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Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation.
The Pinsker sigma algebra is defined as the lower sigma algebra that contains all partition P of measurable sets such that $h(T,P)=0$ ( entropy of T with respect to P).

How calculate the Pinsker sigma algebra of shift Bernoulli $\left(\dfrac{1}{2},\dfrac{1}{2}\right)$?

I think that the Pinsker sigma algebra is the sigma algebra of all measurable sets of measure $0$ or $1$.

And another question is the why is the (SA) Pinsker important for ergodic theory?

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math.univ-toulouse.fr/~bressaud/Cours_2010_Septembre.pdf These course notes by Bressaud claim that the Pinsker $\sigma$-field is degenerate if and only if $T$ is a $K$-automorphism, and it is known that Bernouilli are $K$. They also claim that the Pinsker $\sigma$-field allows to define the bigger factor of $T$ having zero entropy when $T$ is not $K$ (I am not able to explain what does it rigorously means). –  Stéphane Laurent Jan 31 '13 at 19:47
    
@StéphaneLaurent Thanks for Bressaud's notes (well written). So please don't delete your first comment! (or include the link in your answer). –  Davide Giraudo Feb 19 '13 at 19:31

1 Answer 1

Let $T$ be an invertible measure-preserving transformation (automorphism) on a Lebesgue space $(X,{\cal B},m)$.

Pinsker introduced the $\sigma$-algebra ${\cal P}=\{A \in{\cal B} \mid h(T, \{A,A^c\}=0\}$ in his paper M. S. Pinsker. Dynamical systems with completely positive or zero entropy. Soviet Math. Dokl., 1:937-938, 1960. Elementarily, this $\sigma$-field enjoys the following property: a finite partition is ${\cal P}$-measurable if and only if it has zero entropy (which is actually the definition you have given).

Then Pinsker defined the notion of completely positive entropy for $T$ as being the case when ${\cal P}$ is the degenerate $\sigma$-field; in other words $h(T,P)>0$ for all finite partitions $P$. And he proved that every $K$-automorphism has completely positive entropy.

Rohlin and Sinai proved a finer and stronger result in their paper Construction and properties of invariant measurable partitions. Soviet Math. Dokl., 2:1611-1614, 1962. They proved that ${\cal P}$ is the tail-$\sigma$-field of the $(T,P)$-process for a generating measurable partition $P$ (see Rohlin & Sinai's cited paper and/or Glasner's book). As a consequence, the converse of Pinsker's theorem holds true: an automorphism having completely positive entropy is $K$. So finally an automorphism is $K$ if and only if it has completely positive entropy. See how this theorem is powerful by watching the corollaries given in Rohlin & Sinai's paper; two straightforward corollaries are : 1) a factor of a $K$-automorphism is $K$, 2) the inverse of a $K$-automorphism is $K$.

These are the fundational results about the Pinkser algebra. Now I don't know whether there are more modern ergodic theoretic research involving the Pinkser algebra.

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