The Erwin Schrodinger International Pasteurgasse 6/7. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Pasteurgasse 6/7 Institute for Mathematical Physics A-1090 Wien, Austria Invariant Cocycles, Random Tilings and the Super-K and Strong Markov Properties Klaus Schmidt Vienna, Preprint ESI 301 (1996) January 30, 1996 Supported by Federal Ministry of Science and Research, Austria Available via

2 INVARIANT COCYCLES, RANDOM TILINGS AND THE SUPER-K AND STRONG MARKOV PROPERTIES KLAUS SCHMIDT Abstract. We consider 1-cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations we show that the skew product extension dened by the cocycle is ergodic. As an application we obtain an extension of many of the results in [9] to higher-dimensional shifts of nite type, and prove a transitivity result concerning rearrangements of certain random tilings. 1. Introduction Let R be a discrete, nonsingular, ergodic equivalence relation of a standard probability space (;B ; ), and let V be a measure preserving automorphism of (;B ; ) which normalises R, i.e. which sends R-equivalence classes to R-equivalence classes. We consider 1-cocycles c: R 7?! G on R taking values in a locally compact, second countable, abelian group G which are invariant under V, i.e. which satisfy that c(v x; V x 0 ) = c(x; x 0 ) -a.e. on R (for the denitions we refer to Section 2). If the automorphism V is asymptotically central (Denition 2.2), then every V -invariant cocycle has the following property: there exists a null set N 2 B such the closure H in G of the set fc(x; x 0 ) : (x; x 0 ) 2 Rr(NN)g is a subgroup of G, and that c denes an ergodic skew product extension of R by H (Theorem 2.3). This theorem is applied in Section 3 to the Gibbs equivalence relation of a d-dimensional shift of nite type (SFT), where d 1, and to a shift-invariant probability measure which is quasi-invariant and ergodic under the Gibbs relation of (such as Gibbs measures of a suciently rapidly decaying continuous function : 7?! R). For such a measure the shifts m ; 0 6= m 2 Z d, are asymptotically central automorphisms of the Gibbs relation, and the resulting application of Theorem 2.3 yields some surprising properties of the SFT which go well beyond the super-k property discussed in [9]. In [9] it was shown that, if is a one-dimensional SFT, then any Gibbs measure of a function with summable variation on is ergodic under a large family of subrelations of the Gibbs relation, and in particular under the relation in which two points x; x 0 2 are equivalent if their coordinates are nite permutations of each other. Here we extend this result to dimensions d 1, and to any subrelation of the Gibbs relation arising as the kernel of a shift-invariant cocycle c: 7?! G with values in a locally compact, second countable group G (Theorem 3.1 and Corollary 3.3). As an illustration we consider the two-dimensional 2-shift = f0; 1g 2 with uniform (i.e. equidistributed) Bernoulli measure and regard every x 2 as a colouring of the lattice Z 2 with the colours white and black, corresponding to the possible values 0 and 1 of each coordinate of x. The connected monochromatic subsets of Z 2 of such a colouring are known to be nite for -a.e. x 2, and constitute a random tiling of Z 2 by irregularly shaped black and white tiles of nite size. As a consequence of Corollary 3.3 one obtains that, for a typical point x 2, the tiles occurring in this tiling can be rearranged by a nite permutation so that they still t together exactly, that no tile touches another tile of the same colour, and that the resulting point lies in any given subset B of positive measure (Example 3.4). This represents a considerable strengthening of the 1

