Multiplicative Ergodic Theory, Dynamics and Applications

乘法遍历理论、动力学和应用

• 批准号: RGPIN-2018-03761
• 负责人: Quas, Anthony
• 金额: $ 2.04万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712546
• 关键词: Multiplicative; Ergodic; Theory; Dynamics; Applications

My research program uses ideas from ergodic theory (the study of the long-term behaviour of dynamical systems that have an invariant measure) to validate and inform the creation of measurement schemes for environmental data. In environmental dynamical systems, regions that mix slowly with the rest of the system (e.g. ocean gyres and cyclones in the atmosphere) have a significant impact on the remainder of the system. Locating these features and measuring their properties is a major challenge due to the difficulty of obtaining fine-grained data. My work involves mathematical techniques for using coarse-grained data measurements together with sophisticated post-processing to recover information that would otherwise have to be found using much finer (and therefore much more expensive) measurements. These ideas have proved highly effective in practice, although there is a theoretical gap in that there is no mathematical proof that what is being measured corresponds to physical reality. My research is concerned with reducing the gap between theory and practice. Along the way, I anticipate being able to make further improvements in the data reconstruction techniques. The mathematical underpinnings of this research are of very considerable interest in their own right. One of the key tools involved in the study, Lyapunov exponents, are a major component of chaos theory. Lyapunov exponents measure the rates of stretching in dynamical systems (and underlie such questions as for how long we can expect the weather forecast to be accurate). There has been a large amount of research on this field over decades, spurred by the widespread availability of fast computers, and a question of very considerable mathematical interest is the extent to which these Lyapunov exponents are stable. That is, if a small change is made to a system, is it the case that the Lyapunov exponents also undergo a small change? Surprisingly, the answer to this question in some important cases is "no": a very small change to the system can result in radical changes to the Lyapunov exponents. This is problematic if one is trying to reconstruct the Lyapunov exponents in a real-world dynamical system from data. A major part of my research is understanding scenarios and types of changes to the system that do lead to stability of Lyapunov exponents. Indeed, there seems to be an important connection between Lyapunov exponents and the slowly mixing regions of environmental dynamical systems mentioned above.

我的研究计划使用了厄基德理论（对具有不变量度的动态系统的长期行为的研究）来验证和告知创建环境数据的测量方案的想法。在环境动力学系统中，与系统的其余部分（例如，大气中的海洋回旋和旋风）缓慢混合的区域对系统的其余部分产生了重大影响。由于难以获得细粒度数据，因此找到这些功能并测量其性质是一个重大挑战。我的工作涉及使用粗粒数据测量结果以及复杂的后处理来恢复信息的数学技术，否则必须使用更精细的（因此更昂贵）的测量结果找到这些信息。这些想法在实践中被证明是非常有效的，尽管理论上的差距是没有数学证据表明所测量的内容与物理现实相对应。我的研究涉及减少理论与实践之间的差距。一路上，我预计能够进一步改进数据重建技术。 这项研究的数学基础本身具有很大的兴趣。研究中涉及的关键工具之一Lyapunov指数是混乱理论的主要组成部分。 Lyapunov指数衡量动态系统中拉伸的速率（以及我们可以期望天气预报准确多长时间的问题）。几十年来，对这一领域进行了大量研究，这受到快速计算机的广泛可用性的刺激，并且这些Lyapunov指数稳定的程度是非常相当大的数学兴趣的问题。也就是说，如果对系统进行了少量更改，那么Lyapunov指数也会经历很小的变化吗？令人惊讶的是，在某些重要情况下，这个问题的答案是“否”：对系统的很小的更改可能会导致对Lyapunov指数的根本性更改。如果一个人试图从数据中重建Lyapunov指数，这是有问题的。我的研究的主要部分是了解系统的场景和类型，这确实导致了Lyapunov指数的稳定性。确实，Lyapunov指数与上述环境动力学系统的缓慢混合区之间似乎存在着重要的联系。