... in recognition of his paper "Bernoulli shifts with the same entropy are isomorphic" (1970).

Ergodic theory, the study of measure-preserving transformations or flows, arose from the study of the long-term statistical behaviour of dynamical systems. Consider, for example, a billiard ball moving at constant speed on a rectangular table with a convex obstacle. The state of the system (the position and velocity of the ball), at one instant of time, can be described by three numbers or a point in Euclidean 3-dimensional space, and its time evolution by a flow on its state space, a subset of 3-dimensional space. The Lebesgue measure of a set does not change as it evolves and can be identified with its probability. One can abstract the statistical properties (e.g., ignoring sets of probability 0) and regard the state-space as an abstract measure space. Equivalently, one says that two flows are isomorphic if there is a one-to-one measure-preserving (probability-preserving) correspondence between their state spaces so that corresponding sets evolve in the same way (i.e., the correspondence is maintained for all time). It is sometimes convenient to discretize time (i.e., look at the flow once every minute), and this is also referred to as a transformation. Measure-preserving transformations (or flows) also arise from the study of stationary processes. The simplest examples are independent processes such as coin tossing. The outcome of each coin tossing experiment (the experiment goes on for all time) can be described as a doubly-infinite sequence of heads H and tails T. The state space is the collection of these sequences. Each subset is assigned a probability. For example, the set of all sequences that are H at time 3 and T at time 5 gets probability $\large\frac{1}{4}\normalsize$. The passage of time shifts each sequence to the left (what used to be time 1 is now time 0). (This kind of construction works for all stochastic processes, independence and discrete time are not needed.) The above transformation is called the Bernoulli shift B(½, ½). If, instead of flipping a coin, one spins a roulette wheel with three slots of probability $p_{1}, p_{2}, p_{3}$ , one would get the Bernoulli shift B($p_{1}, p_{2}, p_{3}$).

Bernoulli shifts play a central role in ergodic theory, but it was not known until 1958 whether or not all Bernoulli shifts are isomorphic. A N Kolmogorov and Ya G Sinai solved this problem by introducing a new invariant for measure-preserving transformations: the entropy, which they took from Shannon's theory of information. They [used entropy to prove] that not all Bernoulli shifts are isomorphic. The simplest case of the Ornstein isomorphism theorem (1970), states that two Bernoulli shifts of the same entropy are isomorphic.

In 1969, the author solved one of the most difficult problems in ergodic theory when he showed that two Bernoulli shifts with the same entropy are isomorphic. This immediately led to many new results about measure-preserving transformations including proofs that many transformations of physical and mathematical interest are just disguised versions of Bernoulli shifts. In these lectures the author gives a clear and thorough discussion of the ideas which are the basis for this growing new branch of ergodic theory. ... Since the publication in 1974 of this book the subject has continued to grow. There is now an extensive theory of isomorphism of flows, group actions (including Ising models), and even communications channels.