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.
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.
Each states that, under some assumptions, we can construct Rokhlin towers that are arbitrarily high with arbitrarily small error sets.
is ergodic, and the space contains sets of arbitrarily small sizes, then we can construct Rokhlin towers.
is aperiodic, and the space is Lebesgue, and has measure 1, then we can construct Rokhlin towers.
(aperiodic, invertible, independent base) — Assume that
is aperiodic and invertible, and the space is Lebesgue, and has measure 1.
, we can construct Rokhlin towers where each level is probabilistically independent of the partition.
For example, (Section 2.5 [2]) Countable generator theorem (Rokhlin 1965) — Given a dynamical system on a Lebesgue space of measure 1, where
is invertible and measure preserving, it is isomorphic to a stationary process on a countable alphabet.
(Section 4.6 [2]) Krieger finite generator theorem (Krieger 1970) — Given a dynamical system on a Lebesgue space of measure 1, where
be a topological dynamical system consisting of a compact metric space
is called minimal if it has no proper non-empty closed
It is called (topologically) aperiodic if it has no periodic points (
be a topological dynamical system which has an aperiodic minimal factor.
be a topological dynamical system which has an aperiodic factor with the small boundary property.
An ergodic map on an atomless Lebesgue space is aperiodic.
Since the space has finite total measure, there are only finitely many atoms of a certain measure, and they must cycle back to the start eventually.
So we define a “time till arrival” function:
If we can pick a near-zero set with near-full coverage, namely some
Thus, our task reduces to picking a near-zero set with near-full coverage.
Any totally ordered chain contains an upper bound.
So by a simple Zorn-lemma–like argument, there exists a maximal element
It suffices to prove the case where only the base of the tower is probabilistically independent of the partition.
Because of how we defined the equivalence classes, each level in each column
parts, and put one into a new Rokhlin tower base
The only lost mass is due to a small corner on the top right and bottom left of each column, which takes up
Thus, the new Rokhlin tower still has a very small error set.
Even after accounting for the mass lost from cutting off the column corners, we still have