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受运动规律支配的粒子的长期行为。通常情况下，对粒子位置和速度的完全了解是不精确的。当一个粒子在一个单位时间（比如一天）后移动时，新的位置是不知道的。本项目对这类系统进行了研究。目标是表明粒子集合的长期行为是由相空间中的一些概率分布所控制的。分布应该有很好的几何性质。虽然这些系统的“混沌”行为根据定义是不可能被精确预测的，但有些系统是用代数方程描述的，研究人员可以从概率上研究其长期行为。微分同态的随机迭代是求解不同类型动力系统的重要模型。PI计划研究这些随机迭代，并给出与之相关的平稳测量的精确（尽可能精确）描述，特别是其几何特性。一旦建立了几何性质，一些统计性质就可以作为推论推导出来，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），另一个涉及经典动力系统（即一维时间），显示出一些一致的（尽管可能是部分的）双曲性。该研究的统一主题是所使用的技术，涉及测量系统的理论性质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测度，这如何影响拓扑动力学以及最终如何影响环境流形的拓扑。PI还寻求中间相互作用，例如流形的拓扑结构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.