New York Journal of Mathematics

New York J. Math. 3A (1997) 11–14.

Multiple Rokhlin Tower Theorem: A Simple Proof

S. J. Eigen and V. S. Prasad

Abstract. S. Alpern has proved that an invertible antiperiodic measurable measure preserving transformation of a Lebesgue probability space can be represented by k towers of heights n_{1}, …, n_{k}, with prescribed measures, provided that the heights have greatest common divisor 1. In this paper we give a simple proof of Alpern’s theorem. It is elementary in the sense that it involves no limits and uses Kakutani’s easy proof of Rokhlin’s Lemma.

Contents

1. Alpern’s Multiple Rokhlin Tower Theorem 11 2. Proof of Alpern’s Theorem 12 References 13

1. Alpern’s Multiple Rokhlin Tower Theorem

In this paper we show that Kakutani’s proof of Rokhlin’s Lemma [Hal56] can be used to give a short, elementary proof of the following Multiple Rokhlin Tower Theorem of Alpern’s [Alp79, Cor 2].

Theorem 1.1 (Alpern). For any k ≥ 2, let n_{1}, n_{2}, …, n_{k} be relatively prime positive integers, and let q_{1}, …, q_{k} be positive numbers such that n_{1}q_{1} +⋯+n_{k}q_{k} = 1. Then for any antiperiodic invertible measure preserving transformation T of a Lebesgue probability space (X, Σ, μ), there exist sets Q_{i} ∈ Σ, i = 1, …, k with μ(Q_{i}) = q_{i} and such that {T^{j}(Q_{i}) : i = 1, …, k, j = 0, …, n_{i} − 1} is a partition of X (into k columns of heights n_{1}, …, n_{k} and μ-widths q_{1}, …, q_{k}).

Alpern applied this result to prove that in the space of measure preserving homeomorphisms of a compact connected manifold with the topology of uniform convergence, any measure theoretic property which is generic (dense G_{δ}) in the weak topology on the space of invertible measure preserving transformations of the underlying probabilty measure space is also generic (dense G_{δ}) in the uniform convergence topology in the group of measure preserving homeomorphisms. The latter result is a far-reaching generalization of the classical (1940) Oxtoby-Ulam Theorem [OU40], where it is proved that ergodicity is generic in the space of measure

\begin{matrix}c \\ ◯ \end{matrix}1997 State University of New York

ISSN 1076-9803/97

Received July 15, 1997.

Mathematics Subject Classification. Primary 28D05; Secondary 58F11.

Key words and phrases. Rokhlin Tower.

preserving homeomorphisms. Furthermore this contains Katok and Stepin’s 1970 result [KS70] that weak mixing is generic for measure preserving homeomorphisms.

Very recently, N. Ormes [Orm] has used Alpern’s Multiple Rokhlin Tower Theorem as one step in obtaining the following result about realizing an ergodic measure preserving system as a minimal homeomorphism of the Cantor set within a given topological orbit equivalence class: Let Y be a Cantor set, S a minimal homeomorphism of Y and ν a uniquely ergodic S-invariant Borel probability measure. Let T be an ergodic invertible measure preserving transformation of the Lebesgue probability space (X, Σ, μ). Then there is a topological realization (S′, ν) of the ergodic system (T, μ), where S′ is a minimal homeomorphism of the Cantor set Y strongly orbit equivalent to S if and only if the finite rotations which are topological factors of S are measurable factors of T.

Two homeomorphisms, S and S′, of the Cantor set Y, are said to be strongly orbit equivalent if there is a homeomorphism h : Y → Y and integer valued maps m, n : Y → Z such that hS^{m(x)}(x) = S′h(x), hS(x) = (S′)^{n(x)}h(x) and m and n have no more than one point of discontinuity. Strong orbit equivalence of two minimal Cantor homeomorphisms has been identified in the work of Giordano, Putnam and Skau [GPS96, Theorem 2.2] as a necessary and sufficient condition for the isomorphism of the crossed product C^{∗}-algebras (C(Y) ×_{S} Z, and C(Y) ×_{S′} Z) associated with the minimal homeomorphisms .

Multiple Rokhlin Towers arise naturally in the study of minimal homeomorphisms S of the Cantor set in the following manner. Given any clopen set A in the Cantor set Y, consider r_{A} : A → N, the first return time function to A for the homeomorphism S, where r_{A}(x) is the smallest positive integer such that S^{r_{A}(x)}(x) ∈ A. Then the continuity of this function implies there is a finite set of positive integers n_{1}, …, n_{k} which is the range of r_{A}. This gives a Multiple Rokhlin Tower partition of Y into clopen sets (into k towers of heights n_{1}, …, n_{k} over the base A). It is this tower for S which Ormes “copies” for the ergodic T using Alpern’s theorem. While it need not be true that the gcd{n_{1}, …, n_{k}} = 1, Ormes notes [Orm, Cor 4.2] that if gcd{n_{1}, …, n_{k}} = p, then there is periodic clopen set for S of period p. Thus to have a similar tower picture for T, there must also be a T -periodic set in X of order p.

Extensions of Alpern’s Multiple Rokhlin Tower Theorem to denumerably many columns and to nonsingular aperiodic transformations may be found in [Alp81] and [AP90]. In addition applications of these extensions to coding Markov chains and approximate conjugacy theorems can also be found in those papers.

