Problems in Measurable Dynamics

提出了一系列可测动力学的研究，它们一般是轨道等价方法的推广和应用。 在这方面，PI帮助开发了两种一般方法。 第一个是限制轨道等价，其中一个地方的限制，各种轨道保持地图允许。 第二种新方法是“轨道转移法”，其中轨道等价是任意的，但相对于与感兴趣的代数正交的代数是可测量的。 其中thesapplications建议这些方法是：1）一个更深层次的，也许是完整的理解弱Pinsker性质的Pinsker 2）一个版本的理论的等距扩展的伯努利自同构的某些类的限制轨道等价，特别是标准的并矢反向过滤。3）扩展的限制轨道等价超越行动的离散顺从组的空间的标记随机图，不需要离散组和非奇异行动。 这里最重要的是，我们应该寻找好的例子，例如非奇异作用的熵理论，大尺度的现象往往表现出随机的行为。晶体通常包含随机夹杂物。 大型数据库经常（总是？）包括随机误差。 基因组可以被建模为一个无限的碱基对序列，在某些地方是随机的，而在某些地方则不是。 一种是用概率的术语把这种结构建模为一个由相连的节点组成的空间，这些节点上附着着标签，如果你愿意的话，可以用颜色来表示，并赋予某种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，不允许节点及其链接的扭曲，只是将特定节点上的标签编码为数组中其他标签中的同一节点。 或者更一般和有趣的是，可以允许节点和链接也扭曲。 通过对扭曲进行一些控制，人们可以设计一种方法来理解这种系统的各种属性。 虽然这个提议的目标是抽象的，但第一个这样的受控扭曲（称为f-bar）是从遗传学中带来的，它被用来衡量DNA分子之间的进化相关性。 这里提出的是各种应用的方法，涉及控制这种标记阵列的失真，以更深入地了解这种随机现象。

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.