Finite energy foliations, Floer and contact homology in low dimensions, Hamiltonian dynamics, braids

截面是理解动力学系统的有用工具，这是庞加莱在他的三体问题研究中引入的概念。不幸的是，截面的表面通常很难找到。90年代后期，Hofer-Wysocki-Zehnder的著名工作是将哈密顿系统中与截面相关的曲面与来自邻近辛几何域的对象称为伪全纯曲线(这是一类偏微分方程解，是特殊的极小曲面)。这开启了通过一种叫做有限能量叶理的小装置获得截面表面的可能性。申请人最近的结果利用这些联系在遍历理论中的两个问题上取得了进展，其中一个是长期存在的。不幸的是，目前构造有限能量叶的方法严重缺乏与有限能量叶的渐近行为相关的某种控制，我们认为这强烈地限制了动力学的应用。我们的建议探索了一种构造具有理想渐近控制的有限能量叶的新框架。这一建议的第一个目标是为二维哈密顿系统做这件事。一个关键的新想法将是从LeCalvez的工作中提取一个非辛概念，并将其与辛几何中已建立的称为Floer同调的工具相结合。这将导致后者有趣的新混合版本。我们提到的LeCalvez的工作大约是在2000年左右，他在表面动力学中发现了一个物体，它与有限能量的叶状结构共享许多特征，但显著地与伪全纯曲线或微分方程没有直接关系。然而，我们将把他的方法中的一个概念并入辛范畴。除了帮助解决我们的问题外，这也应该是独立的利益。我们建议的第二个目标是将这些研究扩展到一类称为Reeb流的三维系统，这是一类自然的哈密顿系统。这里没有LeCalvez工作的直接类比，但在辛侧，我们将用接触同调代替Floer同调。第三个目标是探索我们提出的Floer和Contact同调的另一种新变体，这在某种意义上与我们上面的研究是横向的。

A surface of section is a useful device for understanding a dynamical system, a concept introduced by Poincaré in his studies of the 3-body problem. Unfortunately, surfaces of section are generally very difficult to find. Celebrated work of Hofer-Wysocki-Zehnder in the late ¿90¿s related surfaces of section in Hamiltonian systems with objects from the neighbouring field of symplectic geometry, called pseudo-holomorphic curves (which are solutions to a certain partial differential equation and are special minimal surfaces). This opened the possibility of obtaining surfaces of section through a gadget called a finite energy foliation. Recent results by the applicant used these connections to make progress on two questions in ergodic theory, one of these long-standing. Unfortunately, current methods for constructing finite energy foliations crucially lack a certain control related to the asymptotic behaviour of finite energy foliations, which we believe strongly limits applications to dynamics. Our proposal explores a new framework for constructing finite energy foliations with the desirable asymptotic control. The first aim of this proposal is to do this for 2 dimensional Hamiltonian systems. A key new idea will be to take a non-symplectic concept from work of LeCalvez¿ and combine it with an established tool from symplectic geometry called Floer homology. This will result in an interesting new hybrid version of the latter. The work of LeCalvez we refer to was around 2000 where he discovered an object in surface dynamics that shares many features with a finite energy foliation, but remarkably has no direct relation to pseudo-holomorphic curves, or differential equations. We will nevertheless incorporate a concept from his approach into the symplectic category. Besides helping to solve our problem, this should also be of independent interest. The second aim of our proposal is to extend these investigations to a class of 3 dim systems called Reeb flows, which is a natural class of Hamiltonian systems to work with. Here there is no direct analogue of LeCalvez¿ work, but on the symplectic side we will replace Floer homology by contact homology. A third aim is to explore another new variant of Floer and contact homology we propose, which is in some sense ¿transverse" to our investigations above.