Rokhlin lemma explained

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.

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.

Terminology

A Lebesgue space is a measure space

(X, l B, \mu) composed of two parts. One atomic part with finite/countably many atoms, and one continuum part isomorphic to an interval on $\backslash R$.

We consider only measure-preserving maps. As typical in measure theory, we can freely discard countably many sets of measure zero.

An ergodic map is a map

T such that if T ^{-1} (A) = A (except on a measure-zero set) then A or X - A has measure zero.

An aperiodic map is a map such that the set of periodic points is measure zero:$$\backslash mu(\backslash cup\_\backslash )\; =\; 0$$A Rokhlin tower is a family of sets $S,\; TS,\; \backslash dots,\; T^S$ that are disjoint. $S$ is called the base of the tower, and each $T^nS$ is a rung or level of the tower. $N$ is the height of the tower. The tower itself is

R : = (S \cup TS \cup ... \cup T ^{N-1} S) . The set outside the tower $X\; -\; R$ is the error set.

There are several Rokhlin lemmas. Each states that, under some assumptions, we can construct Rokhlin towers that are arbitrarily high with arbitrarily small error sets.

Theorems

^{[1] [2]}

Applications

The Rokhlin lemma can be used to prove some theorems. For example, (Section 2.5)

(Section 4.6)

Ornstein isomorphism theorem (Chapter 6).

Topological Rokhlin lemmas

Let

style (X,T) be a topological dynamical system consisting of a compact metric space style X and a homeomorphism style T:X → X . The topological dynamical system style (X,T) is called minimal if it has no proper non-empty closed style T -invariant subsets. It is called (topologically) aperiodic if it has no periodic points (T ^{k} x = x for some x\in X and k\in {Z } implies k =0 ). A topological dynamical system style (Y,S) is called a factor of style (X,T) if there exists a continuous surjective mapping style \varphi:X → Y which is equivariant, i.e., style \varphi(Tx) = S\varphi(x) for all style x\in X .

Elon Lindenstrauss proved the following theorem:^[3]

Theorem: Let

style (X,T) be a topological dynamical system which has an aperiodic minimal factor. Then for integer style n\in\N there is a continuous function style f\colon X → \R such that the set style E = \{x\in X\mid f(Tx) ≠ f(x) +1 \} satisfies style E,TE,\ldots,T ^{n-1} E are pairwise disjoint.

Gutman proved the following theorem:^[4]

Theorem: Let

(X,T) be a topological dynamical system which has an aperiodic factor with the small boundary property. Then for every \varepsilon> 0 , there exists a continuous function f\colon X → \R such that the set style E = \{x\in X \mid f(Tx) ≠ f(x) +1 \} satisfies \operatorname{ocap}( style E)<\varepsilon , where \operatorname{ocap} denotes orbit capacity.

Other generalizations

• There are versions for non-invertible measure-preserving transformations.^{[5] [6]}
• Donald Ornstein and Benjamin Weiss proved a version for free actions by countable discrete amenable groups.^[7]
• Carl Linderholm proved a version for periodic non-singular transformations.^[8]

Proofs

Proofs taken from.

Useful results

Proposition. An ergodic map on an atomless Lebesgue space is aperiodic.

Proof. If the map is not aperiodic, then there exists a number $n$, such that the set of periodic points of period $n$ has positive measure. Call the set $S$. Since measure is preserved, points outside of $S$ do not map into it, nor the other way. Since the space is atomless, we can divide $S$ into two halves, and $T$ maps each into itself, so $T$ is not ergodic.

Proposition. If there is an aperiodic map on a Lebesgue space of measure 1, then the space is atomless.

Proof. If there are atoms, then by measure-preservation, each atom can only map into another atom of greater or equal measure. If it maps into an atom of greater measure, it would drain out measure from the lighter atoms, so each atom maps to another atom of equal measure. 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.

