A short proof of the Ornstein theorem
This paper is another refinement in the development of approaches to the fundamental theorem of \textit{D. Ornstein} [Adv. Math. 4, 337--352 (1970; Zbl 0197.33502)] that two independent identically distributed processes (in the case at hand, of finite entropy) of equal entropy are measure-theoretically isomorphic. Here the approach of Burton and Rothstein [unpublished] exploiting the Baire category theorem in the space of factor maps is refined to give a short and self-contained proof, using only basic results in ergodic theory, of the residual Sinai theorem: the set of factor maps from an ergodic process of entropy \(h\in(0,\infty)\) to an iid process with entropy less than or equal to \(h\) is residual in the space of all ergodic joinings of the two systems. The isomorphism theorem is a direct consequence. While the brevity and lack of prerequisites beyond basic ergodic theory (entropy and the \(L^1\) von Neumann ergodic and McMillan theorems and the marriage lemma) is impressive, this remains a conceptually sophisticated part of the subject.
scientific article; zbMATH DE number 6076542