3 2 KLAUS SCHMIDT super-k property, since every rearrangement of these tiles corresponds, in particular, to a nite permutation of the coordinates of x. Another illustration, related to the strong Markov property of shifts of nite type, appears in Example 3.5: let be a one-dimensional SFT, furnished with the Gibbs measure of a function with summable variation, and let B is a Borel set of positive measure. Then -a.e. x 2 visits B innitely often both in the past and the future, and we write S(B; x) = (n k ; k 2 Z) Zfor the sequence of successive times at which x = (x m ) visits B. For each visit at time n k ; k 2 Z, we consider the string C(x; k) = (x nk ; : : :; x nk+1?1) consisting of the coordinates of x until the next visit to B. If we call two points x; x 0 2 equivalent if their sequences of strings (C(x; k); k 2 Z) and (C(x 0 ; k); k 2 Z) dier only by a nite permutation, then this equivalence relation turns out to be ergodic whenever B can be approximated suciently quickly by closed and open sets. Again we conclude that, for typical points x; x 0 2, the sequence of strings arising from x can be nitely permuted to resemble locally the sequence of strings coming from x 0. This work arose out of discussions with R. Burton and J. Steif during their visit to the Erwin Schrodinger Institute in September I am particularly grateful to them for introducing me to the setting of Example 3.4, and to J. Steif for the reference [5]. This paper is a direct continuation of [9]; as it turns out, the results presented here require little more than stripping down some of the methods in [9] to their bare essentials. 2. Invariant cocycles We begin by recalling a few denitions and results from [4]. Let be a standard Borel space with Borel sigma-algebra B. A discrete Borel equivalence relation R is an equivalence relation which is a Borel set, and for which each equivalence class R(x) = fx 0 2 : (x; x 0 ) 2 Rg; x 2, is countable. Since we only deal with discrete Borel equivalence relations we shall use the term equivalence relation to denote a discrete Borel equivalence relation. The full group [R] of an equivalence relation R is the group of all Borel automorphisms W of with W x 2 R(x) for every x 2. Under our assumption that R is discrete there exists a countable subgroup? [R] with?x = fx : 2?g = R(x) for every x 2 ; in particular, the saturation R(B) = S x2b R(x) of every B 2 B lies in B. A sigma-nite measure on B is quasi-invariant under R if (R(B)) = 0 for every B 2 B with (B) = 0, and ergodic if it is quasi-invariant and either (R(B)) = 0 or ( r R(B)) = 0 for every B 2 B. Every which is quasi-invariant under R is also quasi-invariant under every W 2 [R], and by piecing together the Radon-Nikodym derivatives d=d; 2?, we can dene a Borel map : R 7?! R such that (1) (W x; x) = (dw=d)(x) -a.e., for every W 2 [R], (2) (x; x 0 ) (x 0 ; x 00 ) = (x; x 00 ) for every (x; x 0 ); (x; x 00 ) 2 R. The map is the Radon-Nikodym derivative of under R, and is (R-)invariant if there exists a -null set N 2 B with (x; x 0 ) = 1 for every (x; x 0 ) 2 R r (N N). In order to simplify terminology we say that a property holds (mod ), or -a.e. on R, or for -a.e. (x; x 0 ) 2 R, if there exists a -null set N 2 B such that the property holds everywhere on Rr(N N). Furthermore, if R is an equivalence relation and a sigma-nite measure on B which is quasi-invariant (resp. ergodic) under R, then R is said to be a nonsingular (resp. ergodic) equivalence relation on (;B ; ). A nonsingular equivalence relation R on (;B ; ) is hypernite if there exists a (nonsingular) automorphism V of (;B ; ) such that R = f(v k x; x) : k 2 Zg (mod ). Fix an equivalence relation R on a standard Borel space and a nonatomic, sigma-nite measure on B which is quasi-invariant and ergodic under R. A Borel

4 INVARIANT COCYCLES AND RANDOM TILINGS 3 automorphism V of is a (-nonsingular) automorphism of (R; ) if is quasi-invariant under V and W (R(x)) = R(W (x)) for -a.e. x 2. Write Aut(R; ) for the group of all nonsingular automorphisms of R and observe that [R] Aut(R). Let G be a locally compact, second countable, abelian group. A Borel map c: R 7?! G is a (1-)cocycle of R if c(x; x 0 ) + c(x 0 ; x 00 ) = c(x; x 00 ) for every (x; x 0 ); (x; x 00 ) 2 R. A cocycle c: R 7?! G is a (1-)coboundary if there exists a Borel map b: 7?! G with c(x; x 0 ) = b(x)? b(x 0 ) for -a.e. (x; x 0 ) 2 R, and two cocycles c; c 0 : R 7?! G are cohomologous if they dier by a coboundary. If