Orbit Methods in Ergodic Theory

遍历理论中的轨道方法

• 批准号: 0400491
• 负责人: Daniel Rudolph
• 金额: $ 19.5万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400491&HistoricalAwards=false
• 关键词: Orbit; Methods; Ergodic; Theory

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

PI 将研究可测量的轨道空间。这些空间的共同点是它们都是概率空间；这种空间的点（或状态）被链接成称为“轨道”的类，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结构，也可以是某些有限对一自同态的逆像的二叉树结构。该结构可以采用拓扑的形式或更强烈地采用轨道度量的形式，或者可以是树或图结构，其中状态作为图的节点。 PI 开发了一系列方法来研究此类轨道空间的大规模统计特性，并研究这些允许轨道扭曲或重新排列的结构的等效概念。 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且重要的问题。 这些方法有着重要的历史。他们允许人们将伯努利自同构的奥恩斯坦同构理论扩展到自同态（与 C. Hoffman 合作），并提供一种工具，通过轨道转移方法将 Z 作用理论的大部分提升到一般离散顺应群的作用（与 B. Weiss 合作）。 该提案提出了一系列自然的前进方向。对自同态逆像树的推广可以推动对一般树或图以及此类树相似性的各种几何概念的研究。最近的工作表明了有效推广到非奇异甚至奇异动力学的方法。人们还可以将标记空间和索引空间概括为连续体，从而将布朗运动研究为均匀自同态的连续模拟。 迄今为止的工作以及拟议的工作将继续增进我们对可测量动力学和更广泛的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根源于热力学、天体力学、概率论和泛函分析。它们在数学、代数、组合学、数论几何、概率和统计中都有应用。除了数学之外，它还可以应用于物理、化学、电气工程和遗传学。 研究生的培训也是该提案的核心。 PI、该系和马里兰大学表现出了对多样性以及对学生进行适当培训和指导的坚定承诺。1