Ergodic Theory

历经理论

• 批准号: 9706829
• 负责人: Daniel Rudolph
• 金额: $ 10.2万
• 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: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706829&HistoricalAwards=false
• 关键词: Ergodic; Theory

Abstract Rudolph Over the coming three years Rudolph proposes to focus efforts in three directions. Having proved the multi-term return-times theorem by way of multi-fold genericity of sequences for measures, he now proposes to consider multiple recurrence results. Having just completed with Janet Kammeyer a multi-year project to lay the foundations of restricted orbit equivalence for discrete amenable groups, he proposes to apply this machinery to classical and novel problems. In particular, the application of restricted orbit equivalence methods to Vershik's theory of reverse filtrations has been extremely fruitful and will be continued. Finally, he proposes efforts on constructive problems, in particular exotic Markov randomfields and smooth examples related to the classical T,T-inverse map. Most if not all interesting phenomena have some random character. Ergodic theory, as envisioned in this project, is the attempt to ferret out the nature and origin of this randomness in phenomena modeled abstractly as probabilistic processes. This is done by examining the behavior of the process over ever larger blocks of time and attempting to analyze or recode the information gained. The principle investigator has developed an approach to this analysis that considers controlled distortions of time in the process to attempt to modify it to some simpler form. The degree or manner of distortion one allows will determine what properties of the process are preserved or lost. This allows one to focus on some particular properties to save allowing others to be eliminated. One particularly interesting case of this is when randomness appears spatially in a crystal or tiling and "time" is two or three dimensional. The crystal structure or tile shapes will determine what "distortions" are allowed. This "time distortion" approach, called "restricted orbit equivalence" has already shown itself well-suited to understanding a wide variety of properties of random processes and appears applicable to many others. Pursuing these possibilities is the main focus of this project.

摘要 鲁道夫提出，未来三年鲁道夫将重点在三个方向努力。通过测量序列的多重通用性证明了多项返回时间定理后，他现在建议考虑多重递归结果。他刚刚与 Janet Kammeyer 完成了一个为离散服从群的受限轨道等效性奠定基础的多年项目，他提议将这种机制应用于经典和新问题。特别是，限制轨道等效方法在 Vershik 的反向过滤理论中的应用已经取得了极大的成果，并将继续下去。 最后，他提出了在建设性问题上的努力，特别是奇异的马尔可夫随机场和与经典 T,T 逆映射相关的平滑示例。 大多数（如果不是全部）有趣的现象都有一些随机特征。 正如本项目所设想的，遍历理论试图在抽象建模为概率过程的现象中找出这种随机性的本质和起源。 这是通过在更大的时间段内检查流程的行为并尝试分析或重新编码所获得的信息来完成的。 主要研究者开发了一种分析方法，考虑过程中受控的时间扭曲，试图将其修改为某种更简单的形式。 允许的扭曲程度或方式将决定过程的哪些属性被保留或丢失。 这使得人们可以专注于某些特定的属性，从而避免其他属性被淘汰。 一个特别有趣的例子是当随机性出现在晶体或平铺中的空间中并且“时间”是二维或三维时。 晶体结构或瓦片形状将决定允许的“扭曲”。这种被称为“限制轨道等效”的“时间扭曲”方法已经表明它非常适合理解随机过程的各种属性，并且似乎适用于许多其他过程。 追求这些可能性是该项目的主要重点。