A K -automorphism with non-isomorphic ZxZ2 actions

A K-AUTOMORPHISM WITH NON-ISOMORPHIC Z x Z2 A C T I O N S By KYEWON PARK Abstract. We construct a measure space (X, T) with a K-automorphism. We define two different order two maps on (X, T), S, and S~, These maps commute with T. We show two group actions {T, Sj} and {T, 82} are not isomorphic. 1. I n t r o d u c t i o n D. Rudolph has shown that if T~ and Tz are Bernoulli and isomorphic, then {7"1,Sl} and {T~, $7} are isomorphic where T~Si = S,-Z and S 2 = I for i = 1,2. That is, in the Bernoulli case, an isomorphism between Z-actions forces an isomorphism between Z x Z2-actions [5]. In this paper we want to show that this property does not necessarily hold for K-automorphisms. Hence this counterexample serves the purpose of distinguishing K-automorphisms from Bernoulli automorphisms. Also it is a stepping stone of the forthcoming paper in which we construct two non-isometric ergodic Z-foliations with isomorphic Z-actions, where we need a skew commuting property between T~ and S , S~T~S~ = T~ L. Also this idea can be easily extended to R ~-actions where ergodic foliations look more natural. The outline of the construction is as follows. We first construct a Kautomorphism U using a block structure that is similar to that by Ornstein and Shields [2]. And we make a two point extention to add colors following Rudolph's idea. Hence every point has a double name, the base name and the colored name. We will construct a K-automorphism T which admits two different order two maps St and $2 so that they commute with T and {T, S~} and {T, S2} are not isomorphic. We refer the reader to Ornstein's text [1] or Ornstein and Shield's article [2] for a complete exposition of a geometric construction of a base transformation. We define two different order two maps, S~ and S,. Define St to change base names and $2 to change colored names. Since an isomorphism between {T, St} and {T, $2} satisfies ~$1 = S~lk, if once a long enough n-block, say b, is mapped to a block a (a is not necessarily an n-block), then an n-block with the base name changed by $1 but with the same colored name has to be mapped to a block with the same base name but with the changed colored name from a. But we will show using the block structure and the regularity of colored names that this is impossible. I would like to give my thanks to Professors D. S. Ornstein and Y. Katznelson whose helpful suggestions were vital to this work. 37 JOURNAL D'ANALYSE MATH~MATIOUE, VoL 45 (1985)

38

K. PARK 2. C o n s t r u c t i o n

o f ()~, T, ~ ' )

First we will construct U, a version of Ornstein and Shields' K-automorphism [2]. We refer the reader to [2] and [4] for more rigorous constructions and proofs. Since the construction here is very much parallel to that by D. Rudolph, we will simply highlight the different aspects of constructions. The transformation U will be defined as a shift on a space X of bilateral sequences of four symbols {o, r, [, s} relative to a measure on X. The construction will show how to determine the measure. Each sequence x in X is made up of nested n-blocks. We define two different 1-block names, with equal probability. A 1-block name begins with two s's followed by either 2 ~ consecutive o's or r's named after Ornstein and Rudolph with equal probability, and finally by two s's again (see Fig. 1). Construction of an n-block name from (n - 1)-block names goes as follows. Choose independently a sequence of 22" (n - 1)-block names. Let f(n) be a random integer out of the set { 2 , 4 , 6 , . . . , 2 n } independent of the chosen sequence of (n - 1)-blocks. An n-block name begins with a string of f's, [ ( n ) long, again ends with a string of ]"s, 2n - f(n) (n >-2) long. In between are the chosen 22" (n - 1)-block names separated by spacers, s. The length of spacers between the kth ( n - 1)-block and the (k + 1)st ( n - 1)-block is ks(n)-1. This gives the various n-block names, each with equal probability. We put two s's in front of the first (n - 1)-block and at the end of the last (n - 1)-block to show where (n - 1)-blocks start and end in an n-block. If we let s(n)= 100 n 3 (n = 2 . . . . ) then certainly s(n) - 1 > El=, f(k) q ( 1 ) = 0 ) . . . (1), Let X. be the union of all n-blocks. Since we

2s's

[ 2s's two 1-blocks

2 Ct-VI~::2:::--'T~3

[(n)

(n - l~block

2s(n)- 1

s(n)-I . . . .