V 2 Aut(R; ), then a cocycle c: R 7?! G is invariant under V if c(x; x 0 ) = c(v x; V x 0 ) for -a.e. (x; x 0 ) 2 R. Under pointwise addition, the set of G-valued cocycles on R forms a group Z 1 (R; G), the sets B 1 (R; G) Z 1 (R; G) of coboundaries and Z 1 (R; G) V Z 1 (R; G) of V -invariant cocycles are subgroups, and the quotient group H 1 (R; G) = Z 1 (R; G)=B 1 (R; G) is called the (rst) cohomology group of R with coecients in G. An element g 2 G is an essential value of a cocycle c 2 Z 1 (R; G) if f(x; x 0 ) 2 R \ (B B) : c(x; x 0 ) 2 N(g)g 6=? for every neighbourhood N(g) of g in G and every B 2 B with (B) > 0 (cf. [10]). The set E(c) of essential values of c is a closed subgroup of G. If G is noncompact we write G = G [ f1g for the one-point compactication of G, say that 1 is an essential value of c if f(x; x 0 ) 2 R \ (B B) : c(x; x 0 ) =2 Kg 6=? for every compact set K G and every B 2 B with (B) > 0, and set E(c) = ( E(c) [ f1g if 1 is an essential value of c; E(c) otherwise. An important feature of E(c) is that E(c1 ) = E(c2 ) whenever c 1 ; c 2 2 Z 1 (R; G) are cohomologous, and that c 2 B 1 (R; G) if and only if E(c) = f0g. For later use we also dene the range R(c) of a cocycle c 2 Z 1 (R; ): an element g 2 G lies in R(c) if and only if there exists, for every neighbourhood N(g) of g in G and every -null set N 2 B, an element (x; x 0 ) 2 R r (N N) with c(x; x 0 ) 2 N(g). Proposition 2.1. Suppose that V 2 Aut(R; ) is weakly mixing, i.e. that V has no nonconstant eigenfunction in L 1 (;B ; ). Then B 1 (R; G) \ Z 1 (R; G) V = f0g. Proof. If 0 6= c 2 B 1 (R; G) \ Z 1 (R; G) V, then there exists a Borel map b: 7?! G with c(x; x 0 ) = b(x)? b(x 0 ) = b(v x)? b(v x 0 ) for -a.e. (x; x 0 ) 2 R, and the ergodicity of under R implies that the map b V? b is -a.e. equal to a constant g 2 G. For every character 2 ^G the map b: 7?! S= fz 2 C : jzj = 1g is an eigenfunction of V with eigenvalue (g). As V is mixing, g = 0, b is constant, and c = 0. For the remainder of this section we assume that R is an equivalence relation, and that is a probability measure on B which is quasi-invariant and ergodic under R. Denition 2.2. A measure preserving automorphism V 2 Aut(R; ) is asymptotically central if lim n!1 (B4V n W V?n B) = lim n!1 (V?n B4W V?n B) = 0 (2.1) for every W 2 [R] and B 2 B.

5 4 KLAUS SCHMIDT Theorem 2.3. Let R be a nonsingular and ergodic equivalence relation on a standard probability space (;B ; ), and let V be an asymptotically central automorphism of (R; ). Then V is strongly mixing. Furthermore, if G is a locally compact, second countable, abelian group, then every cocycle c 2 Z 1 (R; G) V satises that E(c) = R(c). In particular, if E(c) is the Haar measure of the closed subgroup E(c) = R(c) G, then E(c) is nonsingular and ergodic under the skew product equivalence relation R (c) = f((x; g + c(x; x 0 )); (x 0 ; g)) : (x; x 0 ) 2 R; g 2 Gg (( E(c)) ( E(c))): (2.2) Proof. In the terminology of [11] and [6], Denition 2.2 can be expressed as saying that a measure preserving automorphism V of (R; ) is asymptotically central if and only if, for every B 2 B, the sequence (V?n B; n 0) is asymptotically invariant under R. In particular, every weak limit of the sequence (1 V?n B; n 0) of indicator functions must be invariant under every W 2 [R] and hence, by ergodicity, constant. Since V preserves, this constant is equal to (B). Hence lim n!1 (B 1 \ V?n B 2 ) = (B 1 )(B 2 ) for all B 1 ; B 2 2 B, so that every asymptotically central automorphism of (R; ) is mixing. If c 2 Z 1 (R; G) then it is clear that E(c) R(c). Conversely, if g 2 R(c), then there exists, for every neighbourhood N(g) of g in G, a set C 2 B with (C) > 0 and an element W 2 [R] such that c(w x; x) 2 N(g) for every x 2 C. Let B 2 B with (B) > 0. Since V is mixing, lim n!1 (B \ V?n C) = (B)(C); and (2.1) guarantees that lim n!1 (B4V n W?1 V?n B) = 0. In particular, lim n!1 (B \ V n W?1 V?n B \ V?n C) = (B)(C) > 0; and every x 2 B \ V n W?1 V?n B \ V?n C satises that x 2 B, V n W V?n x 2 B, V n x 2 C, and c(v n W V?n x; x) = c(w V?n x; V?n