Ergodic Theory

摘要 鲁道夫 在未来的三年里，鲁道夫建议在三个方向上集中努力。在证明了多项返回时间定理的方式多倍genericity序列的措施，他现在建议考虑多个复发的结果。刚刚完成了与珍妮特Kammeyer一个多年的项目，奠定了基础的限制轨道等效离散顺从群体，他建议将这一机制的经典和新颖的问题。特别是，应用限制轨道等价方法Vershik的理论，反向filtrations一直非常富有成效，并将继续下去。 最后，他建议努力建设性的问题，特别是外来的 马尔可夫随机场和光滑的例子有关的经典的T，T-逆映射。 大多数有趣的现象都有某种随机性。 遍历理论，如本项目所设想的，是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。 这是通过在更长的时间段内检查进程的行为并尝试分析或重新编码所获得的信息来完成的。 主要研究者已经开发了一种分析方法，该方法考虑了过程中时间的受控扭曲，试图将其修改为某种更简单的形式。 人们所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来保存，从而消除其他属性。 一个特别有趣的例子是，当随机性在空间上出现在晶体或瓷砖中时，“时间”是二维或三维的。 晶体结构或瓦片形状将决定允许什么样的“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于理解 随机过程的各种性质，并似乎适用于许多其他。 追求这些可能性是该项目的主要重点。

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.