A short proof of the Ornstein theorem
type: Item
instance of: scholarly article
title: A short proof of the Ornstein theorem
zbMATH Open document ID: 1258.37006
DOI: 10.1017/S0143385711000265
author: T. Downarowicz
author: J. Serafin
published in: Ergodic Theory and Dynamical Systems
publication date: 2012-09-04T00:00:00Z
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 6076542
zbMATH Keywords: Ornstein theorem
zbMATH Keywords: entropy
zbMATH Keywords: Sinai theorem
MaRDI profile type: Publication
cites work: Q4065756
cites work: Bernoulli schemes of the same entropy are finitarily isomorphic
cites work: Bernoulli shifts with the same entropy are isomorphic
review text: 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 Here the approach of Burton and Rothstein [unpublished] exploiting the Baire category theorem in the space of factor maps is refined to give a short and self-contained proof, using only basic results in ergodic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.
reviewed by: Thomas B. Ward
description: scientific article; zbMATH DE number 6076542