Measure Theory and Geometric Topology in Dynamics

动力系统理论的目标是描述一个物体或一组服从运动规律的粒子的长期行为。通常情况下，对粒子位置和速度的全面了解并不准确。当粒子在一个时间单位(比方说一天)之后运动时，新的位置是不确切知道的。本课题对这类系统进行了研究。我们的目标是证明粒子集合的长期行为受相空间中的某些概率分布控制。分布应该具有良好的几何属性。虽然这些系统的“混沌”行为从定义上是不可能被准确预测的，但是有一些系统可以用代数方程来描述，研究人员可以在那里以概率的方式研究其长期行为。PI计划研究这些随机迭代，并给出与之相关的平稳度量的精确(尽可能精确)描述，特别是它的几何性质。一旦建立了几何性质，就可以推出一些统计性质作为推论，例如，几乎每个轨道相对于体积的均匀分布。这个项目将向另外两个方向发展。一个涉及具有多维时间的动力系统(即高阶群的作用)，另一个涉及表现出某种一致(虽然可能是部分)双曲性的经典动力系统(即一维时间)。研究中的统一主题是所使用的技术，这些技术涉及系统的测量、理论和拓扑性质之间的相互作用。在所有情况下，都需要描述相关的不变度量，这如何影响拓扑动态，以及最终如何影响环境流形的拓扑。PI还寻求中间交互，例如，流形的拓扑如何对动力学施加限制。

The goal of Dynamical Systems theory is to describe the long run behavior of a body or of a collection of particles subject to laws of motion. It is often the case that full knowledge of the particles' position and velocity are not exact. As a particle moves after a unit of time (say a day), the new position is not known exactly. In this project such systems are studied. The goal is to show that the long run behavior of the collection of particles is governed by some probability distribution in the phase space. The distribution should have nice geometric properties. While the "chaotic" behavior of these system is by definition impossible to be exactly predicted there are systems which are described by algebraic equations where researchers can probabilistically study the long run behavior.The random iteration of diffeomorphisms is an important model for different types of dynamical systems. The PI plans to study these random iterations and gives a precise (as precise as possible) description of the stationary measure associated with it, particularly its geometric properties. Once geometric properties are established, some statistical properties can be deduced as corollaries, for example, the equidistribution of almost every orbit with respect to volume. This project will develop in two additional directions. One concerns dynamical systems with multidimensional time (i.e. actions of higher rank groups) and the other concerns classical dynamical systems (i.e. one dimensional time) displaying some uniform (though possibly partial) hyperbolicity. The unifying theme in the study is the techniques used which involve the interactions between measure theoretical and topological properties of a system. In all cases a description of relevant invariant measures is desired, how this impacts the topological dynamics and ultimately how this affects the topology of the ambient manifold. The PI also seeks intermediate interactions, for instance how the topology of the manifold imposes restrictions on the dynamics.