2. Proof of Alpern’s Theorem

Step 1: Prescribing return times: We find a set A ⊂ X such that the set of first return times to A under T are exactly n_{1}, …, n_{k}.

Proof Step 1: Since gcd{n_{1}, …, n_{k}} = 1, let R be a positive integer such that every integer r ≥ R, can be expressed as nonnegative integer multiples of n_{1}, …, n_{k}.

Following Kakutani’s easy proof of Rokhlin’s Lemma, let E be a sweep out set (i.e., X = ∪_{i=0}^{∞}T^{i}(E)), such that the set of first return times back to E are all greater than R (in Step 2 we will require that E has small measure). This is elementary in the case that T is ergodic. See [Hal56] for a proof when T is antiperiodic. Write E = ∪_{m≥R}E_{m}, where E_{m} = {x ∈ E : the first return time to E is m}. The column of height m over E is the set C(E_{m}) = ∪_{i=1}^{m}T^{i−1}E_{m}.

For each m ≥ R write m = qN + r where R ≤ r < R + N and N = n_{1}n_{2} ⋯ n_{k}. Then we can break up the column of height m over E, (C(E_{m})), into q new subcolumns of height N and one new subcolumn of height r as follows: The first N floors of C(E_{m}) are labeled (N, 1), …, (N, N), as are the next N floors. After labeling the first qN floors of C(E_{m}) into q many N-columns in this manner, the last r floors are labelled to form an r-column (i.e, as (r, 1), …, (r, r)). After doing this for each m ≥ R and combining all of the new columns of the same height (i.e, grouping all sets with the same label), we end up with one (large) column of height N and at most N “remainder” columns of heights R, R + 1, … R + N − 1.

For each r, R ≤ r < R+N, since we can write r = r_{1}n_{1} +⋯+r_{k}n_{k} where the r_{i} are nonnegative integers, we can break up each remainder r-column as follows: the first r_{1}n_{1} floors are broken up into r_{1} columns of height n_{1}; the next block of r_{2}n_{2} floors are grouped into r_{2} columns of height n_{2}; continuing, the last r_{k}n_{k} floors of the r-column is broken into r_{k} columns of height n_{k}.

This partitions the space into a group of columns, one of height N and the others of heights n_{1}, …, n_{k}. The set A which is the base of these columns, has return times N and n_{1}, …, n_{k}. Note that the column of height N can be decomposed (labeled) into as many columns of height n_{1}, …, n_{k} as we wish, since N is a multiple of n_{i} for each i.

Step 2: Controlling the distribution: We note this has been a measure free construction thus far. If we wish to prescribe the distribution of the return times we note that the total measure of the “remainder” columns of height r, R ≤ r < R +N is less than (R + N)μ(E) where E was the base of the original skyscraper. By choosing E small enough so that (R + N)μ(E) < min {n_{i}q_{i} : i = 1, …, k} we can guarantee that after breaking up the remainder columns of height r into columns of heights n_{1}, …, n_{k} no column of height n_{i} has used up more than its “total allowance” of measure n_{i}q_{i}.

Now partition the N-column into k disjoint vertical columns, one for each i = 1, …, k, in the following manner: Suppose that the measure of the n_{i}-column used in paving all the “remainder” r-columns (R ≤ r < R+N) totals up to p_{i}. From the column of height N we take a vertical strip of measure n_{i}q_{i} − p_{i}. Then this part of the N-column is partitioned into columns of height n_{i}. Choosing disjoint vertical strips from the N-column for the different i’s, the partition of X into columns of height n_{i}, i = 1, …, k, has the required distribution.

References

[Alp79] S. Alpern, Generic properties of measure preserving homeomorphisms, Ergodic Theory, Springer Lecture Notes in Mathematics 729 (1979) 16–27.

[Alp81] S. Alpern, Return times and conjugates of an antiperiodic transformation, Ergodic Theory and Dynamical Systems 1 (1981) 135–143.

[AP90] S. Alpern and V.S. Prasad, Return times of nonsingular transformations, Journal of Mathematical Analysis and its Applications, 152 (1990) 470–487.

[GPS96] T. Giordano, I. Putnam, and C. Skau, Topological orbit equivalence and C^{∗}-crossed products, J. fur Reine. Angew. Math, 469 (1996) 51–110.

[Hal56] P. Halmos, Lectures on Ergodic Theory, Publications of the Mathematical Society of Japan, No. 3, Chelsea Publishing Company (1956).

14 S. J. Eigen and V. S. Prasad

[KS70] A. Katok and A. Stepin, Metric properties of measure preserving homeomorphisms, Uspekhi Mat. Nauk 25:2 (1970) 193–220. (Russian Math. Surveys 25 (1970) 191–220.)

[Orm] N. Ormes, Strong orbit realization for minimal homeomorphisms, preprint.

[OU40] J. C. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity, Annals of Mathematics 2 (1940) 874–920.

Northeastern University, Boston MA 02115 eigen@neu.edu http://www.math.neu.edu/˜ eigen/

University of Massachusetts, Lowell MA 01854 vidhu prasad@uml.edu