Orbit Methods in Dynamics

轨道方法试图通过组合、几何和统计技术研究轨道的大尺度结构来理解动力系统。首席调查员开发了一系列工具来探索这种轨道空间，并研究这些结构的等价性概念，以便测量和控制这些结构的扭曲或重新排列。该计划的目标是将这些轨道方法扩展到尽可能广泛的视角，并将它们应用于回答有趣的问题。这些方法在研究保存概率度量的系统方面有着重要的历史。两个活跃的新方向是将这一理论推广到几乎连续的情形，以及康托极小理论。这两种方法都涉及改进任何超出可测性的允许同构的正则性，在第一种情况下是以概率1连续的，在第二种情况下实际上是连续的。推广这些方法的另一个广泛尝试是研究散布在叶片上的概率度量。扩散量度给出了叶片上的质量分布。一个去掉了所有的条件，尽管将动力学和度量联系在一起。对于一维树叶，可以在这个一般的上下文中证明一个遍历定理，并且将其推广到更高的维度似乎是可能的。对于将轨道方法扩展到相当一般的大型结构而言，这是一个潜在的非常富有成效的领域。人们还在寻求各种其他方向，包括开始开展应用流体力学方面的合作，使用为测量轨道大规模扭曲而开发的度量标准，创建一系列工具来评估数值模型的准确性。从基因组学到数字成像到地质学，许多背景下都会出现大量数据。类似的阵列出现在数学环境中，如动力学系统状态空间中的轨迹或轨道。轨道方法涉及这样的阵列的失真大小的概念，或者测量两个这样的阵列有多相似的不同方法。人们将扭曲或距离的概念与问题的背景相协调。在这一背景下，工程学和应用科学领域与抽象动力学之间存在着深度交叉的可能性。例如，物种之间的进化距离可以通过它们的基因组相似程度来衡量，在这里，人们必须调整自己所说的相似是什么意思，才能在生物学上说得通。作为另一个例子，在压缩视觉图像中的数据时，只要图像没有严重失真，人们可能愿意损失一些精度。严重的扭曲是什么意思？作为第三个例子，在对河流的曲折进行建模时，必须建立一个模型来捕捉这一复杂现象的真实行为意味着什么。同样，这可以采取经过深思熟虑的模式贴近性概念的形式。这个项目的一个主旨是寻求合作，寻找这种想法的交叉滋养。这已经在基因组学和流体流动领域开始了。

Orbit methods seek to understand dynamical systems by investigating the large-scale structure of their orbits through combinatorial, geometric, and statistical techniques. The principal investigator has developed a range of tools to explore such spaces of orbits and to study notions of equivalence of these structures that allow for measuring and controlling distortions or rearrangments of them. The goal of this program is to extend these orbit methods to as broad a perspective as possible and to apply them to answer interesting questions. These methods have a significant history in the study of systems that preserve probability measures. Two active new directions of work are extension of this theory to the almost continuous case, and the Cantor minimal theory. Both involve improving the regularity of any allowed isomorphisms beyond measurability, in the first case to be continuous with probability one and in the second actually to be continuous. Another broad attempt at extending these methods is to study probability measures diffused on the leaves of a foliation. The diffused measure gives a mass distribution on the leaves. One removes all conditions, though linking the dynamics and the measure. For one-dimensional leaves an ergodic theorem can be proved in this general context, and extending it to higher dimensions appears possible. This is a potentially very fruitful domain for extending orbit methods to quite general large-scale structures. A variety of other directions are pursued, including the beginnings of a collaboration in applied fluid dynamics that uses the metrics developed for measuring large-scale distortion of orbits to create a bag of tools for assessing the accuracy of numerical models.Large arrays of data arise in many contexts, from genomics to digital imaging to geology. Similar arrays arise in a mathematical context as trajectories or orbits in the space of states of a dynamical system. Orbit methods involve notions of the size of distortion of such arrays or different ways of measuring how similar two such arrays are. One tunes the notion of distortion or distance to the context of the problem. There is the possibility of deep cross-fertilization between areas of engineering and applied science and abstract dynamics in this context. For example, the evolutionary distance between species can be measured in terms of how similar their genomes are, where one must tune what one means by similar to make biological sense. As another example, in compressing the data in a visual image one is perhaps willing to lose some precision so long as the picture is not significantly distorted. What does one mean by a significant distortion? As a third example, in modeling the meanderings of rivers one must establish what it would mean for a model to capture the real behavior of this complex phenomenon. Again, this can take the form of a thoughtfully tuned notion of closeness of patterns. One thrust of this project is to seek collaborations that look for such cross-fertilization of ideas. This has already begun in the areas of genomics and of fluid flow.