ABSTRACT

A powerful theory due to Ornstein and his collaborators has been successfully applied to many random systems to show that they are isomorphic to independent and identically distributed systems. That is, in disguise, up to a change of coordinates, Ornstein theory tells us that many random and seemingly deterministic and mechanical systems of interest have the same behavior as sequences of dice roll; these random systems are said to be Bernoulli. Ornstein theory is the analogue of a dictionary which can be used tell us that the sentences “雪是白色的” and “The snow is white” have the same meaning. The isomorphism, the dictionary provided by Ornstein’s theory, may not be finitary, that is, effectively realizable in practice. Ornstein's dictionary is not a simple one; in fact, it is more like a mysterious black box. If we were to use Ornstein's dictionary to translate an essay, one would have to read the entire essay and understand its complete meaning just to translate one word. Despite the large number of systems known to be Bernoulli, there are only a handful of cases where explicit finitary isomorphisms have been constructed. In this talk, we will discuss classical and recent constructions, and some long standing open problems