Rokhlin lemma


In mathematics, the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. It was proven by Vladimir Abramovich Rokhlin and independently by Shizuo Kakutani. The lemma is used extensively in ergodic theory, for example in Ornstein theory and has many generalizations.

Terminology

Rokhlin lemma belongs to the group mathematical statements such as Zorn's lemma in set theory and Schwarz lemma in complex analysis which are traditionally called lemmas despite the fact that their roles in their respective fields are fundamental.

Statement of the lemma

Lemma: Let be an invertible measure-preserving transformation on a standard measure space with. We assume is aperiodic, that is, the set of periodic points for has zero measure. Then for every integer and for every, there exists a measurable set such that the sets are pairwise disjoint and such that.
A useful strengthening of the lemma states that given a finite measurable partition, then may be chosen in such a way that and are independent for all.

A topological version of the lemma

Let be a topological dynamical system consisting of a compact metric space and a homeomorphism. The topological dynamical system is called minimal if it has no proper non-empty closed -invariant subsets. It is called aperiodic if it has no periodic points. A topological dynamical system is called a factor of if there exists a continuous surjective mapping which is equivariant, i.e., for all.
Elon Lindenstrauss proved the following theorem:
Theorem: Let be a topological dynamical system which has an aperiodic minimal factor. Then for integer there is a continuous function such that the set satisfies are pairwise disjoint.
Gutman proved the following theorem:
Theorem: Let be a topological dynamical system which has an aperiodic factor with the small boundary property. Then for every, there exists a continuous function such that the set satisfies, where denotes orbit capacity.

Further generalizations