Classification problems in hyperbolic dynamics

双曲动力学中的分类问题

• 批准号: 1266282
• 负责人: Andriy Gogolyev
• 金额: $ 12.4万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266282&HistoricalAwards=false
• 关键词: Classification; problems; hyperbolic; dynamics

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.

拟议的项目是一个广泛的计划，旨在促进我们对Anosov和部分双曲系统结构的理解。这些动力学系统是混乱动力学系统的主要例子，到目前为止，我们对它们的统计特性有很好的了解。另一方面，双曲动力学系统的基础空间的拓扑尚不清楚。例如，所有已知的Anosov二型示例都属于InfranilManifolds，但是Anosov diffemoilthism的出色分类问题仍然开放。 PI提出了一个广泛的程序，以在双曲线动力学中处理分类问题。该程序包括：（1）搜索Anosov差异性存在的更高同质障碍； （2）寻找几何障碍物（例如存在负曲率的存在），以使Anosov差异性存在； （3）在PL-异性属性构造上构建扩展的地图和Anosov差异； （4）搜索部分双曲线差异的新例子； （5）根据其对同胞的作用，部分双曲线差异分类； （6）根据中心叶子空间上的诱导动力学对部分双曲线差的分类。混合动力学系统丰富。示例包括：双摆的运动，在矩形盒子中弹性碰撞硬球的自由运动（理想气体），大气对流（著名的洛伦兹吸引子）。对自然产生的例子的研究非常具有挑战性，而且仍然有许多谜团。 PI建议研究的双曲动力系统是由自然产生的混乱动力学系统的简化数学模型。从统计学上讲，双曲线系统的长期行为相对充分理解。但是，双曲线动力系统的基础空间的结构知之甚少。 PI提出了一个广泛的程序，该程序将提高我们对支持双曲系统的空间的了解。这项研究可能会影响物理，生物学，化学，工程和混乱动态的其他领域。