Talk:Measure-preserving dynamical system

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[edit] Redirect

Why am I redirected from Kolmogorov entropy to this page? (User:128.227.48.154 on 17 an 2006)

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"? -- 130.94.162.61 23:51, 27 February 2006 (UTC)

[edit] 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.

[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

[edit] Discussion

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 $μ(T (A)) = μ(A)$ 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)