Cutting and Independent Stacking o f Intervals

by

PAUL SHIELDS

Mathematics Department Stanford University

Stanford, California 94305

ABSTRACT

Methods of cutting and stacking intervals have been frequently used in ergodic theory to construct transformations with special properties. We show that for independent stacking the partition into subintervals is a Markov partition. In particular, if the resulting transformation is mixing it must be a Bernoulli shift.

Definitions. The following definitions are similar to those described in Friedman [2, Section 6].

An n-column is an ordered set of intervals C = (B I, B2,'" ", B,), all of the same width, which will be called the width of C and denoted by w(C). The base of C is B1, the roof of C is B,, and the height of C, which will be denoted by h(C), is n. Associated with the column C is the map T(C) which maps B~ linearly onto B~+I for t _< i < n - 1 , and is undefined on B,. It is usually convenient to think of Bi+~ as being directly above B~, so that T(C)just maps a point in Bi directly upwards to B,+ 1.

Columns are stacked in the following way. Given

C = (B1, Bz, '" , B.)

c ' = (B'~, B ; , . . . , B; , )

with w(C) = w(C'), then

c • c ' = (B,, B2,-", B., B', B;,- • -, B').

Note that T(C. C') extends both T(C) and T-1(C ') and maps B, linearly onto B'I.

A q-tower J- is a collection of columns {C 1, C2, . . .,Cq}, all of the same width, such that ~ h(Ci)w(C i) = 1. Thus a q-tower is obtained by cutting the unit interval into subintervals of a fixed length, then stacking these into q separate columns. The width of a tower, w(Y), is the sum of the widths of the bases of its columns, that is w(J-) = ~%lw(C i) if f = {C 1, C 2 , " . . , cq). The roof of J - is the union of the roofs of its columns and the base of g - is the union of the bases of its columns. The transformation T(J-) is the common extension of the T(C ~) to all but the roof of ~--.

1

MATHEMATICAL SYSTEMS THEORY, Vol. 7, No. 1. © 1973 by Springer-Verlag New York Inc.

2 PAUL SHIELDS

Cutting and independent stacking is done as follows. Given a tower ~ - = ( C x, cZ, . . . , Cq}, cut each column C i into 2q columns C~ of equal width and form

C~'J = C~ * CJq+ i 1 <_ i < q, 1 < j < q.

The tower 3 - ' consists of the columns E ~' J. Note that

(1) w(Y-') = ½w(Y-). (2) The number of columns in J " is the square of the number of columns

of ~-'.

Repetition of cutting and independent stacking will define a measure- preserving transformation on the unit interval with Lebesgue measure /~. Suppose ~ '0 is a given tower. Form J ' t = ~-~, ~ '2 = 9-'t," " " and in general ~-.+1 = ~ " , n _> 0. The set of'all points x which are in the roof of all the . 7 . is a set of measure zero, since w(3",) ~ 0; hence the T(3- , ) have a common extension T, defined for almost all x in the unit interval. Clearly T preserves Lebesgue measure. Where necessary, we shall indicate its dependence upon ~--o by writing T = ~'(J-o).

Markov Partitions. Suppose T = z ( J 0 ) and pC,) is the partition of the unit interval into the intervals which make up if--,. We shall prove the following theorem.

THEOREM 1. pC,) is a Markov partition for T. Proof. This is the statement that for any finite sequence Ao, A t , . . . , A,, of

atoms in pC,) the following holds

(6o) (3) tz T - i A i tz(At) = tz(Ao n T - t A l ) t z T - i A i . i i

The intersection ~m= o T - i A , will be non-empty only if, for each i, T-XAz+ t =A s or Ai+ t is a base interval and A i a roof interval of 3" . (since T -1 maps down each column of J ' , ) . Let us suppose that this holds, that k is the number of indices 0 < i < m for which Ai+ t is a base interval of J-n and that ~-". = {C t, C2, . . . , Cq}. We will prove that

(4) /z(~o T - i A , ) = ql-~/~(Ao).

Without loss of generality we can suppose that A o is a base interval and that Am is a roof interval of J - . . Thus we can find a sequence 0 = il < i2 < "'" < ik+t = m such that each string of sets (Air, Ai . . . . . . . , Ai,+l ) is a column, say C j, of ~--.. In summary,

(5) -,40, A t , ' ' ' , A m can be grouped in natural order to give the sequence of columns C jl, CJ2, • • ", C jk of J - . .

Proceeding with the proof of (4), we let K be large relative to k and let M = 2 r. A column C of ~--n+K is made up of vertical pieces of columns of 3- , , that is, there is a one-to-one map

(6) ~o(C) = (i~, i 2 , ' " , iM), ij ~ { 1, 2," " ", q},

Cutting and Independent Stacking of Intervals 3

and the bottom h(Cit) intervals of C are subintervals of C~1, the next h(C~2 ) intervals of C are subintervals of C~2, etc. As K ---> ~ , the frequency of occur- rence of (Jl, J2, ' ' ' ,Jk) among k-blocks of M-sequence (6) converges to 1/q k and this establishes (4).

