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，包括它们的周期行为和长时间行为。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，在给定一组初始条件的情况下，动力学预测未来的行为。首席研究员将研究一类环境空间是局部三维的动力系统。特别是，该项目将研究被称为部分双曲微分同胚的东西。这些系统的重要之处在于它们满足强稳定性条件：邻近系统也是部分双曲的，它们是三维空间中唯一满足这些性质的系统。该项目将分析三维部分双曲微分同胚的结构，这是目前非常活跃的一个领域。该项目还将涉及研究生的培训。部分双曲微分同胚(在三维中)是双射光滑映射，它允许不变的收缩方向和扩张方向，以及介于两者之间的另一个不变方向，称为中心方向。这个项目的一个目标是理解这些微分同胚，当它们与恒等式同伦时。特别地，这包括双曲3-流形中的所有这类映射，直到迭代。到目前为止，双曲三维流形是三维中最常见的流形。该项目还将分析最近由PI与T.Barthelme和R.Potrie一起引入的新一类崩溃的Anosov流。当基本群不可解时，这个类包含了三维部分双曲微分同胚的所有已知传递例子。一个目的是证明在某些流形类中，包括所有具有双曲基的Seifert流形中，折叠Anosov流族包含所有部分双曲微分同胚。该项目还将分析这种微分同胚的遍历性的基本性质。PI最近证明了一大类这样的微分同态是遍历的。这个项目的目的是证明某些流形中所有部分双曲微分同胚的遍历性，包括所有具有双曲基的Seifert流形。该项目还旨在显示所有崩溃的阿诺索夫流不等同于停职的遍历性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.