
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 formula_1 be an invertible measure-preserving transformation on a standard measure space formula_2 with formula_3. We assume formula_4 is (measurably) aperiodic, that is, the set of periodic points for formula_4 has zero measure. Then for every integer formula_6 and for every formula_7, there exists a measurable set formula_8 such that the sets formula_9 are pairwise disjoint and such that formula_10.
A useful strengthening of the lemma states that given a finite measurable partition formula_11, then formula_8 may be chosen in such a way that formula_13 and formula_11 are independent for all formula_15.
A topological version of the lemma.
Let formula_16 be a topological dynamical system consisting of a compact metric space formula_17 and a homeomorphism formula_18. The topological dynamical system formula_16 is called minimal if it has no proper non-empty closed formula_4-invariant subsets. It is called (topologically) aperiodic if it has no periodic points (formula_21 for some formula_22 and formula_23 implies formula_24). A topological dynamical system formula_25 is called a factor of formula_16 if there exists a continuous surjective mapping formula_27 which is equivariant, i.e., formula_28 for all formula_29.
Elon Lindenstrauss proved the following theorem:
Theorem: Let formula_16 be a topological dynamical system which has an aperiodic minimal factor. Then for integer formula_6 there is a continuous function formula_32 such that the set formula_33 satisfies formula_9 are pairwise disjoint.
Gutman proved the following theorem:
Theorem: Let formula_35 be a topological dynamical system which has an aperiodic factor with the small boundary property. Then for every formula_36, there exists a continuous function formula_37 such that the set formula_38 satisfies formula_39, where formula_40 denotes orbit capacity.
See also.
Rokhlin's lemma should not be confused with Rokhlin's theorem.