论文标题
$ \ mathbb {r} $的轨道等价 - 覆盖了Anosov流动和类似双曲线的动作
Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line
论文作者
论文摘要
我们证明了该线上的小组动作的刚性结果，其元素具有我们所谓的“双曲线样”动力学。使用此功能，我们为$ \ mathbb {r} $ - 覆盖的Anosov在3个manifolds上提供了一个光谱刚度定理，以周期性轨道表示的基本组的元素来表征轨道等效流的表征。因此，我们给出了一个有效的标准，以确定$ \ mathbb {r} $ - 覆盖的Anosov流量的自轨道等效的同位素类别，并证明了在任何给定的歧管上的触点流动的有限性。 在与乔纳森·鲍登（Jonathan Bowden）的附录中，我们证明了接触的轨道等效度Anosov流完全对应于相关接触结构的同构。这提供了一个强大的工具，可以将结果转换为Anosov流以接触几何形状，反之亦然。我们通过在接触几何形状中给出两个新的结果来说明它的使用：具有许多不同不同的Anosov接触结构的歧管的存在，回答了Foulon-Hasselblatt-vaugon的问题，以及对接触Anosov结构的触点转换组的虚拟描述，从而推广了Giroux和Massot的结果。
We prove a rigidity result for group actions on the line whose elements have what we call "hyperbolic-like" dynamics. Using this, we give a spectral rigidity theorem for $\mathbb{R}$-covered Anosov flows on 3-manifolds, characterizing orbit equivalent flows in terms of the elements of the fundamental group represented by periodic orbits. As consequences of this, we give an efficient criterion to determine the isotopy classes of self orbit equivalences of $\mathbb{R}$-covered Anosov flows, and prove finiteness of contact Anosov flows on any given manifold. In the appendix with Jonathan Bowden, we prove that orbit equivalences of contact Anosov flows correspond exactly to isomorphisms of the associated contact structures. This gives a powerful tool to translate results on Anosov flows to contact geometry and vice versa. We illustrate its use by giving two new results in contact geometry: the existence of manifolds with arbitrarily many distinct Anosov contact structures, answering a question of Foulon--Hasselblatt--Vaugon, and a virtual description of the group of contact transformations of a contact Anosov structure, generalizing a result of Giroux and Massot.