x) 2 N(g). This proves that g 2 E(c). Since E(c) = R(c) we may assume (after modifying c on R \ (N N) for some null set N 2 B, if necessary) that c(x; x 0 ) 2 E(c) for every (x; x 0 ) 2 R, so that the skew product relation R (c) in (2.2) is well dened. The ergodicity of E(c) under R (c) is a well known and easy consequence of the denition of E(c) (cf. e.g. [10]). Corollary 2.4. Let R be an equivalence relation, and let be a probability measure on B which is quasi-invariant and ergodic under R. If there exists an asymptotically central automorphism of (R; ), then is either R-invariant, or (R; ) is of type III for some 2 (0; 1] (i.e. E(log ) 6= f0g whenever is not invariant under R). Proof. By assumption, the cocycle c = log : R 7?! R lies in Z 1 (R; R) V, and Theorem 2.3 guarantees that either E(c) = f0g (in which case is R-invariant), or that E(c) = R(c) is a closed, nonzero subgroup of R. In the latter case E(c) is either equal to R or to Z for some 0 < 2 R, and (R; ) is therefore either of type III 1 or of type III exp(?). Corollary 2.5. Let R be an equivalence relation, a probability measure on B which is quasi-invariant and ergodic under R, V an asymptotically central automorphism of (R; ), (G; ) a locally compact, metric, abelian group with identity element 0 G, and c 2 Z 1 (R; G) V. Then there exists a -null set N 2 B with the following property: for every x 2 rn, every " > 0, and every B 2 B with (B) > 0 there exists an x 0 2 R(x) \ B with (c(x; x 0 ); 0 G ) < ". In particular, if E(c) is a discrete subgroup of G then is ergodic under the equivalence relation R (c;0) = f(x; x 0 ) 2 R : c(x; x 0 ) = 0g R: (2.3)

6 INVARIANT COCYCLES AND RANDOM TILINGS 5 Proof. The assertion of this corollary is equivalent to the ergodicity of under R (c;0). The equivalence relation R (c;0) in (2.3) is called the kernel of the cocycle c: R 7?! G. If the automorphism V of (R; ) is not asymptotically central the conclusion of Theorem 2.3 and its corollaries cannot be expected to hold, as the following example shows. Example 2.6. Let (;B ; ) be a standard probability space, and let S; T be commuting ergodic, measure preserving automorphisms of (;B ; ) such that the Z 2 -action (n 1 ; n 2 ) 7! S n 1 T n 2 is free in the sense that (fx 2 : Sn 1 T n 2 x = xg) = 0 whenever (0; 0) 6= (n 1 ; n 2 ) 2 Z 2. If R = f(x; S n x) : x 2 ; n 2 Zg, then T 2 Aut(R; ) has innite outer period (i.e. ft n x : n 2 Zg \ R(x) =? for -a.e. x 2 cf. [2]), but T?n ST n = S for every n 0. In particular, (2.1) cannot hold for any B 2 B with 0 < (B) < 1. The cocycle c 2 Z 1 (R; Z) T given by c(s n x; x) = n for every x 2 and n 2 Zsatises that E(c) = f0; 1g. In fact, if G is a locally compact, second countable group, then every cocycle c 0 2 Z 1 (R; G) T looks essentially like this: the function x 7! c 0 (Sx; x) is T -invariant, since c 0 (Sx; x) = c 0 (T Sx; T x) = c 0 (ST x; x); and therefore -a.e. equal to a constant g 0 2 G. This shows that c 0 (S n x; x) = ng 0 for every n 2 Zand x 2. If G is compact and S is weakly mixing, the cocycle c 0 does have the property that E(c 0 ) = R(c 0 ), but if R(c 0 ) = fng 0 : n 2 Zg G in noncompact, then E(c 0 ) 6= R(c 0 ). We end this section with another example which shows that the inverse of an asymptotically central automorphism of an ergodic equivalence relation need not be asymptotically central. Example 2.7. Unstable and homoclinic relations of ergodic toral automorphisms. Let n 2, and let A 2 GL(n; Z) be an ergodic automorphism of = T n = R n =Z n. We regard A as a linear map on R n, write k k for the Euclidean norm on R n, and denote by F? = fv 2 R n : lim k!1 ka k vk = 0g; F + = fv 2 R n : lim k!1 ka?k vk = 0g the contracting and expanding subspaces of A. Under the factor map R n 7?! T n the spaces F? and F + are sent to the unstable group E? and stable group E + of A, which are A-invariant, dense subgroups of. In order to remain within the framework of discrete equivalence relations we x a countable, dense, A-invariant subgroup? E? and denote by R = R? = f(x + ; x) : x 2 ; 2?g the equivalence relation on generated by the orbits of?. Let be an A-invariant probability measure on B which is quasi-invariant and ergodic under translation by?. Then we claim that A is an asymptotically central automorphism of (R; ). Indeed, denote by T x = x + the translation by an element 2?, and let B 2 B with (B) > 0. As is A-invariant, the sequence of Radon-Nikodym derivatives (dt A k =d; k 0) is uniformly integrable. In particular, if d is a translation invariant metric on, we set, for every > 0, B d (B; ) = fx 2 : inf d(x; y) < g; y2b and conclude that there exists, for every " > 0, a > 0 with (T A k (B d (B; ) r B)) < "

