Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
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DOI：
10.1007/bf01692494
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发表时间：
1967-03
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影响因子：
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通讯作者：
H. Furstenberg
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作者：
H. Furstenberg
遍历理论的对象--具有保测变换群的测度空间--将被称为过程，拓扑动力学的对象--具有同胚群的紧致度量空间--将被称为流。我们将研究这些对象类的所谓”算术”。人们可以形成过程和流动的产品，也可以谈论要素过程和要素流动。与整数类比，我们可以说两个过程是互质的，如果它们没有共同的非平凡因子。另一个条件是，当这两个过程表现为第三个过程的要素时，它们的产品也表现为要素。在我们的理论中，这两个条件是否等价是未知的。我们选择这些条件中的第二个更有用，并将其称为不相交性。不相交性概念的第一个应用是过程和流的分类。显然，某些类别的过程（流）可以由与其他类别的过程（流）的成员不相交的属性来表征。例如，熵为0的过程就是那些与所有伯努利流不相交的过程。进程不相交性的另一个应用是下面的过滤问题。如果{xn}和{Yn}表示两个平稳随机过程，那么什么时候{xn}可以从{Xn+ Yn}中完全滤出？我们将发现（第一部分，第9节），一个充分条件是所讨论的过程的不相交性。对于流，不相交性的主要应用是研究极小集的性质（第三部分）。考虑单位圆K={z：[zl= 1}上由变换z--~ z 2产生的流。关于这个流的极小集的”大小”，即在z~ z下K不变的闭子集，但不包含具有这些性质的真子集，可以说什么呢？在K中存在无数个这样的极小集。写作z= exp（2~ ri Ean/2n），an= 0，1，我们看到这相当于研究迷你l。这项研究部分由空军科学研究办公室和斯隆基金会资助。
The objects of ergodic theory-measure spaces with measure-preserving transformation groups-will be called processes, those of topological dynamics-compact metric spaces with groups of homeomorphisms-will be called flows. We shall be concerned with what may be termed the" arithmetic" of these classes of objects. One may form products of processes and of flows, and one may also speak of factor processes and factor flows. By analogy with the integers, we may say that two processes are relatively prime if they have no non-trivial factors in common. An alternative condition is that whenever the two processes appear as factors of a third process, then their product too appears as a factor. In our theories it is unknown whether these two conditions are equivalent. We choose the second of these conditions as the more useful and refer to it as disjointness. Our first applications of the concept of disjointness are to the classification of processes and flows. It will appear that certain classes of processes (flows) may be characterized by the property of being disjoint from the members of other classes of processes (flows). For example the processes with entropy 0 are just those which are disjoint from all Bernoulli flows. Another application of disjointness of processes is to the following filtering problem. If {xn} and {Yn} represent two stationary stochastic processes, when can {xn} be filtered perfectly from {Xn+ Yn}? We will find (Part I, § 9) that a sufficient condition is the disjointness of the processes in question. For flows the principal application of disjointness is to the~ tudy of properties of minimal sets (Part III). Consider the flow on the unit circle K={z:[zl= 1} that arises from the transformation z--~ z 2. What can be said about the" size" of the minimal sets for this flow, that is, closed subsets of K invariant under z~ z, but not containing proper subsets with these properties. Uncountably many such minimal sets exist in K. Writing z= exp (2~ ri Ean/2n), an= 0, 1, we see that this amounts to studying the mini-l This research was sponsored in part by the Air Force Office of Scientific Research and by a grant from the Sloan Foundation.