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 isomorphism theorem still holds if the group G is a countably infinite amenable group. Bernoulli automorphism An invertible, measure-preserving transformation of a standard probability space (Lebesgue space) is called a Bernoulli automorphism if it is isomorphic to a Bernoulli shift. Loosely Bernoulli A system is termed "loosely Bernoulli" if it is Kakutani-equivalent to a Bernoulli shift; in the case of zero entropy, if it is Kakutani-equivalent to an irrational rotation of a circle. See also Shift of finite type Markov chain Hidden Bernoulli model References Markov models Ergodic theory Symbolic dynamics