T

(n - l~block

?--V-- ...... (n - l~block

L-2:;:I:II;:2;;

(2 ~" - 1)s(n)- 1

l

...s..

l (n - l)-block

an n-block Fig. I.

2n -/(n)

K-AUTOMORPHISM

39

chose the 1-block long enough, it is not hard to see that the total amount of s's and ['s added to X,.I from X, is less than p.(X,)/26" [4]. H e n c e / z ( X , ) are uniformly bounded. We note here that the number of ['s at the beginning of each block is equidistributed over the even integers from 2 to 2n. If we let k ( n ) be the number of different n-blocks, then we have k ( n ) = ( k ( n - 1))22" 9 n. Also it is clear from the construction that an (n - 1)-block occurs at all possible (n - 1)-block places in an n-block with equal probability, independent of each other. If we call X the space where U is well defined, then we have a partition on X according to the symbols, P = {Po, P,, Pr, P,}. To construct the K-automorphism T, let .,~ be the direct product of X with the two point space {B, W}. Partition .~ into

{po x{w}, po • ={Wo,Bo,

t', x{w}, P, x{B}, Pr x{B, W}, P, x{B, W}}

W , , B , , P f , P,}.

We call this partition/5. Since we often do not have to differentiate the two symbols o and r, we will say that a point is in W (colored white) if it is in Wo tJ W, and in B (colored block) if it is in Bo U B,. We will define T for a point in .,~ by specifying changes of colors at each n-block. For each n there are two different colored n-block names whose base names are the same. There are two colorings in 1-blocks which makes 4 different 1-blocks. Suppose we have shown how to color (n - 1 ) blocks. We call an (n - 1)-block white (black) if its color starts with white (black). We will show how to construct white n-blocks. Choose independently a sequence of 22" ( n - 1)-blocks according to base names. Choose the color of the 1st (n - 1)-block to be white, the 2nd to be black, the 3rd to be black, the 4th to be white and the 5th to be white . . . . , so that the colors of (n - 1)-blocks in a white n-block is W B B W W B B W W B B 9 9 9 W. And a black n-block is an n-block which is colored just the opposite, B W W B B W W B B . . 9 B. Hence T may be regarded as a skew product of U with an interchange map. We want to define two different automorphisms of order two, $1 and $2. Define S~(x, to) = ( x ' , to) where x and x' have their symbols o's and r's interchanged in their base names. Note that they have the same colored name. Define

S2(x,

to

)

(x, B )

if to = W,

i I. (x, W)

if to = B,

i.e., $2 maps a point in )~ to the point whose base name is the same but its colored name is the opposite. It is clear that S~ = I and S~TSi = T for i = 1 and 2. 3. T is a K - a u t o m o r p h i s m To prove that T is a K-automorphism, we note three major differences in the construction of U from the general construction of a K-automorphism.

40

K. PARK

(i) We start with two 1-blocks for U. (ii) We put f's at the end of n-blocks instead of e's. (iii) We put only an even number of f's at each step. (This is not quite relevant for our purpose.) From the construction it is clear that (i) and (ii) do not change the rigidity of blocks and the randomness of the occurrences of blocks in their bigger blocks. To prove that T is a K-automorphism, we need to show that the distant past has little ettect on the future. We will use the following definition of the K-automorphism. For given l and e > 0, there is an N such that V~ T~/3 is e-independent of V ~§ T~/3 for any k and any n > N. T h e o r e m 1.

T is a K-automorphism.

