Classification problems in hyperbolic dynamics

拟议的项目是一个广泛的计划，旨在促进我们对Anosov和部分双曲系统的结构的理解。这些动力系统是混沌动力系统的主要例子，到目前为止，我们已经很好地了解了它们的统计性质。另一方面，双曲动力系统的底层空间的拓扑结构并没有得到很好的理解。例如，所有已知的Anosov微分同态的例子都存在于次流形上，但Anosov微分同态的分类问题仍然悬而未决。PI提出了一个广泛的计划来解决双曲动力学中的分类问题。该程序包括：(1)寻找Anosov微分同态存在的高次同伦障碍；(2)寻找Anosov微分同态存在的几何障碍(如负曲率的存在)；(3)构造准奇异次流形上的扩张映射和Anosov微分同态；(4)寻找部分双曲微分同态的新例子；(5)根据部分双曲微分同态在上同调上的作用对部分双曲微分同态进行分类；(6)根据中心叶空间上的诱导动力学对部分双曲微分同态进行分类。这些例子包括：双摆的运动，弹性碰撞的硬球在矩形盒子中的自由运动(理想气体)，大气对流(著名的洛伦兹吸引子)。对自然界中产生的例子的研究是非常具有挑战性的，仍然有许多谜团。PI建议研究的双曲动力系统是源于自然界的混沌动力系统的简化数学模型。从统计学上讲，双曲型系统的长期行为是相对容易理解的。然而，人们对双曲动力系统的基本空间结构知之甚少。PI提出了一个范围广泛的计划，将促进我们对支持双曲线系统的空间的了解。这项研究可能会对物理、生物、化学、工程和其他出现混沌动力学的领域产生影响。

The proposed project is a broad program that aims at advancing our understanding of the structure of Anosov and partially hyperbolic systems. These dynamical systems are the prime examples of chaotic dynamical systems and, by now, we have good understanding of their statistical properties. On the other hand the topology of the underlying spaces of hyperbolic dynamical systems is not well understood. For example, all known examples of Anosov diffeomorphism live on infranilmanifolds, but the outstanding classification problem of Anosov diffeomorphisms is still open. The PI proposes a wide-ranging program to approach classification questions in hyperbolic dynamics. This program includes: (1) search for higher homotopy obstructions to existence of Anosov diffeomorphism; (2) search for geometric obstructions (such as presence of negative curvature) to existence of Anosov diffeomorphism; (3) construction of expanding maps and Anosov diffeomorphisms on PL-exotic infranilmanifolds; (4) search for new examples of partially hyperbolic diffeomorphism; (5) classification of partially hyperbolic diffeomorphisms according to their action on cohomology; (6) classification of partially hyperbolic diffeomorphisms according to the induced dynamics on the space of center leaves.Chaotic dynamical systems are abundant. The examples include: the motion of the double pendulum, free motion of elastically colliding hard balls in a rectangular box (ideal gas), atmospheric convection (the famous Lorenz attractor). The study of the examples that arise from the nature is extremely challenging and there are still many mysteries. Hyperbolic dynamical systems that the PI proposes to study are simplified mathematical models of chaotic dynamical systems that arise from the nature. Statistically the long term behavior of hyperbolic systems is relatively well understood. However, the structure of the underlying spaces for hyperbolic dynamical systems is poorly understood. The PI proposes a wide-ranging program that will advance our knowledge of spaces that support hyperbolic systems. Potentially this research may have impact in physics, biology, chemistry, engineering and other areas where chaotic dynamics arises.