的辛类似，并对通量理论作了一些评论。

英文摘要：

This paper continues to carry out a foundational study of Banyaga topologies of a closed symplectic manifold [3]. Our intension in writing this paper is to provide several symplectic analogues of some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out the impact of the $L^\infty$ version of Banyaga Hofer-like metric in the investigation of the symplectic nature of the $C^0-$limit of a sequence of symplectic maps. This result is the symplectic analogue of a result that was proved in Hofer-Zehnder [8] (for compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$), and then reformulated in Oh-Müller [10] for Hamiltonian diffeomorphisms in general. Furthermore, we extend to symplectic isotopies the regularization procedure for Hamiltonian paths introduced in Polterovich [11], and then we use it to prove the equality between the two versions of Banyaga Hofer-like norms defined on the identity component in the group of symplectomorphisms. This result was announced in [2]. It shows the uniqueness of Banyaga Hofer-like geometry, and then yields the symplectic analogue of a result that was proved in Polterovich [11]. Finally, we elaborate the symplectic analogues of some approximation results found in Oh-Müller [10], and make some remarks on flux theory.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文