To complete the proof of Theorem 1 we must show that for any finite sequence Ao, A1, . . . , Am of atoms in p(,) the following holds.

(3) /~(,~0 T-iA~)/~(A,) = /z(Ao n T-1A~)/L(,~I T- 'A~) .

where we can assume that both sides are non-zero. Thus (4) gives

tz(iN=oT-iA~) = q~k/z(Ao) •

If A~ is a base piece, then

/z(i~x T_ iA0 = 1 A o); t,(,40 n T-1.4~) = _1 ~(Ao), q

while if At is not a base piece, then

t~ T-~Ai = /~(Ao); tz(Ao ~ T - I A I ) = tZ(Ao). \i=1 q'k

In both these cases the relation (3) holds, so that Theorem 1 is proved. One can now use the theory of Markov shifts (Billingsley [1]) to obtain

several theorems. Theorems 2 and 3 are due to Friedman and Ornstein and can be found in Friedman's book [2, Section 6].

THEOREM 2. T is ergodic. Proof. If A, B E P("), then TkA will meet B for some k > 0 so that T is

ergodic on the cr-field Z, generated by the TiP ("), - ~ < i < ~ . The ergodic theorem therefore holds for the restriction of T to Z, and since Z, ~ Z,+l, it will hold for the restriction of T to the or-algebra generated by { Z, }. The E, generate Z so that T must be ergodic.

THEOREM 3. I f two columns of some ~--, have height differing by 1, then T is mixing. More generally, i f for each n the greatest common divisor d, of the column heights in ~--n is 1, then T is mixing.

Proof. Certainly dk = 1 if for some n, J - , . has two columns with height differing by 1. If some d R = 1, then T on Z k is mixing for if A is a base piece, then T-mA n A v~ ;5 if m is any column height in p(k), and hence the greatest common divisor of the cycle lengths is 1. If d k = 1, then T is mixing on each E k and therefore mixing.

Now we can utilize the recent results of Friedman and Ornstein [4], which show that a mixing Markov shift is Bernoulli, and the result of Ornstein [6], which shows that a transformation is Bernoulli on the union of an increasing sequence of Bernoulli subalgebras, to obtain the following result.

4 PAUL SHIELDS

THEOREM 4. I f the hypotheses o f Theorem 3 hold, then T is Bernoulli. We also have the following result which answers a query made by Friedman

[2, p. 130].

THEOREM 5. I f J-o has at least two columns, then for a dense class of sets A the induced transformation T a can be embedded in a flow.

Proof. Except for using Theorem 4 instead of Theorem 3 and the fact that Bernoulli shifts can be embedded in flows (Ornstein [5]), the proof is identical with that given in Friedman [2, pp. 127-130].

The following theorem gives the entropy of T. Using other methods, Friedman [3] has also calculated the entropy of T.

THEOREM 6. The entropy of T = r(J-o) is w(~-"o) log qo, where qo is the number of columns of J-o.

Proof. From Billingsley [1] the entropy of the restriction T. of T to Z. is

E(T.) = - ~ /~(TA n B) log (/~(TA n B)/t~(A)). A,BEP(n)

Each term will be zero unless A is a roof interval and B a base interval of Y . , in which case Lemma 3 gives

t~(TA n B) = _1 I~(A), q,

where q, is the number of" columns of ~'-,. Since the roof and base of Y , each contain q, pieces and q,l~(A)= w(9-'n) for any roof interval A, we have E(T,) = w(J-,) log q,. Note that q, = q~" and w(J'n) = 2-'w(.Y--o) so we obtain E(T,) = w(J'o) log qo. Since E(T) = lira E(T,), the theorem follows.

REFERENCES

[1] P. BILLINGSLEY, Ergodie Theory and Information, John Wiley, New York, 1965. [21 N. A. FRIEDMAN, Introduction to Ergodic Theory, Van Nostrand Reinhold, New York,

1970. [3l N. A. FRIEDMAN, On mixing, entropy and generators, J. Math. Anal. Appl. 26 (1969), 512-

528. [4] N. A. FRIEDMAN and D. S. ORNSTEIN, On isomorphism of weak Bernoulli transformations,

Advances in Math. 5 (1970), 365-394. [5] D. S. ORNSTEIN, Imbedding Bernoulli shifts in flows, Lecture Notes in Mathematics,

Springer-Verlag, Berlin, 1970, pp. 178-218 (Proc. of 1st Midwestern Conference on Ergodic Theory, Ohio State University, Mar. 27-30, 1970).

[6] D. S. ORNSTEIN, TWO Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339-348.

(Received 12 April 1971, and in revised form, 17 November 1971)