Partially Hyperbolic Diffeomorphisms, Foliations, and Flows

动力系统领域研究函数在重复迭代下的性质，包括它们的周期性和长时间行为。动力系统有许多应用，例如预测天气行为，如飓风。这些系统遵循一定的规律，在给定的初始条件下，动态预测未来的行为。主要研究者将研究一类周围空间是局部三维的动力系统。特别是，该项目将研究什么是所谓的部分双曲型同态。这些系统的重要性在于它们满足强稳定性条件：附近的系统也是部分双曲的，并且它们是三维中唯一满足这些性质的系统。该项目将分析三维中的部分双曲同构的结构，这是一个目前非常活跃的领域。部分双曲同态（三维）是一种双射光滑映射，它允许不变的收缩和扩张方向，以及介于两者之间的另一个不变方向，称为中心方向。该项目的一个目标是了解这些同伦时，他们是同伦的身份。特别地，这包括双曲3-流形中的所有这样的映射，直到迭代。双曲三维流形是迄今为止三维空间中最常见的流形。该项目还将分析新的崩溃Anosov流类，最近介绍了PI与T。Barthelme和R.波特里这个类包括所有已知的传递的例子部分双曲型同态在三维，当基本组是不是几乎可解的。其中一个目标是证明在某些流形类中，包括所有具有双曲基的塞弗特流形，坍缩的Anosov流族包括所有部分双曲型同态。这个项目也将分析遍历的基本属性，这样的仿射。PI最近证明了一个大类的这种同构是遍历的。该项目旨在证明某些流形中所有部分双曲同态的遍历性，包括所有具有双曲基的塞弗特流形。该项目还旨在显示所有崩溃的Anosov流不等于suspensation.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The field of dynamical systems studies properties of functions under repeated iterations, including their periodic and long-time behavior. Dynamical systems have many applications, for example to predict weather behavior, such as hurricanes. These systems follow certain laws and the dynamics predict future behavior given an initial set of conditions. The principal investigator will study a certain class of dynamical systems where the ambient space is locally three dimensional. In particular, the project will study what are called partially hyperbolic diffeomorphisms. The importance of these systems is that they satisfy strong stability conditions: nearby systems are also partially hyperbolic, and they are the only ones in dimension three satisfying these properties. The project will analyze the structure of partially hyperbolic diffeomorphisms in dimension three, an area that is currently extremely active. The project will also involve the training of graduate students.Partially hyperbolic diffeomorphisms (in dimension three) are bijective smooth maps that admit invariant contracting and expanding directions, and another invariant direction that is in between, called the center direction. One goal of the project is to understand these diffeomorphisms when they are homotopic to the identity. In particular, this includes all such maps in hyperbolic 3-manifolds, up to iterates. Hyperbolic 3-manifolds are by far the most common manifolds in dimension three. The project will also analyze the new class of collapsed Anosov flows, recently introduced by the PI together with T. Barthelme and R. Potrie. This class includes all known transitive examples of partially hyperbolic diffeomorphisms in dimension three, when the fundamental group is not virtually solvable. One goal is to prove that in certain classes of manifolds, including all Seifert manifolds with hyperbolic base, the family of collapsed Anosov flows includes all partially hyperbolic diffeomorphisms. The project will also analyze the fundamental property of ergodicity for such diffeomorphisms. The PI recently showed that a large class of such diffeomorphisms is ergodic. The project aims to show ergodicity for all partially hyperbolic diffeomorphisms in certain manifolds, including all Seifert manifolds with hyperbolic base. The project also aims to show ergodicity for all collapsed Anosov flows not equivalent to suspensions.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.