Talk:Measure-preserving dynamical system
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- Because its the only page on WP at this time that even partially defines the Kolmogorov entropy (down near the bottom of the article). linas 18:17, 17 January 2006 (UTC)
- Then it would be better a good idea to put the term Kolmogorov entropy into the article to reduce confusion.
- I thought it was Kolmagorov-Sinai entropy or KS entropy, if it had to be named after somebody. What about "topological entropy"? -- 18.104.22.168 23:51, 27 February 2006 (UTC)
problem with definition of generator
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.
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
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 . Even if one asked there would still exist many such 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 which (i) preserves intersections, unions and complements and (ii) sends to (observe that these are exactly the properties we need) is of type for some surjective while the construction via does not cover all such possible set to set maps.--22.214.171.124 (talk) 10:14, 14 March 2008 (UTC)
Example of Measure-theoretic Entropy
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 126.96.36.199 (talk) 06:15, 17 April 2012 (UTC)
- 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)