Problems in Measurable Dynamics

可测量动力学问题

• 批准号: 0070511
• 负责人: Daniel Rudolph
• 金额: $ 19.17万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070511&HistoricalAwards=false
• 关键词: Problems; Measurable; Dynamics

A range of studies in measurable dynamics of are proposed.They lie generally in extensions and applications of orbit equivalence methods. The PIhas helped to develop two general approaches here. The first is Restricted Orbit Equivalence, whereone places restrictions on the sorts of orbit preserving maps allowed. The second,and new, method is an "orbit transference method" where the orbit equivalence is arbitrarybut measurable with respect to an algebra orthogonal to the algebra of interest. Among theapplications proposed for these methods are:1) a deeper and perhaps complete understanding the the Weak Pinsker property of Thouvenot2) a version of the theory of Isometric Extensions of Bernoulli automorphism for some class of restricted orbit equivalences, in particular for standard dyadic reverse filtrations.3) extensions of restricted orbit equivalence beyond actions of descrete amenable groups to spaces of labeled random graphs, to not necessarilly descrete groups and to non-singular actions. Most important here is that one should look for good examples, for example an entropy theory for nonsingular actions.Large scale phenomena can often exhibit a random behavior. Crystals often include random inclusions. Large data bases often (always?) include random errors. The genome can be modeled as an infinite sequence of base pairs that in places is random and places not so. One models such structures in probabilistic terms as a space of linked nodes to which are attached labels, colors if you will, endowed with some notion of the probability of occurance of some particular local labeling or picture occuring within an infinite picture. It is natural to look for ways of coding and comparing such random arrays among themselves. One can seek extremely rigid encodings that allow no distortion of the nodes and their links, just an encoding of the label at a particular node to that same node in the other labelingof the array. Or more generally and interestingly one can allow the nodes and links to distort as well. By setting some control on the distortions one can craft an approach to understanding a variety of properties of such systems. Although the goals of this proposal are of an abstract nature, one of the first such controlled distortions (called f-bar) was brought over from genetics where it was used as a measure of evolutionary relatedness among DNA molecules. What is proposed here are a variety of applications of methods involving controlled distortion of such labeled arrays to understand such random phenomena more deeply.

提出了一系列可测量动力学的研究。它们通常是轨道等效方法的扩展和应用。pii帮助开发了两种通用方法。第一种是受限轨道等效，即对允许的轨道保留图的种类加以限制。第二种新方法是“轨道转移法”，其中轨道等效是任意的，但相对于与感兴趣的代数正交的代数是可测量的。这些方法的应用包括：1)对thouvenot的弱Pinsker性质的更深入和完整的理解2)对某些受限轨道等价的伯努利自同构的等距扩展理论的一个版本，特别是对标准二进反滤波。3)将离散可服从群作用之外的受限轨道等价扩展到标记随机图空间、不一定离散群和非奇异作用。这里最重要的是，我们应该寻找好的例子，比如非奇异行为的熵理论。大尺度现象往往表现出随机行为。晶体通常含有随机夹杂物。大型数据库经常（总是）包含随机错误。基因组可以被建模成一个无限的碱基对序列，其中有些地方是随机的，有些地方则不是。一种是用概率的方式来模拟这样的结构就像一个连接节点的空间，上面有标签，如果你愿意，也可以是颜色，赋予一些特定的局部标签或图像在无限图像中出现的概率的概念。寻找编码和比较这些随机数组的方法是很自然的。人们可以寻求非常严格的编码，这种编码不允许扭曲节点及其链接，只是将特定节点上的标签编码为数组中其他标记中的同一节点。或者更普遍和有趣的是，可以允许节点和链接扭曲。通过对扭曲设置一些控制，人们可以设计一种方法来理解这种系统的各种特性。虽然这一提议的目标是抽象的，但第一个这样的受控扭曲（称为f-bar）是从遗传学中引入的，它被用来衡量DNA分子之间的进化关系。这里提出的是各种方法的应用，包括这种标记阵列的控制畸变，以更深入地理解这种随机现象。