7 6 KLAUS SCHMIDT for every k 0. As lim k!1 A k = 0 there exists an integer K 0 with T A k (B) = B A k B d (B; ) for every k K. It follows that, for k K, so that (T A k B r B) (B d (B; ) r B) < "; (B r T A k B) (T A k (B d (B; )) r T A k B) < "; (B4T A kb) = (B4A k T A?k B) < 2": This proves that A is asymptotically central on (R; ). Now assume that = is the normalised Lebesgue measure on. If A?1 is an asymptotically central automorphism of (R; ), then Denition 2.2 implies that lim (B4(B + n!1 A?n )) = 0 for every B 2 B and 2?. As the set B is arbitrary, we conclude that, for every 2?, 0 is the only limit point of the sequence (A?n ; n 0), so that every 2? is homoclinic to 0 (i.e. 2 F? \ F + ). However, if A 2 GL(n; Z) is irreducible and ergodic, but not expansive (= hyperbolic), then A has no nonzero points which are homoclinic to 0, so that A?1 cannot be an asymptotically central automorphism of (R; ). Finally assume that the toral automorphism A 2 GL(n; R) is irreducible and expansive. In this case the group = E? \ E + of points which are homoclinic to 0 is dense in, and we choose a countable, dense, A-invariant subgroup? and dene R = R? as before. The above proof shows that both A and A?1 are asymptotically central automorphisms of (R; ) for every probability measure on B which is quasi-invariant and ergodic under translation by?. In this case it is easy to construct A-invariant cocycles c: R 7?! R: take a Holder continuous map : 7?! R, and put, for every x 2 ; 2?, c(x + ; x) = k2z (Ak ( + x))? (A k x): (2.4) Since lim jkj!1 d(a k ; 0) = 0 exponentially fast as jkj! 1 we see that the cocycle c in (2.4) is well dened, and obviously A-invariant. From [8] we know that c = 0 if and only if there exists a constant a 2 R and a Holder continuous function b: 7?! R with = a + b A? b (cf. also [12]). Via Markov partitions this example is closely related to the examples discussed in the next section. 3. Shifts of finite type, random tilings and the super{k and strong Markov properties This section is devoted to a particular class of examples of asymptotically central automorphisms obtained by considering the shift-action of Z d on the Gibbs equivalence relation of a d-dimensional shift of nite type. Let A be a nonempty nite set, d 1, and let AZ d be the space of all maps x: Z d 7?! A. For every set F Z d we denote by F : AZ d 7?! A F the projection map which restricts every x 2 AZ d to the set F. The space AZ d is compact in the product topology. We write a typical point x 2 AZ d as x = (x m ) = (x m ; m 2 Z d ), where x m denotes the value of x at m, and dene, for every n 2 Z d, a homeomorphism n of AZ d by ( n (x)) m = x m+n (3.1) for every x = (x m ) 2 AZ d. The map : n 7?! n is the shift-action of Z d on AZ d, and a subset AZ d is shift-invariant if n () = for every n 2 Z d. A closed, shiftinvariant subset AZ d is a subshift, and is a shift of nite type (SFT) if there

