Entropies and the Isomorphism Problem for Bernoulli Shifts

Abstract

<p> In 1970, D. S. Ornstein introduced some new approximation concepts which enabled him to establish that the Shannon entropy of endomorphism was a complete invariant for a class of transformations known as Bernoulli shifts. This work of Ornstein contains powerful, deep and elegant techniques which have opened up a new period in the theory of measure preserving transformations, or as it is usually called, in ergodic theory.</p> <p> This thesis contains the study of two new classes of entropies, the γ-entropy and the δ-entropy, where each of these two classes have Shannon's entropy as a member. The algebraic and analytic properties of these entropies and their characterizations are discussed. Finally, the δ-entropy of endomorphism is defined and it has been used in solving the isomorphism problem for Bernoulli shifts. Thus, it is shown that the isomorphism problem for Bernoulli shifts holds not only for one entropy, but for an infinite class of entropies introduced in this thesis.</p>