Proposition. If $T$ is ergodic, then any set $A\; >\; 0$ satisfies (up to a null set)$$X\; =\; \backslash cup\_\; T^k\; A\; =\; \backslash cup\_\; T^k\; A$$Proof. $T^(\backslash cup\_\; T^k\; A)$ is a subset of $\backslash cup\_\; T^k\; A$, so by measure-preservation they are equal. Thus $\backslash cup\_\; T^k\; A$ is a factor of $T$, and since it contains $A>\; 0$, it is all of $X$.

Similarly, $T(\backslash cup\_\; T^k\; A)$ is a subset of $\backslash cup\_\; T^k\; A$, so by measure-preservation they are equal, etc.

Ergodic case

Let $A$ be a set of measure . Since $T$ is ergodic, $X\; =\; \backslash cup\_\; T^k\; A$, almost any point sooner or later falls into $A$. So we define a “time till arrival” function: $$f(x)\; :=\; \backslash min\backslash$$ with $f(x)\; :=\; +\backslash infty$ if $x$ never falls into $A$. The set of $\backslash$ is null.

Now let $S\; =\; \backslash$.

Aperiodic case

Simplify

By a previous proposition,

X is atomless, so we can map it to the unit interval $(0,\; 1)$.

If we can pick a near-zero set with near-full coverage, namely some $A\; =\; O(\backslash epsilon)$ such that $X\; -\; \backslash cup\_\; T^k\; A\; =\; O(\backslash epsilon)$, then there exists some $n$, such that $X\; -\; \backslash cup\_\; T^k\; A\; =\; O(\backslash epsilon)$, and since

T ^{-i} (T ^{n} A) \supset T ^{n-i} A for each i = 0 , 1 , 2 , ... , we have$$X\; -\; \backslash cup\_\; T^k\; (T^nA)\; =\; O(\backslash epsilon)$$Now, repeating the previous construction with T ^{n} A , we obtain a Rokhlin tower of height $N$ and coverage $1-O(\backslash epsilon)$.

Thus, our task reduces to picking a near-zero set with near-full coverage.

Constructing A

Pick $M\; >\; 1/\backslash epsilon$. Let $S$ be the family of sets $A$ such that $A,\; T^A,\; \backslash dots,\; T^A$ are disjoint. Since $T$ preserves measure, any $A\; \backslash in\; S$ has size .

The set $S$ nonempty, because $\backslash emptyset\; \backslash in\; S$. It is preordered by iff $\backslash mu(B-A)\; =\; 0$. Any totally ordered chain contains an upper bound. So by a simple Zorn-lemma–like argument, there exists a maximal element $A$ in it. This is the desired set.

We prove by contradiction that $X\; =\; \backslash cup\_T^k\; A$. Assume not, then we will construct a set $I\backslash cap\; E\; >\; 0$, disjoint from $A$, such that $A\; \backslash cup\; (I\; \backslash cap\; E)\; \backslash in\; S$, which makes $A$ no longer a maximal element, a contradiction.

Constructing E

Since we assumed $X\; -\; \backslash cup\_T^k\; A=\; \backslash epsilon\text{'}\; >\; 0$, with positive probability,

x \not\in \cup _{k\in} T ^{k} A .

Since $T$ is aperiodic, with probability 1,$$(x\; \backslash neq\; Tx)\; \backslash wedge(x\; \backslash neq\; T^2x)\; \backslash wedge\; \backslash dots\; \backslash wedge\; (x\; \backslash neq\; T^Mx)$$And so, for a small enough $\backslash delta$, with probability $>\; 1-\; \backslash epsilon\text{'}/2$,$$(|x\; -\; Tx|\; >\; \backslash delta)\; \backslash wedge(|x\; -\; T^2\; x|\; >\; \backslash delta)\; \backslash wedge\; \backslash dots\; \backslash wedge\; (|x\; -\; T^M\; x|\; >\; \backslash delta)$$And so, for a small enough $\backslash delta$, with probability $>\; \backslash epsilon\text{'}/2$, these two events occur simultaneously. Let the event be $E$.