8 exists a non-empty, nite set F Z d with INVARIANT COCYCLES AND RANDOM TILINGS 7 = fx 2 A Zd : F n (x) 2 F () for every n 2 Z d g: (3.2) If AZ d is a SFT then we may change the `alphabet' A, if necessary, and assume without loss in generality that F = f0; 1g d Z d : (3.3) The restriction of the shift-action to will again be denoted by. For the remainder of this section we x a nite alphabet A and a SFT AZ d of the form (3.2){(3.3). The Gibbs (or homoclinic) equivalence relation is dened by = f(x; x 0 ) 2 : x m 6= x 0 m for only nitely many m 2 Zd g: (3.4) We shall be interested is shift-invariant probability measures on B which are quasiinvariant and ergodic under. The standard examples of such measures are the Gibbs measures of suitable functions : 7?! R. In order to describe these measures and some related ideas we assume that G is a locally compact, second countable group with a distinguished translation invariant metric. For every continuous map : 7?! G and every k 0 we set where! k () = max f(x;x 0 )2: B(k) (x)= B(k) (x 0 )g ((x); (x0 )); B(k) = fn = (n 1 ; : : :; n d ) : jn i j k for i = 1; : : :; dg: The map has l-summable variation, l 0, if!() = n0 n l! n () < 1: Maps with 0-summable variation are simply said to have summable variation. If G = R, (a; a 0 ) = ja? a 0 j, and if : 7?! R is a map with (d? 1)-summable variation we call a probability measure on B a Gibbs measure of if is quasi-invariant under with Radon-Nikodym derivative log (x; x 0 ) = m2z d ( m (x))? ( m (x 0 )) (3.5) for -a.e. (x; x 0 ) 2. The assumption that has (d? 1)-summable variation is sucient but not necessary to guarantee that the sum on the right hand side of (3.5) converges for every (x; x 0 ) 2 ; in many interesting cases this sum converges under much weaker assumptions on, in which case one can still speak of Gibbs measures in the sense of (3.5) for such functions. We write M 1 () for the set of Gibbs measures of and observe exactly as in the case where d = 1 that M 1 () is nonempty (cf. e.g. [9]). In particular, the set M 1 () of shift-invariant Gibbs measures of is nonempty. We are interested in the case where M 1 () contains nonatomic measures which are ergodic under. For such measures to exist, obviously has to be reasonably large; however, even if is topologically transitive or minimal (i.e. if (x) is dense in for some or every x 2 ), the existence and possible uniqueness of such measures is a dicult and intriguing problem (cf. e.g. [1]) The following theorem is an immediate consequence of Denition 2.2 and Theorem 2.3.

9 8 KLAUS SCHMIDT Theorem 3.1. Let d 1, A a nite set, AZ d be a SFT, and let be a shift-invariant probability measure on B which is nonsingular and ergodic under. Then every shift m ; 0 6= m 2 Z d, is an asymptotically central automorphism of ( ; ). In particular, m is a K-automorphism (with respect to ) whenever 0 6= m 2 Z d. Proof. The only statement going beyond Theorem 2.3 is that m is a K-automorphism (and not just mixing). However, if P 0 = f?1 f0g (fag) : a 2 Ag is the state partition of, then the ergodicity of under T W guarantees W that, for every k 0, every set in the two-sided tail-sigma-algebra n0 jljn m2b(k)?m?ln(p 0 ) is saturated under and hence of measure zero or one. Corollary 3.2. Suppose that the assumptions of Theorem 3.1 are satised, and that G is a locally compact, second countable, abelian group and c: 7?! G a cocycle of which is shift-invariant, i.e. invariant under every m ; m 2 Z d. Then the skew product relation (c) dened in (2.2) is ergodic. In particular, if G is discrete, then is ergodic under the subrelation (c;0) dened in (2.3). Proof. Apply Theorem 2.3 and Corollary 2.5. As a special case of Corollary 3.2 we obtain the following statement. Corollary 3.3. Suppose that the assumptions of Theorem 3.1 are satised, and that G is a discrete, abelian group, : 7?! G a continuous map, and c : 7?! G the cocycle dened by c (x; x 0 ) = m2z d ( m (x))? ( m (x 0 )) (3.6) for every (x; x 0 ) 2. Then is ergodic under the subrelation (c ;0). In [9], Corollary 3.3 was proved for d = 1. The distinguishing feature of the subrelations covered by Corollary 3.3 is that they are dened by means of continuous functions taking values in discrete, abelian groups. The next example uses a similar construction of subrelations, but with the help of a discontinuous function. Example 3.4. The relation associated with nite permutations of monochromatic sets. Let d 1, A a nite set, AZ d a SFT, the Gibbs equivalence relation of, and let be a shift-invariant probability measure on B which is quasi-invariant and ergodic under. For every m = (m 1 ; : : :m d ) 2 Z P d