P r o o f . We will sketch the proof. (For details, see [4].) We fix k and choose K such that for most x E X, {x, Tx, T2x,..., T~x} are completely in a K-block. Let h(n) be the length of an n-block. We choose L so large that 2L > h(K + 1). We let N = h(L)+ 1 and n > N. It is clear that {T'x, T'§ . . . . . Tn§ are completely contained in a K-block for most x E X. Let x be a point satisfying this condition and be in one of the atoms A in V of T,/3. Since n > N = h (L) + 1, an L-block starts somewhere between x and T'x. We make the following observations in our construction (i) The number of ['s at the beginning of an L-block is equidistributed over all even integers between 2 and 2L, independent of the choice of the sequence of (L - 0-blocks. (ii) Any (K + 1)-block can occur at an allowed position in an L-block with equal probability. (iii) Half of the K-blocks start at an odd distance from the beginning of their (K + 1)-block and half at even. The probability that a certain K-block starts at an odd distance is equally likely as at an even distance. Hence we see T" (x) at all possible places in all possible K-blocks with equal probability. This shows that the distribution of U (i.e., T on uncolored names) conditioned on A is the same as the unconditioned distribution of U. Hence U is a K-automorphism. Since half of the K-blocks located at even (or odd) positions in the (K + 1)-block are black and half white, and since T'(x) can lie in any K-block in a ( K + l ) - b l o c k with equal probability, it follows that (T,/3) is a Kautomorphism. 4. Rigidity of b l o c k s We wish to show that the P-names of points have a rigid block structure. Hence any isomorphism between {T, S~} and {T, $2} has to preserve much of the block structure of P-names. As usual, we want to prove that if two names agree in most places, then their n-blocks are not too far apart. Let a and b be two n-blocks in

K-AUTOMORPHISM

41

P-names of x and y respectively. Let nl and n2 be integers such that T ' , x is the first term of a and T'~(y) is the first term of b. We say two n-blocks, a and b, are close if In, - n2[ < ~ : ~ , f ( k ) . We write a - b = In, - n21. Also if a and b are close and of the same order in their (n + D-blocks which we n a m e a" and b" respectively, then a" and b" are close and every n-block in a" is close to an n-block of the same order (corresponding n-block) in b". It is clear from the construction that an n-block a in the p - n a m e of x is close to at most one n-block b in the P - n a m e of y for any x and y. We also note that if any (n - 1)-block a ' in a is close to an (n - 1)-block b' in b and their orders in a and b respectively are different, then there is at most one other (n - 1)-block in a that is close to an (n - 1)-block in the P - n a m e of y. Since h(n - 1) is much bigger than the sum of lengths of spacers, if the ith (n - D-block a', from a ' has a close (n - 1)-block in b, then it has to be the ith (n - 1)-block b~ from b'. But their respective orders in a and b are different. H e n c e they are at least s ( n ) - E ~ , , / ( k ) ( > E k - , , / ( k ) ) apart. If a has an (n - 1)-block ,i' which is close to an (n - 1)-block/~' in the P - n a m e of y, then b' has to lie in an n-block/~ which is different from b. Also it is clear that their orders in a and/~ are different. From the previous argument there is no other ( n - 1 ) - b l o c k in a that is close to an (n - 1)-block in/~. Hence there are at most two (n - D-blocks in a that have close (n - D - b l o c k s in the /5-name of y.

L e m m a 1. There exists an ~ > 0 such that i[ a is an n-block in the P - n a m e of x, then ]'or any y, either (i) there is an n-block in the P - n a m e o[ y close to a, or (ii) there are more than g h ( n ) places i such that x i # y~. Proof.

