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

一个表面的部分是一个有用的设备，了解一个动力系统，一个概念介绍了庞加莱在他的研究的3体问题。不幸的是，截面的表面通常很难找到。著名的工作霍费尔-Wysocki-Zehnder在90年代后期的相关表面部分在汉密尔顿系统与对象从邻近领域的辛几何，所谓的伪全纯曲线（这是解决方案，以一定的偏微分方程和特殊的极小曲面）。这开启了通过一种叫做有限能量叶理的小工具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，其中一个问题由来已久。不幸的是，目前的方法，用于构建有限的能量叶关键缺乏一定的控制有限能量叶的渐近行为，我们相信强烈限制应用动态。我们的建议探索了一个新的框架，用于构建有限能量叶理所需的渐近控制。这个建议的第一个目的是这样做的2维哈密顿系统。一个关键的新想法将是采取非辛概念的工作LeCalvez联合收割机它与一个既定的工具从辛几何称为弗洛尔同源。这将导致一个有趣的新的混合版本的后者。我们所指的勒卡尔韦兹的工作是在2000年左右，他发现了一个对象的表面动力学，共享许多功能与有限的能源叶理，但显着没有直接关系的伪全纯曲线，或微分方程。不过，我们将纳入一个概念，从他的方法到辛范畴。除了有助于解决我们的问题外，这也应该是独立的利益。我们建议的第二个目的是将这些调查扩展到一类称为Reeb流的3维系统，这是一个自然的Hamilton系统。这里没有直接类似的勒卡尔韦兹工作，但在辛的一面，我们将取代弗洛尔同调接触的同源性。第三个目的是探索我们提出的Floer和接触同源性的另一个新变体，它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.