d we set jmj = i=1 jm ij, and we call two points m; n 2 Z d adjacent if jm? nj = 1. A subset F Z d is connected if there exists, for all m; n 2 F, a nite sequence m 0 = m; m 1 ; : : :; m k = n in F such that m j and m j+1 are adjacent for j = 0; : : :; k? 1. The connected component C F (m) of a point m 2 F is the largest connected subset of F containing m. If d = 1, then C F (m) is the longest interval in F containing m. If we interpret the elements of A as colours, then every point x 2 represents a colouring of Z d, and we write F (x) = fm 2 Z d : x m = x 0 g for the set of coordinates with the same colour as the origin 0 = (0; : : :; 0) and denote by C(x) the connected component of 0 in F (x). Let F be the collection of all nite, connected subsets of Z d containing 0, denote by G = Z 1 the free abelian group generated by the set fe(a; F ) : a 2 A; F 2 Fg, and dene a Borel map f : 7?! G by setting ( e(x0 ; C(x)) if C(x) is nite f(x) = 0 otherwise.

10 INVARIANT COCYCLES AND RANDOM TILINGS 9 We dene a shift-invariant cocycle c: 7?! G by setting c(x; x 0 ) = m2z d f( m (x))? f( m (x 0 )) (3.7) for every (x; x 0 ) 2 (note that the sum on the right hand side of (3.7) contains only nitely many nonzero dierences). According to Corollary 3.2, is ergodic under the equivalence relation (c;0) dened as in (2.3). For an intuitive interpretation of this ergodicity property we again consider a typical point x 2 as a colouring and assume that x 0 2 is a second colouring which diers from x only in a nite region of space (i.e. x 0 2 (x)). Suppose that we know that x 0 has (up to permutation) exactly the same nite, connected, monochromatic sets as x. Can one draw any conclusions as to what colour any particular coordinate (or set of coordinates) of x 0 is? (We know by assumption the colour of each coordinate of x.) According to Corollary 3.2, our knowledge of x reveals no information whatsoever about x 0. If d = 1 this may not be too surprising, since one can t together the alternating monochromatic intervals of a point x 2 in many dierent ways (note that under our assumptions each monochromatic interval has nite length for -a.e. x 2 ). If d > 1, the sets C(x) will in general no longer be nite. However, in some interesting examples, like the full two-dimensional two-shift = f0; 1gZ 2 with uniform Bernoulli measure, the monochromatic component C(x) is nite for -a.e. x 2 (cf. [5]), but generally has an irregular shape. Nevertheless these monochromatic `tiles' of x can be tted together in many dierent ways; in particular one can rearrange, for typical points x; x 0 2, the tiles dened by x in a nite region of space so that the tiles in this rearrangement still alternate (i.e. no tile of a given colour `touches' any other tile of the same colour), and that this new arrangement looks locally like the tiling arising from x 0. Figure 1 shows a typical partial conguration in = f0; 1gZ 2, in which the zeros and ones are represented by white and black squares, and gives an impression of the random tiling arising from the connected, monochromatic subsets of Z 2 dened by a point x 2 f0; 1gZ 2. Any pair of points (x; x 0 ) 2 (c;0) also has the property that, for every a 2 A and every suciently large k 0, N(x; a; k) = jfm 2 Z d : jmj k and x m = agj = jfm 2 Z d : jmj k and x 0 m = agj = N(x0 ; a; k): In particular, the coordinates of x and x 0 dier by a nite permutation. The ergodicity of the subrelation of consisting of all pairs (x; x 0 ) 2 which dier by such nite permutations (which was called the super-k property in [9]) is a consequence of the ergodicity of the relation (c;0) dened above. For further discussion of the super-k property and related topics we refer to [9]. Example 3.5. Marker relations and the strong Markov property. Let A be a nite set, AZ a SFT, and let be the (unique) Gibbs measure of a function : 7?! R with summable variation (cf. [3]). We x a Borel set B with (B) > 0 and set, for every x 2, ( m? min fn 0 :?n (x) 2 Bg if x 2 [ (x) = n0 n (B); ; 1 otherwise, ( min fn 1 : m + (x) = n (x) 2 Bg if x 2 [ n0?n (B); 1 otherwise.

