The Structure of Bernoulli Systems

In the wake of D. Ornstein's proof five years ago that Bernoulli shifts with the same entropy are isomorphic came a flood of new results concerned in one way or another with Bernoulli shifts. Perhaps most surprising has been the discovery that so many of the classical examples of ergodic theory, as well as many "physical" systems, are isomorphic to Bernoulli shifts. We shall not discuss this aspect in any detail and merely mention some of the systems that have been shown to be Bernoullian: (a) ergodic automorphisms of T [4], T°° [7]; (b) the geodesic flow on surfaces of negative curvature [13]; (c) Anosov flows with smooth measures [16]; (d) two-dimensional billiards with a convex scatterer [3]. Even though many innocent looking questions still remain unanswered, such as [20, Problem 7.2], it seems that the time is ripe to focus on other aspects of Bernoulli systems. In particular, having shown that some physical system is Bernoullian, what does that allow one to say about the system itself? To answer such questions one must dig deeper and gain a better understanding of a Bernoulli system, and here the work has in some sense just begun. While some progress has been made in understanding properties that are universally true for all partitions of a Bernoulli shift, there are many questions, such as the central limit theorem, whose answers depend upon the particular partition chosen. To concretize this last point let us look at the property of weak Bernoulli (WB) (as general references I shall use [12], [17] where the terms not defined here, as well as further references, may be found). It is known that not every partition of a Bernoulli shift is WB, and there are some properties that hold for them that do not hold in general (for example the generators a mentioned at the end of §2(a) cannot be WB, since for WB ß, f]n V m>n <p'ß is trivial). The problem is on the one hand—what else is true specifically for WB partitions and on the other hand—how are we to recognize them? For example if an Anosov flow