Ergodic Properties of Smooth Systems on Manifolds

本课题研究光滑动力系统的混沌特性。光滑动力系统可能表现出混沌行为，这是系统的未来进化与当前状态强烈独立的地方，进化的行为就像一系列独立的抛硬币。首席研究员计划在广泛的自然动力系统中研究这种混沌行为。所开发的方法将导致对基本动力学现象的理解取得进展，并可能应用于其他数学领域，如几何和数论，以及自然科学。首席研究员还计划参与协同活动，包括组织会议和与研究生和博士后一起工作。（均匀）双曲系统的混沌性质现在已经被很好地理解了。对于部分双曲系统，特别是那些在中心空间上具有非平凡（快速）增长的系统，我们所知的要少得多。首席研究员计划建立一个研究部分双曲系统的K和伯努利性质的一般框架，并计划研究抛物系统的混合和光谱性质，试图进一步发展多项式轨道增长系统的一般理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project concerns the chaotic properties of smooth dynamical systems. Smooth dynamical systems may exhibit chaotic behavior, this is where the future evolution of the systems is strongly independent of its present state and the evolution behaves like a sequence of independent coin tosses. The principal investigator plans to study such chaotic behavior for a wide class of natural dynamical systems. The developed methods will result in progress in the understanding of fundamental dynamical phenomena, with possible applications to other mathematical fields such as geometry and number theory, and also to the natural sciences. The principal investigator also plans to be involved in synergistic activities including organizing conferences and working with graduate students and postdocs.Chaotic properties of (uniformly) hyperbolic systems are by now well understood. Much less is known for partially hyperbolic systems, especially those with non-trivial (fast) growth on the center space. The principal investigator plans to create a general framework for studying K and Bernoulli properties for partially hyperbolic systems, and also plans to study mixing and spectral properties of parabolic systems, trying to further develop a general theory for systems of polynomial orbit growth.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.