Measure Theory and Geometric Topology in Dynamics

动力学系统理论描述了粒子服从一个定律的长期行为。为了实现这一点，人们需要理解，在大多数系统中，粒子的位置和速度的全部知识，甚至定律的知识都是不精确的。这种不确定性可以通过引入一些随机性来建模。当一个粒子在一个单位时间后移动时，新的位置只知道一个随机误差。在这个项目中，研究了这种类型的系统，目的是表明粒子的长期行为是由面空间中的一些分布决定的，这些分布具有良好的几何甚至代数性质，从而将一种纯粹的数学和非常强大的技术引入到一个复杂问题的研究中。这项研究的另一个重要特点是分类问题。找到系统的不变量有助于系统的分类，完整的不变量集将给出系统的完整分类。这组不变量将先验地取决于人们想要对系统进行分类的粗糙度。对于混沌系统，期望一组先验非常粗糙的不变量可以自动导致更精细的分类并不罕见。这个项目将利用这一现象，并给出一些刚性的结果。几何不变的措施，系统显示一些双曲去很长的路要走，当试图描述其动态。在该项目中，PI打算通过理解具有一些良好几何性质的不变测度、如何使它们出现以及何时期望它们出现一些唯一性现象来加深对动力系统的了解。这种唯一性伴随着一些一致的性质，特别是关于轨道的等分布信息和系统的良好统计性质。当这项研究与几何对象的分析或更大级别的群体的行动联系起来时，人们通常期望出现良好的刚性特性。首席研究员打算在几个不同的框架中发展这一哲学，但在所有情况下，一些双曲性都会以这样或那样的方式出现，有时来自内在动力学性质，有时来自群体作用，或者来自随机动力学的随机性。在所有情况下，主要研究者试图描述相关的不变测度，并使用此描述对动力系统进行分类，或获得测度支持的几何/拓扑信息，或获得系统的强统计特性。在某些情况下，所有这些都是可以实现的。这个奖项反映了NSF的法定使命，并被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。

Dynamical Systems theory describes the long run behavior of particles subject to a law. In order to achieve this, one needs to understand that, in most systems, the full knowledge of the particle's position and velocity, and even that of the law are not exact. This uncertainty can be modeled by the introduction of some randomness. As a particle moves after a unit of time, the new place is only known up to a random error. In this project this type of systems are studied and the aim is to show that the long run behavior of particles is governed by some distribution in the face space, which has nice geometric and even algebraic properties, thus introducing a purely mathematical and very powerful technology into the study of an otherwise intricate problem. Another important feature of this research is the classification problem. Finding invariants for a system helps its classification and a full set of invariants will give a full classification of the system. This set of invariants will depend a priori on the roughness one wants to classify the system. For chaotic systems it is not uncommon to expect that a set of invariants that is a priori very rough can lead to a finer classification automatically. This project will take advantage of this phenomenon and give some rigidity results.The geometry of invariant measures for systems displaying some hyperbolicity goes a long way when trying to describe its dynamics. In this project, the PI intends to deepen the knowledge of dynamical systems through the understanding of invariant measures with some nice geometric properties, how to make them appear and when to expect some uniqueness phenomena for them. This uniqueness comes attached with some uniform properties especially on information of equidistribution of orbits and nice statistical properties of the system. When this study gets in connection with the analysis of geometric objects or the actions by groups of larger rank, one often expects nice rigidity properties to show up. The principal investigator intends to develop this philosophy in several different frameworks, but in all cases some hyperbolicity will show up one way or another, sometimes from an intrinsic dynamical property, sometimes it will show up from the group acting, or from the randomness in the case of random dynamics. In all cases the principal investigator attempts to have a description of the relevant invariant measures and use this description to classify the dynamical systems or to get information of the geometry/topology of the support of the measure or to get strong statistical properties of the system. In some cases all these can be achieved.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.