# Talk:Ornstein theory

The article gives a good summary of the Ornstein Theory.

I think one should distinguish between the "Ornstein Theory" and "Ornstein's Theorem". The latter usually refers to his theorem to the effect that entropy completely classifies Bernoulli actions. The former includes much more such as various criteria for being isomorphic to Bernoulli actions. It was the former that had an enormous impact on ergodic theory lasting to the present and really forms the subject of the article.

Here are some more specific comments:

1. In the Background the sentence beginning with "We need to add a probability structure..." is confusing. If it refers to the Liouville measure then I don't see the relevance of the "non reversibility of real phenmenon". If it refers to something else that should be made more explicit.

2. In the last sentence of the Background the phrase "corresponding points are transformed into corresponding points" is obscure. Replacing it with something like "the action of T is taken to the action of \hat T" would be more informative.

3. In the formula defining the Shannon entropy a minus sign is missing.

4. Matters would be clearer to the general reader if the THEOREM were first stated explicitly in discrete time and then in continuous time. Including the statement as a parenthetical remark in part b. of the THEOREM in continuous time gives a misleading impression.

5. In the list of consequences of the criteria #6 MIXING should be added to Markov Processes to make the statement correct.

6. At the end of the section on Structural Stability the reference should be to [OW1] not [OW2]

7. There is another characterization of Bernoulli - Almost Block Independence - that perhaps is worth mentioning since it is often used in the theory of random fields.

## Second reviewer

This is a very nice presentation, although the going gets tougher towards the end. I second the opinions by the first referee.

The statement of the main theorem is a little awkward - I had a double-take on the first sentence. I suggest: There is an abstract flow \(B_t\) "with the following properties". (Without this, it sounds like we have a trivial theorem followed by remarks about the fact that there exists a flow.) The statement for the infinite-entropy case looks like it could be made more clear.

The "footnotes" in the "Criteria..." section should probably just be parentheses in the text. By contrast, the first footnote (to a. in the finite-entropy statement) should be part of an introductory sentence to the section on "The Ornstein Theorem".

The section on structurel stability (i.e., the explanation of statistical stability) is likely unclear to nonexpert readers.

The "join" notation in the section on "Finitely Determined (F.D.)" should be introduced, or there should be a reference to another scholarpedia entry that introduces it. The title "Finitely Determined (F.D.)" should be completed to something like "Finitely Determined (F.D.) systems". Likewise, the next section title should be something like "very weak Bernoulli (V.W.B.) systems".

"Lebesgue" seems misspelled consistently, and "criterion" (singular) and "criteria" (plural) are mixed up. "ingenuous" should be "ingenious". "more that zero-entropy" should be "more than zero-entropy". There are numerous other little items to be smoothed, maybe there is a volunteer willing to do this.

Most of my coments on the first version have been addressed but the article

still needs a careful proofreading. Here are a few examples:

1. In the contents "foototes" appears instead of "footnotes. 2. The footnote 7 refers to example 7 and not example 6. 3. Footnote 8 is a very vague reference and probably does not refer to the first sentence - where it is currently placed. The attribution of what Kolmogorov did and what Rokhlin-Sinai did should be made a little clearer. Finally while there are several transliterations of Rokhlin' name in Latin characters "Rocklin" is not one of them. 4. In Orbit Equivalence it is worth adding a specific reference to the ORW memoir - since it already appears in the sparse list of technical sources.

Most of my coments on the first version have been addressed but the article

still needs a careful proofreading. Here are a few examples:

1. In the contents "foototes" appears instead of "footnotes. 2. The footnote 7 refers to example 7 and not example 6. 3. Footnote 8 is a very vague reference and probably does not refer to the first sentence - where it is currently placed. The attribution of what Kolmogorov did and what Rokhlin-Sinai did should be made a little clearer. Finally while there are several transliterations of Rokhlin' name in Latin characters "Rocklin" is not one of them. 4. In Orbit Equivalence it is worth adding a specific reference to the ORW memoir - since it already appears in the sparse list of technical sources.