Pinsker $\sigma$-Algebra

Question

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

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

And another question: Why is the Pinsker $\sigma$-algebra important in ergodic theory?

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

Answer 2

Here are two examples of important applications of the Pinsker partition in smooth ergodic theory, the second one building on the first one.

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Let $M$ be a compact $C^\infty$ manifold with a $C^\infty$ Riemannian metric, $r\in\mathbb{R}_{>1}$, $f:M \to M$ be a $C^r$ diffeomorphism (fractional regularity is to be interpreted as $\text{floor}(r)$-th derivatives satisfying local Hölder estimates as discussed at Definition of Hölder Space on Manifold). Denote by $d_M$ the distance function induced by the Riemannian metric and define for any $x\in M$

$$\mathcal{S}_x(f)=\left\{y\in M\,\left|\, \limsup_{n\to\infty}\dfrac{1}{n}\log d_M(f^n(y),f^n(x))<0\right.\right\}\text{ and }$$

$$\mathcal{U}_x(f)=\mathcal{S}_x(f^{-1})=\left\{y\in M\,\left|\, \limsup_{n\to\infty}\dfrac{1}{n}\log d_M(f^{-n}(y),f^{-n}(x))<0\right.\right\}.$$

By Oseledets' Multiplicative Ergodic Theorem, there is an $f$-invariant measurable subset $\text{Osel}(f)\subseteq M$ of full measure with respect to any $f$-invariant Borel probability measure on $M$ such that for any $x\in\text{Osel}(f)$, both $\mathcal{S}_x(f)$ and $\mathcal{U}_x(f)$ are injectively immersed $C^r$ submanifolds of $f$ passing through $x$, and further, $x\mapsto \mathcal{S}_x(f)$ and $x\mapsto \mathcal{U}_x(f)$ are both measurable functions into spaces of $C^r$ immersions. $\mathcal{S}_x(f)$ and $\mathcal{U}_x(f)$ are called the global stable and unstable manifolds of $f$ at $x$, respectively, and $\mathcal{S}(f)$ and $\mathcal{U}(f)$ are called the stable and unstable ae-foliations of $f$ (despite lacking local triviality as is commonly assumed of foliations, the stable and unstable ae-foliations admit Hölder charts w/r/t which they are locally trivial when the basepoint is restricted to be in special sets called Lusin-Pesin sets).

In any event, $\mathcal{S}(f)$ and $\mathcal{U}(f)$ are ae-partitions of $M$ into measurable subsets with respect to any $f$-invariant probability measure on $M$ (though often they fail to be "measurable partitions" (as mentioned at The supremum in the Kolmogorov-Sinai entropy can be attained by finite measurable partitions, Topology on the set of partitions, Pinsker $\sigma$-algebra is $T$ invariant)). Then Ledrappier & Young, generalizing Pesin's earlier result, showed in their paper "The Metric Entropy Of Diffeomorphisms - Part I - Characterization Of Measures Satisfying Pesin's Entropy Formula" (see Thm.B on p.513) the following:

Theorem (Ledrappier-Young): For any $f$-invariant Borel probability measure $\mu$ on $M$,

$$\widehat{\mathcal{S}(f)}=_\mu\text{Pinsker}(\mu,f)=_\mu\widehat{\mathcal{U}(f)}.$$

(Here $\text{Pinsker}(\mu,f)$ is the Pinsker $\sigma$-algebra of $f$ w/r/t $\mu$, $\widehat{\mathcal{S}(f)}$ is the sub-$\sigma$-algebra of the Borel $\sigma$-algebra of $M$ generated by the stable ae-foliation of $f$, $\widehat{\mathcal{U}(f)}$ is the sub-$\sigma$-algebra of the Borel $\sigma$-algebra of $M$ generated by the unstable ae-foliation of $f$, and $=_\mu$ means that for any element $L$ of the LHS there is an element $R$ of the RHS with $\mu(L\triangle R)=0$ (so that the sub-$\sigma$-algebras coincide as sub-$\sigma$-algebras of the measure algebra of $\mu$).)

One can equivalently formulate the statement in terms of partitions: the measurable hull of the stable ae-foliation, the Pinsker partition, and the measurable hull of the unstable ae-foliation all coincide $\mu$-ae.

Heuristically the importance of the Pinsker $\sigma$-algebra then is that it allows one to switch between (measure theoretical counterparts) of geometric objects that are defined only using the future (stable ae-foliation) and only using the past (unstable ae-foliation). More specifically, the theorem works in tandem with a Hopf argument (which is a classical argument that concludes ergodicity from constancy along stables and unstables). For further details and references see Appendix C by Obata in Brown et al's "Entropy, Lyapunov exponents, and rigidity of group actions" (https://arxiv.org/abs/1809.09192).

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The second example is the "Pinsker partition trick" in the smooth ergodic theory of actions of compactly generated abelian groups (with no compact factors) by diffeomorphisms. This is commonly attributed to Katok and Spatzier (see their paper "Invariant Measures for Higher Rank Hyperbolic Abelian Actions", the chain of inequalities at the end of the proof of Lem.6.1 on p.767; see also the paper "Nonuniform Measure Rigidity" (p.383) by Kalinin, Katok & Rodríguez Hertz (https://arxiv.org/abs/0803.3094)).

Very roughly speaking, the Pinsker partition trick in higher rank is used to carry some ergodicity along hyperplanes in the acting group along which one direction has vanishing Lyapunov exponents. A set being invariant under more diffeomorphisms is more restrictive than being invariant under fewer diffeomorphisms, whence ergodicity in higher rank is weaker. Still, one needs to relate the ergodic components of one diffeomorphism of the action with the ergodic components of another one, and this is where the Pinsker partition trick is used.