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都是概率空间；这样一个空间中的点（或状态）被连接成我们称之为“轨道”的类，这些轨道具有大规模的几何或组合结构。这种结构可以是可数或连续群作用的轨道结构，也可以是有限到一个自同态的逆象的二叉树结构。这种结构可以是拓扑结构，或者轨道上的度规结构，也可以是树结构或图结构，状态作为图的节点。PI已经开发了一系列方法来研究这些轨道空间的大规模统计特性，并研究允许轨道扭曲或重排的这些结构的等效概念。本研究的目标是将这些方法扩展到尽可能广泛的视角，并应用它们来回答有趣和重要的问题。这些方法有着悠久的历史。他们允许将Bernoulli自同构的Ornstein同构理论扩展到自同构（与C. Hoffman合作），并提供了一个工具，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可服从群的作用（与B. Weiss合作）。这个建议提出了一系列自然的前进方向。自同态逆象对树的推广可以推广到一般树或图的研究以及这些树的各种几何相似性概念。最近的工作指出了有效推广到非奇异和甚至奇异动力学的方法。我们也可以将标记空间和指标空间推广为连续空间，从而将布朗运动作为一致自同态的连续模拟来研究。迄今为止的工作和提出的工作将继续推进我们对可测量动力学和可测量轨道结构的理解。这种结构在数学和其他硬科学中很常见。遍历理论和可测量动力学的根源在于热力学、天体力学、概率论和功能分析。它们在数学中的应用包括代数、组合学、数论、几何、概率论和统计学。在数学之外，在物理、化学、电子工程和遗传学中也有应用。研究生的培养也是这个提议的核心。PI、系里和马里兰大学已经表现出对多样性的坚定承诺，并对学生进行适当的培训和指导

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