Orbit Methods in Ergodic Theory

PI将研究可测量的轨道空间。这些空间的共同之处在于它们是概率空间；这样的空间的点(或状态)被连接到人们所称的“轨道”类中，并且这些轨道具有大规模的几何或组合结构。该结构可以是可数或连续群作用的轨道的结构，或者可以是某个有限到一个自同态的逆像的二叉树的结构。这种结构可以采取拓扑的形式，或者更强烈地采取轨道上的度量的形式，或者可以是以状态作为图的节点的树或图结构。国际和平研究所开发了一系列方法来研究这种轨道空间上的大规模统计特性，并研究这些结构的等价性概念，这些结构允许轨道扭曲或重新排列。这项研究的目标是将这些方法扩展到尽可能广阔的视角，并将它们应用于回答有趣和有意义的问题。这些方法有着重要的历史。它们允许人们将Bernoulli自同构的Ornstein同构理论推广到自同态(与C.Hoffman共同工作)，并通过轨道转移方法将Z的作用理论的大部分提升到一般离散顺从群的作用(与B.Weiss联合工作)。这项提议提出了一系列自然的前进方向。将自同态的逆象推广到树上，可以推广到研究树或图以及这类树的各种几何相似概念。最近的工作指出了有效地推广到非奇异甚至奇异动力学的方法。我们还可以将标号空间和指标空间推广为连续空间，从而将布朗运动研究为一致自同态的连续模拟。迄今为止的工作，拟议的工作将继续推动我们对可测量的动力学和更一般的可测量的轨道结构的理解。这种结构在数学和其他硬科学中很常见。遍历理论和可测动力学源于热力学、天体力学、概率论和泛函分析。它们在数学中应用于代数、组合学、数论几何、概率和统计学。除了数学，还有在物理、化学、电气工程和遗传学方面的应用。研究生的培训也是这项提议的核心。国际和平研究所、新闻部和马里兰大学表现出对多样性以及对学生的适当培训和指导的坚定承诺。

The PI will study measurable orbit spaces. What is common to these spaces is that they are probability spaces; the points (or states) of such a space are linked into classes which one calls "orbits" and these orbits have a large-scale geometric or combinatorial structure. This structure could be the structure of the orbit of a countable or continuous group action or could be the structure of a binary tree of inverse images of some finite to one endomorphism. This structure could take the form of a topology or more strongly a metric on the orbits or could be a tree or graph structure with the states as nodes of the graph. The PI has developed a range of methods to investigate large-scale statistical properties on such spaces of orbits and to study notions of equivalence of these structures that allow for distortion or rearrangement of the orbits. The goal of this study is to extend these methods to as broad a perspective as possible and to apply them to answer interesting and significant questions. These methods have a significant history. They have allowed one to extend the Ornstein Isomorphism theory for Bernoulli automorphisms to endomorphisms (joint work with C. Hoffman) and give a tool for lifting large parts of the theory of actions of Z to actions of general discrete amenable groups (joint work with B. Weiss) via an orbit transference method. This proposal suggests an array of natural directions to proceed. The generalization to trees of inverse images of endomorphisms can be pushed to a study of trees or graphs in general and various geometric notions of similarity of such trees. Recent work indicates ways to effectively generalize to non-singular and even singular dynamics. One can also generalize both the labeling space and the index space to be continua, leading to a study of Brownian motion as the continuous analogue of a uniform endomorphism. Work to date, and the work proposed will continue to advance our understanding of measurable dynamics and of measurable orbit structures more generally. Such structures are common in mathematics and other hard sciences.Ergodic theory and measurable dynamics have their roots in thermodynamics, celestial mechanics, probability theory, and functional analysis. They have applications within mathematics to algebra, combinatorics, and number theory geometry, probability and statistics. Outside of mathematics, there are applications in physics, chemistry, electrical engineering and genetics. The training of graduate students is central to this proposal as well. The PI, the Department and the University of Maryland have demonstrated a strong commitment to diversity and to the proper training and mentoring of students.1