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New Directions in Partially Hyperbolic Dynamics

部分双曲动力学的新方向

基本信息

批准号：
1823150
负责人：
Andriy Gogolyev
金额：
$ 11.74万
依托单位：
Ohio State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-18 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1823150&HistoricalAwards=false
关键词：
New Directions Partially Hyperbolic Dynamics

项目摘要

This project seeks to gain a deeper understanding of chaotic dynamical systems. Chaotic dynamics is abundant in the real world and appears throughout sciences and engineering. It can appear in simple mechanical mechanisms such as a double pendulum or in complicated natural phenomena such as convection currents in the atmosphere. The study of such naturally arising examples is extremely challenging and we are very far from a good understanding of them. The partially hyperbolic dynamical systems that will be studied in this project are simplified mathematical models of chaotic dynamical systems that arise in nature. The emphasis of the project will be on a wide range of problems which are new in the field or have received only marginal attention in the past. Potentially, this research may have impacts in physics, biology, chemistry, engineering and other areas where chaotic dynamical systems arise.Anosov and partially hyperbolic diffeomorphisms are the prime examples of chaotic dynamical systems. In the past decades the study of ergodic and chaotic properties in partially hyperbolic dynamics has flourished, while geometric aspects haven't received as much attention. This will be a wide-ranging program to develop geometric structure theory for partially hyperbolic dynamics. The program includes: (1) further development of the Anosov bundle theory; (2) development of topological and global structural stability for partially hyperbolic diffeomorphisms; (3) initiation of the smooth conjugacy program for 3-dimensional partially hyperbolic diffeomorphisms. Further, the project will to bring partially hyperbolic vision beyond the dominated splitting paradigm and take first steps into the realm of homological and coarse hyperbolicity. Finally, the project will expand work on new examples in partially hyperbolic dynamics. New examples have intrinsic value and can be used as additional ground for the structure theory to be applied and tested. The project will use a diverse blend of dynamics and three and higher dimensional topology.

该项目旨在更深入地了解混沌动力系统。混沌动力学广泛存在于真实的世界中，并贯穿于科学和工程领域。它可以出现在简单的机械机制，如双摆或复杂的自然现象，如大气中的对流。对这些自然产生的例子的研究极具挑战性，我们离很好地理解它们还很远。本项目将研究的部分双曲动力系统是自然界中出现的混沌动力系统的简化数学模型。该项目的重点将放在该领域的新问题或过去只受到很少注意的广泛问题上。混沌动力系统的研究对物理、生物、化学、工程等领域都有潜在的影响，Anosov和部分双曲同态是混沌动力系统的典型例子。在过去的几十年中，部分双曲动力学的遍历性和混沌性的研究已经蓬勃发展，而几何方面没有得到太多的关注。这将是一个广泛的计划，以发展几何结构理论的部分双曲动力学。该方案包括：（1）进一步发展了Anosov丛理论;（2）发展了部分双曲同态的拓扑稳定性和整体结构稳定性;（3）提出了三维部分双曲同态的光滑共轭程序。此外，该项目将使部分双曲视觉超越占主导地位的分裂范式，并采取第一步进入同调和粗双曲的领域。最后，该项目将扩展部分双曲动力学的新示例。新的例子具有内在价值，可以作为应用和检验结构理论的额外依据。该项目将使用多种动力学和三维及更高维度的拓扑结构。