Let e,-1 be the least n u m b e r such that if (i) is not true, then there are

more than e,_~h(n - 1) places where x~ and yi do not agree. We put el = l/h(1). We will determine ~ inductively. If (i) is not true, then a has at most two (n - D-blocks close to some ( n - D - b l o c k s in the P - n a m e of y. H e n c e there are at least (22" - 2 ) h ( n - 1)e,-I places such that x~# y,. Since h ( n ) < (22" + 1)h(n - 1), e,, ~' (22" - 2)h(n - 1)e.-1 > 1 - 2 -2"§ = h(n) = 1 + 2 - 2 , e,-l. Since I I ~ t [(1 - 2-2k+~)/(1 + 2-'k)] is bounded away from 0 for any n, we have (ii). C o r o l l a r y 1. Let a be an n-block (n >=6) in the P - n a m e o[ x and y be any point. Let K > h ( n ) / 2 " - ' . If we have x, ~ y, on less than ~K/4 places for a segment of K consecutive x,'s in a, then there exists an n-block b in the P - n a m e o / y such that a and b are close. Proof. Since K > 2"h(n - 1), any consecutive K-long names will contain m a n y complete names of (n - 1)-blocks. More than half of these names of (n - 1)-blocks

42

K. PARK

must have an error in less than gh(n - 1) places. Hence these (n - 1)-blocks have close (n - 1)-blocks in the/5-names of y. This can happen only when the n-block a has a close n-block b in the /5-name of y. Thus if two n-blocks agree even on a small fraction, then these two blocks must be close. Next we want to show that if two blocks in t h e / 5 - n a m e of x and in the /5-name of y agree in most places, then these two blocks have to be of the same color. For this purpose, we will show that if two blocks in the/5-names agree across a small segment, then 1-blocks have to match very well. L e m m a 2. Let a and b be two n-blocks that are close in the P - n a m e of x and y respectively. Let K > h (n)12"-'. Given e > O, if x~~ y, in less than egK places [or a segment o[ K consecutive xi's in the overlap, then the fraction o[ 1-blocks in this segment o[ K consecutive x~' s that are close to the 1-blocks o[ the same order in b is at least 1 - 2e. Proof. Let I be the place (index) where this K-long consecutive overlap starts. We note here that if an m-block in a is not close to an m-block of the same order in b and the (m + 1)-block in a containing this m-block is close to the corresponding (m + 1)-block in b, then there is a large error in this (m + D-block. If the kth m-block in a is not close to the kth m-block in b, then none of the (m - 1)-blocks in this m-block is close to the (m - 1)-block of the same order in the m-block in b. Hence inductively none of the 1-blocks in the k th m-block in a is close to the 1-block of the same order in the k th m-block in b. Also none of the m-blocks in the (m + D-block containing this kth m-block in a is close to its corresponding m-block in b (but it is possible for this (m + D-block to be close to an (m + 1)-block in b). Let ~ be the collection of m-blocks located between positions 1, l + 1 , . . . , l + K which are not close to the corresponding m-blocks in b, but whose (m + 1)blocks are close to the corresponding (m + 1)-blocks in b. Since a and b are close, any block (especially any 1-block) that is not close to its corresponding block in b is contained in one of these m-blocks. In each of these m-blocks, by Lemma 1 there exists at least ~h (m) places where names do not match. How many such blocks can we afford in places 1, 1 + 1. . . . . 1 + K ?

,,, b,o~ s i, ~ gh (m ) < EgK. Hence 9,, b , ~ i, ~ h ( m ) < eK. However, .,,b,o~,,.~h(m)> h(1) N where N is the number of 1-blocks contained in one of these m-blocks in r~. Hence

K-AUTOMORPHISM

43

N < EK/h(1). But any K-long sequence has at least K/2h(1) 1-blocks. Hence the fraction of good matching 1-blocks is bigger than 1

_ eK/h(1) _ K/2h(1)- 1-2e.

