Orbit Methods in Dynamics

轨道方法试图通过组合、几何和统计技术研究动力系统轨道的大尺度结构来理解动力系统。首席研究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效概念，以便测量和控制它们的扭曲或反射。这个程序的目标是将这些轨道方法扩展到尽可能广泛的视角，并应用它们来回答有趣的问题。这些方法在研究保留概率测度的系统中有着重要的历史。两个活跃的新方向的工作是扩展这个理论的几乎连续的情况下，和康托极小理论。 两者都涉及到提高任何允许的同构超越可测性的规律性，在第一种情况下是连续的概率为1，在第二种情况下实际上是连续的。 扩展这些方法的另一个广泛的尝试是研究在叶理的叶子上扩散的概率测度。扩散测量给出了叶片上的质量分布。一个删除所有的条件，虽然连接的动力学和措施。 对于一维叶遍历定理可以证明在这个一般的情况下，并将其扩展到更高的维度似乎是可能的。 这是一个潜在的非常富有成效的域扩展轨道方法相当一般的大型结构。 他们还在探索其他各种方向，包括开始在应用流体动力学领域开展合作，利用为测量大规模轨道扭曲而开发的指标，创建一系列工具，用于评估数值模型的准确性。从基因组学到数字成像，再到地质学，在许多情况下都会出现大量数据。 类似的阵列出现在数学背景下的轨迹或轨道的状态空间中的动力系统。轨道方法涉及这样的阵列的失真大小的概念或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调整到问题的背景。在这种情况下，工程和应用科学以及抽象动力学领域之间存在着深入交流的可能性。 例如，物种之间的进化距离可以用它们的基因组有多相似来衡量，人们必须调整相似的含义才能有生物学意义。 作为另一个例子，在压缩可视图像中的数据时，人们可能愿意损失一些精度，只要图像没有显著失真。 严重扭曲是什么意思？ 作为第三个例子，在对河流的蜿蜒进行建模时，必须确定模型捕捉这种复杂现象的真实的行为意味着什么。同样，这可以采取经过深思熟虑调整的模式接近度概念的形式。这个项目的一个重点是寻求合作，寻找这种想法的交叉施肥。 这已经在基因组学和流体流动领域开始。

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.