New Directions in Partially Hyperbolic Dynamics

该项目旨在对混沌动力系统有更深入的了解。混沌动力学在现实世界中大量存在，并出现在科学和工程中。它可以出现在简单的机械机制中，如双摆，也可以出现在复杂的自然现象中，如大气中的对流。研究这些自然产生的例子是极具挑战性的，我们离很好地理解它们还很远。本课题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。这个项目的重点将放在广泛的问题上，这些问题是该领域的新问题或过去只受到很少注意的问题。这项研究可能会对物理、生物、化学、工程和其他混沌动力系统出现的领域产生影响。Anosov和部分双曲微分同态是混沌动力系统的主要例子。在过去的几十年里，对部分双曲动力学的遍历和混沌性质的研究蓬勃发展，而几何方面却没有得到足够的重视。这将是一个发展部分双曲动力学几何结构理论的广泛课程。该计划包括：(1)进一步发展Anosov束理论；(2)发展了部分双曲微分同态的拓扑稳定性和全局结构稳定性；(3)建立了三维部分双曲微分同胚的光滑共轭规划。此外，该项目将带来部分双曲的视觉超越主导的分裂范式，并迈出第一步，进入同源和粗糙双曲的领域。最后，该项目将在部分双曲动力学的新例子上扩展工作。新的算例具有内在价值，可以作为结构理论应用和检验的补充依据。该项目将使用动力学和三维及高维拓扑结构的多种混合。

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.