Invertible case

Simplify

It suffices to prove the case where only the base of the tower is probabilistically independent of the partition. Once that case is proved, we can apply the base case to the partition $P\; \backslash vee\; T^\; P\; \backslash vee\; \backslash dots\; \backslash vee\; T^P$.

Since events with zero probability can be ignored, we only consider partitions where each event $P\_k$ has positive probability.

The goal is to construct a Rokhlin tower $R\text{'}$ with base $S\text{'}$, such that

\mu(S' \cap P _{i} ) = (1- \epsilon)/(N) \mu(P _{i)} for each $i\; \backslash in\; 0:K-1$.

Given a partition $P$ and a map $T$, we can trace out the orbit of every point $x$ as a string of symbols $a\_0(x),\; a\_1(x),\; a\_2(x),\; \backslash dots$, such that each $T^i\; x\; \backslash in\; P\_$. That is, we follow $x$ to $T^ix$, then check which partition it has ended up in, and write that partition’s name as $a\_i(x)$.

Given any Rokhlin tower of height $N$, we can take its base $S$, and divide it into $K^N$ equivalence classes. The equivalence is defined thus: two elements are equivalent iff their names have the same first-$N$ symbols.

Let $E\; \backslash subset\; S$ be one such equivalence class, then we call $E,\; TE,\; \backslash dots,\; T^E$ a column of the Rokhlin tower.

For each word $a\_\backslash in\; (0:K-1)^N$, let the corresponding equivalence class be $E\_$.

Since $T$ is invertible, the columns partition the tower. One can imagine the tower made of string cheese, cut up the base of the tower into the $K^N$ equivalence classes, then pull it apart into $K^N$ columns.

First Rokhlin tower R

Let $\backslash delta\; \backslash ll\; \backslash epsilon$ be very small, and let $M\; \backslash gg\; N$ be very large. Construct a Rokhlin tower with $M$ levels and error set of size $\backslash delta$. Let its base be $S$. The tower $R\; =\; S\backslash cup\; TS\; \backslash cup\backslash dots\backslash cup\; T^S$ has mass $1-\backslash delta$.

Divide its base into $K^N$ equivalence classes, as previously described. This divides it into $K^N$ columns $\backslash \_$ where $a$ ranges over the possible words $(0:K-1)^N$.

Because of how we defined the equivalence classes, each level in each column $T^nE\_a$ falls entirely within one of the partitions $P\_0,\; \backslash dots,\; P\_$. Therefore, the column levels $\backslash \_$ almost make up a refinement of the partition $P$, except for an error set of size $\backslash delta$.

That is,$$\backslash mu(R\backslash cap\; P\_i)\; =\; \backslash sum\_\backslash mu(T^nE\_a)\; =\; \backslash mu(P\_i)\; +\; O(\backslash delta)$$The critical idea: If we partition each $T^nE\_a$ equally into $N$ parts, and put one into a new Rokhlin tower base $S\text{'}$, we will have$$\backslash mu(S\text{'}\backslash cap\; P\_i)\; =\; \backslash frac\backslash mu(P\_i)\; +\; O(\backslash delta)$$

Second Rokhlin tower R

Now we construct a new base $S\text{'}$ as follows: For each column based on $E\_a$, add to $S\text{'}$, in a staircase pattern, the sets$$E\_,\; TE\_,\; \backslash dots,\; T^E\_$$then wrap back to the start:$$T^NE\_,\; T^E\_,\; \backslash dots,\; T^E\_$$and so on, until the column is exhausted.The new Rokhlin tower base $S\text{'}$ is almost correct, but needs to be trimmed slightly into another set

S'' , which would satisfy \mu(S''\cap P _{i)} = (1- \epsilon)/(N) \mu(P _{i)} for each $i\; \backslash in\; 0:K-1$, finishing the construction. (Only now do we use the assumption that there are only finitely many partitions. If there are countably many partitions, then the trimming cannot be done.)