Next we want to prove their colors should match well. L e m m a 3. There is an g > 0 such that if a is an n-block in the if-name of x and x,# y, for at most ~h(n) places, then there is an n-block b in the if-name of y which is close to a and their colors are the same. Proof. Let ff = g/lO. Since g < g, it follows that a and b are close. By Lemma 2, at least half of the 1-blocks in a have to be close to their corresponding 1-blocks in b. If two close 1-blocks have different colors, their error is in at least ~h(1) places. If a and b have opposite coloring, then all 1-blocks in a and the corresponding 1-blocks in b have opposite coloring. Then there are at least ~h(1). (number of good matching 1-blocks in a) places where their colors are opposite. Hence the error has to be in more than 9 h(1) h ( n ) ( 1 _ ] ) = 36 h ( n ) 1--6 2h(1) places. Therefore a and b have the same color. C o r o l l a r y 2. Let a be an n-block (n is sufficiently large) in the if-name of x and y be any point. If we have x, # y, in less than gKI4 places for a segment of K consecutive x,'s in a where K > h(n)/2 n-4, then there exists an n-block b in the if-name of y such that a and b are close and of the same coloring. 5. M a i n T h e o r e m Suppose that ~b is an isomorphism between {T, S,} and {T, $2}. We will derive a contradiction from the existence of ~b. Given e > 0, we can find N ( e ) such that there exists a partition Q each set of which is a union of atoms in V_N~]~)T~p and X~=t ~(O, AO-'(/5,)) < e. Hence there exists a finite coding ~, such that the/5-name of x from - N ( e ) + l to N ( e ) + l determines the atom of the partition O which contains Tt~b,(x). By the ergodic theorem, for a.e. point x, the O-name and /5-name of ~b(x) match for all but at most a set of density e if we watch long enough names. We fix an x satisfying this property. The following observation will provide the key to the theorem. Let a be an n-block of the/5-name of x and ~i be the corresponding n-block of the if-name of St(x). That is, a and ri are n-blocks that have the same coloring but their o's and r's in their base names are interchanged. We let a ' and ~i' be strings of names mapped by ~, of a and ~i respectively. We also let a" be a string of names

44

K. PARK

whose coloring is the opposite of a ' and whose base name is the same as a'. If we assume n to be large enough, then 4' and a" will agree in at least a set of density 1-2e.

Let e be ~'~/100 where e is given in Lemrna 3. We say an n-block a in the /5-name of x is good if (i) there is an n-block b in t h e / 5 - n a m e of O(x) that overlaps a in more than ~h(n) places. (ii) a is not one of the first four or last four n-blocks in its (n + 1)-block. (iii) The O-name of O(x) and/5-names Df 0 ( x ) do not match in at most eh(n) places. Choose n large enough so that / z ( X - ? ( , ) < l - e 2 and 2N(e)/h(n)< e 2. We choose m > n such that the fraction of good n-blocks in an m-block is bigger than 1 - e and every n-block appears with proper frequency in an m-block.

4. Let a and b be good n-blocks in the ~5-name of x and b be a translate of a (b = T~(a)). Let a' and b' be n-blocks of ~(x) such that [a - a'] < ~h(n) and I b - b ' [ < ~ h ( n ) . Then Lemma

] T~(a') - b'[ < ~nf(i)

and a' and b' have the same coloring. Notice that the O-names of the blocks a and b are the same. Also since the O - n a m e and /3-name of y = ~ ( x ) differ in a set of at most density e, two h(n)-long names (y~, yz+l. . . . . yl+h(,)-~) and (yt+~, yz§ . . . . , y~§ of y = 0(X) differ in at most 2eh(n) places. Because of the positions of a ' and b', we have I T-k(b') - a'l< 89 ). Hence there is at most 8 e h ( n ) errors in this overlap. By Corollary 2, T-~(b ') and a ' have to be close and have the same coloring. We call a finite sequence a~, a 2, a ~. . . . , a k of n-blocks in the /5-names of x adjacent if they are contained in one (n + 1)-block and separated only by spacers. Proof.

Theorem.

{T, S,} and {T, $2} are not isomorphic.

P r o o f . Let a, b, c, d be adjacent good n-blocks in the/5-name of x, all of which are mapped into the same (n + 0-block in the/5-name of y = 0(x). Since there are at most density e of bad n-blocks, there are at most density 4e of adjacent quadruples which have at least one bad block. Also if n is sufficiently large, then there exist at least density 1 - e of adjacent quadruples all of thich are mapped into the same (n +1)-block. Hence there are at least density 1-5e of adjacent quadruples which are good and are mapped into the same (n + 1)-block. Let a', b', c', d' be the n-blocks in the /5-name of y which satisfy

[a'-al