Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation

Abstract

0. Summary. The objects of ergodic theory -measure spaces with measure-preserving transformation groups-wil l 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-