Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems
Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems
复制标题
矢量场的螺旋性保留了规则的接触形式和拓扑共轭光滑动力系统
DOI：
--
复制
发表时间：
2011
影响因子：
0.9
通讯作者：
Peter W. Spaeth
中科院分区：
文献类型：
--
作者：
S. Müller;Peter W. Spaeth
本文计算了三维闭流形上保持正则切触形式的向量场的螺旋度，改进了Gambaudo和Ghys [Enlacements渐近]的结果。Topology 36（6）（1997），1355-1379]将表面合痕的悬置的螺旋度与合痕的卡拉比不变量联系起来。基于这些结果，我们对Arnold在[渐近Hopf不变量及其应用]中提出的两个问题给出了肯定的回答。数学精选苏联5（4）（1986），327-345]。在存在一个也被保持的正则接触形式的情况下，螺旋性扩展到保体积同胚的合痕的不变量，并且在保体积同胚的共轭下是不变的。类似的陈述也适用于表面同位素和表面同形的悬置。这就需要Banyaga和Spaeth开发的拓扑Hamilton和接触动力学技术[关于拓扑严格接触同位素生成Hamilton的唯一性。预印本，2012年]，Buhovsky和Seyfaddini [连续Hamilton流的生成Hamilton的唯一性。J.辛几何。出现，arXiv：1003.2612v2]，Müller [在$L^infty $-范数的哈密尔顿同胚群。J. Korean Math.Soc.45（6）（2008），1769-1784]，Müller and Oh [The group of Hamiltonian homeomorphisms and $C^0$-辛拓扑. 5（2）（2007），167-219]，Müller和Spaeth [Topological contact dynamics I：symplectization and applications of the energy-capacity inequality. Preprint，2011，arXiv：1110.6705v2]和维泰博[关于生成哈密顿流的连续极限的唯一性.国际数学研究所。（2006），34028; Erratum，Int. Math. Res. Not.（2006），38748]。此外，我们推广了一个例子Furstenberg [严格遍历性和变换的环面。Amer. J. Math. 83（1961），573-601]讨论了两个环面到平凡$T^2$-丛的拓扑共轭但非$C^1$-共轭的保面积双同态，并构造了拓扑共轭但非$C^1$-共轭的Hamilton向量场和切触向量场的例子。高维螺旋被认为是简要的文件的最后。
Abstract We compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the isotopy. Based on these results, we provide positive answers to two questions posed by Arnold in [The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4) (1986), 327–345]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume-preserving homeomorphisms, and is invariant under conjugation by volume-preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed by Banyaga and Spaeth [On the uniqueness of generating Hamiltonians for topological strictly contact isotopies. Preprint, 2012], Buhovsky and Seyfaddini [Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symplectic Geom. to appear, arXiv:1003.2612v2], Müller [The group of Hamiltonian homeomorphisms in the $L^infty $-norm. J. Korean Math. Soc.45(6) (2008), 1769–1784], Müller and Oh [The group of Hamiltonian homeomorphisms and $C^0$-symplectic topology. J. Symplectic Geom. 5(2) (2007), 167–219], Müller and Spaeth [Topological contact dynamics I: symplectization and applications of the energy-capacity inequality. Preprint, 2011, arXiv:1110.6705v2] and Viterbo [On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. (2006), 34028; Erratum, Int. Math. Res. Not. (2006), 38748]. Moreover, we generalize an example of Furstenberg [Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601] on topologically conjugate but not $C^1$-conjugate area-preserving diffeomorphisms of the two-torus to trivial $T^2$-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not $C^1$-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.