Some New Results in the Kolmogorov-sinai Theory of Entropy and Ergodic Theory

ergodic theory : Bernoulli shifts. There is a class of transformations that play a central role in ergodic theory, namely the Bernoulli shifts. The reason for this is partly because they are in a certain sense the simplest examples of measure-preserving transformations and partly because of their role in applications. Definition of Bernoulli shifts. If we are given positive numbers Pu ' ' ' » Pkj where 23*-i £»l> w e define the Bernoulli shift (Pu ' * • , pk) as follows: Let Y be a set with k elements and let us give 880 D. S. ORNSTEIN [November the ith element measure pi. Let Yi, — < i < + °°, be copies of F and let X be the product of the F» with the product measure. Thus each point in X is a doubly infinite sequence of points in F. T will act by shifting each of the above sequences. Tha t is, T will take the sequence {yl } into the sequence {yl } where y I ~yi+v There is a theorem that says that any invertible, ergodic, measurepreserving transformation can be obtained if we modify the above construction by taking F to be countable and by taking some other measure invariant under the shift, instead of the product measure. Thus Bernoulli shifts are simplest in the sense that the product measure is the simplest measure invariant under the shift. Let P be the partition of X into k sets which is obtained as follows : {yi} and {yl } will be in the same atoms of P if yo = yó • I t is easy to see that P , T corresponds to the process of spinning a roulette wheel with k slots of widths pi. Thus from the probability point of view Bernoulli shifts correspond to independent processes. There is a more geometric way of describing Bernoulli shifts, which goes as follows: We will start by describing the Bernoulli shift ( | , | ) . Let X be the unit square. T will send the point (x, y) into (2x, %y) if 0Sx<% and (2# — 1 , f;y+J) if | ^ x < l . We can picture T as follows: We first squeeze down the unit square and we then translate the part that has left the square back onto the part of the square that is now empty.