Bernoulli Scheme - Properties

Properties

Ya. Sinai demonstrated that the Kolmogorov entropy of a Bernoulli scheme is given by

This may be seen as resulting from the general definition of the entropy of a Cartesian product of probability spaces, which follows from the asymptotic equipartition property. For the case of a general base space (i.e. a base space which is not countable), one typically considers the relative entropy. So, for example, if one has a countable partition of the base Y, such that, one may define the entropy as

In general, this entropy will depend on the partition; however, for many dynamical systems, it is the case that the symbolic dynamics is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition.

The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic. The result is sharp, in that very similar, non-scheme systems, such as Kolmogorov automorphisms, do not have this property.

The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different measure-preserving dynamical systems can be judged to be isomorphic to Bernoulli schemes. The result was surprising, as many systems previously believed to be unrelated proved to be isomorphic. These include all finite stationary stochastic processes, subshifts of finite type, finite Markov chains, Anosov flows, and Sinai's billiards: these are all isomorphic to Bernoulli schemes.

For the generalized case, the Ornstein isomorpism theorem still holds if the group G is a countably infinite amenable group.