The new Rokhlin tower $S\text{'},\; TS\text{'},\; \backslash dots,\; T^S\text{'}$, contains almost as much mass as the original Rokhlin tower. The only lost mass is due to a small corner on the top right and bottom left of each column, which takes up $\backslash leq\; \backslash frac$ proportion of the whole column’s mass. If we set $M\; \backslash gg\; N/\backslash delta$, this lost mass is still $O(\backslash delta)$. 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

\end

Since there are only finitely many partitions, we can set $\backslash delta\; =\; o(\backslash frac)$, we then have$$\backslash mu(S\text{'}\backslash cap\; P\_i)\; =\; \backslash frac\backslash mu(P\_i)\backslash times\; (1\; +\; o(1)\; \backslash epsilon)$$In other words, we have real numbers $c\_0,\; c\_1,\; \backslash dots,\; c\_\; =\; o(1)$ such that

\mu(S'\cap P _{i)} = (1- c _{i} \epsilon)/(N) \mu(P _{i)} .

Now for each column $i\; =\; 0,\; 1,\; \backslash dots,\; K-1$, trim away a part of $S\text{'}\backslash cap\; P\_i$ into $S$\cap P_i, so that

\mu(S''\cap P _{i)} = (1- \epsilon)/(N) \mu(P _{i)} . This finishes the construction.

Notes

• Vladimir Rokhlin. A "general" measure-preserving transformation is not mixing. Doklady Akademii Nauk SSSR (N.S.), 60:349–351, 1948.
• Shizuo Kakutani. Induced measure preserving transformations. Proc. Imp. Acad. Tokyo, 19:635–641, 1943.
• Benjamin Weiss. On the work of V. A. Rokhlin in ergodic theory. Ergodic Theory and Dynamical Systems, 9(4):619–627, 1989.
• . Some old and new Rokhlin towers. Contemporary Mathematics, 356:145, 2004.

Notes and References

1. Book: Shields, Paul . The theory of Bernoulli shifts . The University of Chicago Press . 1973 . Chicago Lectures in Mathematics . Chicago, Illinois and London . Chapter 3.
2. Book: Kalikow, Steven . An outline of ergodic theory . McCutcheon . Randall . 2010 . Cambridge Univ. Press . 978-0-521-19440-2 . 1. publ . Cambridge studies in advanced mathematics . Cambridge . 2.4. Rohlin tower theorem.
3. Mean dimension, small entropy factors and an embedding theorem. Publications Mathématiques de l'IHÉS. 1999-12-01. 0073-8301. 227–262. 89. 1. 10.1007/BF02698858. Elon. Lindenstrauss. 2413058. Elon Lindenstrauss.
4. Gutman, Yonatan. "Embedding ℤk-actions in cubical shifts and ℤk-symbolic extensions." Ergodic Theory and Dynamical Systems 31.2 (2011): 383-403.
5. Kornfeld . Isaac . 2004 . Some old and new Rokhlin towers . Contemporary Mathematics . 356 . 145–169 . 10.1090/conm/356/06502 . 9780821833131.
6. Avila . Artur . Artur Avila . Candela . Pablo . 2016 . Towers for commuting endomorphisms, and combinatorial applications . Annales de l'Institut Fourier . 66 . 4 . 1529–1544 . 1507.07010 . 10.5802/aif.3042 . free.
7. Ornstein . Donald S. . Donald Samuel Ornstein . Weiss . Benjamin . Benjamin Weiss . 1987-12-01 . Entropy and isomorphism theorems for actions of amenable groups . . 48 . 1 . 1–141 . 10.1007/BF02790325 . 0021-7670 . 120653036 .
8. Ionescu Tulcea . Alexandra . Alexandra Bellow . 1965-01-01 . On the Category of Certain Classes of Transformations in Ergodic Theory . . 114 . 1 . 261–279 . 10.2307/1994001 . 1994001.