Talk:Measure-preserving dynamical system

message: There seems to be a problem with the following defnition:
"A partition Q is called a generator if μ-almost every point x has a unique symbolic name."
No matter what partition is chosen, every point has a unique symbolic name.
[edit]
Consider the function f defined on the integers by f(x)=x+1 for x odd and f(x)=x-1 for x even. Let Q be the partition into even and odds on Z. We have that each element has the symbolic name EOEOEO... or OEOEOE....(On a side note: this is my first post, and I am not sure if this is the proper way to post on a Talk Page, any help would be appreciated:)
Phoenix1177
:Well, then f fails to generate!? linas (talk) 04:04, 17 July 2012 (UTC)
thread_title:  problem with definition of  ''generator'' 

message: What is written there in Discussion is completely missleading; it is definitelty not the reason why one defines measure preserving transformations via the inverse of T. Even if one asked \mu(T(A))=\mu(A) there would still exist many such T and many of them are even ergodic (or even mixing). For example, have a look at certain interval exchange transformations; as they are all bijections, they preserve the underlaying measure (Lebesgue in this case) in both directions. The only true reason why we define the measure preserving via the inverse images is that any mapping \mathcal{T}\colon\mathcal{P}(X)\to\mathcal{P}(X) which (i) preserves intersections, unions and complements and (ii) sends X to X (observe that these are exactly the properties we need) is of type \mathcal{T}(A)=T^{-1}(X) for some surjective T\colon X\to X while the construction via \mathcal{T}(A)=T(X) does not cover all such possible set to set maps.--140.78.94.103 (talk) 10:14, 14 March 2008 (UTC)
replies: text: I've removed this nonsense. Arcfrk (talk) 10:38, 21 March 2008 (UTC)
replies: text: Perhaps some variant of above comments should be added to the article. linas (talk) 03:43, 17 July 2012 (UTC)
replies: text: Done. 67.198.37.16 (talk) 20:15, 18 September 2020 (UTC)
thread_title: Discussion

message: In the section measure-theoretic entropy it says that the KS entropy of the bernoulli process is log 2. Shouldn't this be the bernoulli map?  — Preceding unsigned comment added by 130.216.209.138 (talk) 06:15, 17 April 2012 (UTC)
replies: text: The map and the process are the same thing, just using different notation, right? Well, almost the same thing, the process is explicitly defined on the cantor set, whereas the map is defined for the reals. But the reals are just a quotient space of the cantor set, so more-or-less the same thing, they differ by a set of measure zero... linas (talk) 03:39, 17 July 2012 (UTC)
thread_title:  Example of Measure-theoretic Entropy 

message: "since every real number has a unique binary expansion"  —Preceding unsigned comment added by 132.72.45.223 (talk) 11:16, 9 November 2010 (UTC)
replies: text: Performed lazy-mans fix of linking almost every instead.linas (talk) 03:48, 17 July 2012 (UTC)
thread_title:  Incorrectness  

message: This article defines measure-entropy, but not metric entropy. 67.198.37.16 (talk) 20:16, 18 September 2020 (UTC)
thread_title:  TODO: Metric entropy 

message: The definition of measure-theoretic entropy relies on the function f(x) = -x log(x).
It would be very helpful if someone knowledgeable on the subject would include in the article an explanation of why this particular function is used. 2601:200:C000:1A0:65EE:1F3C:5AB:B9C0 (talk) 01:12, 10 August 2021 (UTC)
thread_title:  This would be helpful