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

edit

A Lebesgue space is a measure space   composed of two parts. One atomic part with finite/countably many atoms, and one continuum part isomorphic to an interval on  .

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   such that if   (except on a measure-zero set) then   or   has measure zero.

An aperiodic map is a map such that the set of periodic points is measure zero: A Rokhlin tower is a family of sets   that are disjoint.   is called the base of the tower, and each   is a rung or level of the tower.   is the height of the tower. The tower itself is  . The set outside the tower   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

edit

[1][2]

(ergodic) — If   is ergodic, and the space contains sets of arbitrarily small sizes, then we can construct Rokhlin towers.

(aperiodic) — If   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.

Given any partition of   into finitely many events  , we can construct Rokhlin towers where each level is probabilistically independent of the partition.

Applications

edit

The Rokhlin lemma can be used to prove some theorems. 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   is invertible, measure preserving, and ergodic.

If its entropy is less than  , then the system is generated by a partition into   subsets.

Ornstein isomorphism theorem (Chapter 6 [2]).

Topological Rokhlin lemmas

edit

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 (topologically) aperiodic if it has no periodic points (  for some   and   implies  ). 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:[3]

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:[4]

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.

Other generalizations

edit
  • 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

edit

Proofs taken from.[2]

Useful results

edit

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

Proof. If the map is not aperiodic, then there exists a number  , such that the set of periodic points of period   has positive measure. Call the set  . Since measure is preserved, points outside of   do not map into it, nor the other way. Since the space is atomless, we can divide   into two halves, and   maps each into itself, so   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   is ergodic, then any set   satisfies (up to a null set) Proof.   is a subset of  , so by measure-preservation they are equal. Thus   is a factor of  , and since it contains  , it is all of  .

Similarly,   is a subset of  , so by measure-preservation they are equal, etc.

Ergodic case

edit

Let   be a set of measure  . Since   is ergodic,  , almost any point sooner or later falls into  . So we define a “time till arrival” function:   with   if   never falls into  . The set of   is null.

Now let  .

Aperiodic case

edit

Simplify

edit

By a previous proposition,   is atomless, so we can map it to the unit interval  .

If we can pick a near-zero set with near-full coverage, namely some   such that  , then there exists some  , such that  , and since   for each  , we have Now, repeating the previous construction with  , we obtain a Rokhlin tower of height   and coverage  .

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

Constructing A

edit

Pick  . Let   be the family of sets   such that   are disjoint. Since   preserves measure, any   has size  .

The set   nonempty, because  . It is preordered by   iff  . Any totally ordered chain contains an upper bound. So by a simple Zorn-lemma–like argument, there exists a maximal element   in it. This is the desired set.

We prove by contradiction that  . Assume not, then we will construct a set  , disjoint from  , such that  , which makes   no longer a maximal element, a contradiction.

Constructing E

edit

Since we assumed  , with positive probability,  .

Since   is aperiodic, with probability 1, And so, for a small enough  , with probability  , And so, for a small enough  , with probability  , these two events occur simultaneously. Let the event be  .

Proof that E works
Proof

Since  , there exists an interval   of length   such that  .

By construction,   is disjoint from  . It remains to check that the   preimages of   are disjoint.

By construction,   is disjoint from  , so the preimages of   are disjoint from the preimages of  .

Since  , the   preimages of   are disjoint.

If the   preimages of   are not disjoint, then there exists  , such that  . In other words, there exists  , such that  .

However, by construction,   implies   is repelled by   to at least   distance away, so  , contradiction.

Invertible case

edit

Simplify

edit

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  .

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

The goal is to construct a Rokhlin tower   with base  , such that   for each  .

Given a partition   and a map  , we can trace out the orbit of every point   as a string of symbols  , such that each  . That is, we follow   to  , then check which partition it has ended up in, and write that partition’s name as  .

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

Let   be one such equivalence class, then we call   a column of the Rokhlin tower.

For each word  , let the corresponding equivalence class be  .

Since   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   equivalence classes, then pull it apart into   columns.

First Rokhlin tower R

edit

Let   be very small, and let   be very large. Construct a Rokhlin tower with   levels and error set of size  . Let its base be  . The tower   has mass  .

Divide its base into   equivalence classes, as previously described. This divides it into   columns   where   ranges over the possible words  .

Because of how we defined the equivalence classes, each level in each column   falls entirely within one of the partitions  . Therefore, the column levels   almost make up a refinement of the partition  , except for an error set of size  .

That is, The critical idea: If we partition each   equally into   parts, and put one into a new Rokhlin tower base  , we will have 

Second Rokhlin tower R'

edit

Now we construct a new base   as follows: For each column based on  , add to  , in a staircase pattern, the sets then wrap back to the start:  and so on, until the column is exhausted. The new Rokhlin tower base   is almost correct, but needs to be trimmed slightly into another set  , which would satisfy   for each  , 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.)

Trimming the new Rokhlin tower base

The new Rokhlin tower  , 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   proportion of the whole column’s mass. If we set  , this lost mass is still  . 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 

Since there are only finitely many partitions, we can set  , we then have In other words, we have real numbers   such that  .

Now for each column  , trim away a part of   into  , so that  . This finishes the construction.

References

edit
  1. ^ Shields, Paul (1973). The theory of Bernoulli shifts (PDF). Chicago Lectures in Mathematics. Chicago, Illinois and London: The University of Chicago Press. pp. Chapter 3.
  2. ^ a b c d e Kalikow, Steven; McCutcheon, Randall (2010). "2.4. Rohlin tower theorem". An outline of ergodic theory. Cambridge studies in advanced mathematics (1. publ ed.). Cambridge: Cambridge Univ. Press. ISBN 978-0-521-19440-2.
  3. ^ Lindenstrauss, Elon (1999-12-01). "Mean dimension, small entropy factors and an embedding theorem". Publications Mathématiques de l'IHÉS. 89 (1): 227–262. doi:10.1007/BF02698858. ISSN 0073-8301. S2CID 2413058.
  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. doi:10.1090/conm/356/06502. ISBN 9780821833131.
  6. ^ Avila, Artur; Candela, Pablo (2016). "Towers for commuting endomorphisms, and combinatorial applications". Annales de l'Institut Fourier. 66 (4): 1529–1544. arXiv:1507.07010. doi:10.5802/aif.3042.
  7. ^ Ornstein, Donald S.; Weiss, Benjamin (1987-12-01). "Entropy and isomorphism theorems for actions of amenable groups". Journal d'Analyse Mathématique. 48 (1): 1–141. doi:10.1007/BF02790325. ISSN 0021-7670. S2CID 120653036.
  8. ^ Ionescu Tulcea, Alexandra (1965-01-01). "On the Category of Certain Classes of Transformations in Ergodic Theory". Transactions of the American Mathematical Society. 114 (1): 261–279. doi:10.2307/1994001. JSTOR 1994001.

Notes

edit