11 10 KLAUS SCHMIDT Figure 1 S Denote by (A) = k1 Ak the disjoint union of the sets A k ; k 1, write G for the free abelian group with generators fe(!) :! 2 (A)g, and dene a map : 7?! G by setting ( e(x?m? (x); : : :; x m+ (x)?1) 2 A m? (x)+m + (x) (A) if m? (x) + m + (x) < 1; (x) = 0 otherwise: Again we would like to dene a cocycle c B : 7?! G by setting c B (x; x 0 ) = k2z (k (x))? ( k (x 0 )) (3.8) for every (x; x 0 ) 2. However, if B is an arbitrary Borel set, there is no guarantee that the right hand side of (3.8) is well dened -a.e. on. In order to ensure that (3.8) makes sense -a.e. we write D(m) B ; m 0, for the nite algebra of sets generated by all cylinder sets of the form [i?m ; : : :; i m ]?m = fx = (x n ) 2 : x j = i j for j =?m; : : :; mg with (i?m ; : : :; i m ) 2 A 2m+1, and call B well approximable if min E2D(m) m0 (B4E) < 1: (3.9) In particular, if (E n ; n 0) is a sequence of sets such that E m 2 D(m) for every m 0 and P m0 (E m) < 1, then the sets S m0 E m or lim n0 E 0 4 : : :4E n are well approximable. Such sets occur, for example, in the study of isomorphisms of SFT's with nite expected code lengths (cf. [7]). Suppose that B 2 B is well approximable. Let? be the countable group of homeomorphisms of for which there exists an N() 0 with x n = (x) n for every x = (x n ) 2 and every n N(), and observe that?x = (x) for every x 2 (the elements of? are sometimes called uniformly nite dimensional homeomorphisms of ).

12 INVARIANT COCYCLES AND RANDOM TILINGS 11 For every 2? and every m 2 Zwith jmj N() we obtain that (cf. (3.5)), so that ( m (B)4 m (B)) (1 + e (2N()+1)max jj ) m2z (m (B)4 m (B)) < 1: min E2D(jmj?N()) (B4E) A straightforward calculation shows that the map x 7! c B (x; x) in (3.8) is well dened - a.e. for every 2?, since the right hand side of (3.8) contains only nitely many nonzero dierences. By setting c B = 0 on \ (N N) for a suitable null set N 2 B we obtain from (3.8) a well dened, shift-invariant cocycle c B : 7?! G. By Corollary 2.5, is ergodic under the equivalence relation (c;0) appearing in (2.3). If B = [i 0 ; : : :; i l?1 ] = fx = (x m ) 2 : x j = i j for j = 0; : : :; l?1g is a cylinder set, and if is an l-step Markov measure, then the ergodicity of (c;0) is an immediate consequence of the strong Markov property, i.e. of the fact that the strings between successive visits to B form an innite state Bernoulli process. The ergodicity of the relation (c;0) implies a weaker form of the strong Markov property for every Gibbs measure and every well approximable `marker' set B 2 B : the equivalence relation dened by all possible (allowed) rearrangements of the times between successive visits to B is ergodic (but in general not measure preserving). References [1] R. Burton and J.E. Steif, Non-uniqueness of measures of maximal entropy for subshifts of nite type, Ergod. Th. & Dynam. Sys. 14 (1994), 213{235. [2] A. Connes and W. Krieger, Measure space automorphisms, the normalizers of their full groups, and approximate niteness, J. Funct. Anal. 24 (1977), 336{352. [3] Z. Coelho and A.N. Quas, Criteria for Bernoullicity, preprint (1995). [4] J. Feldman and C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289{324. [5] Y. Higuchi, Coexistence of the innite () clusters; a remark on the square lattice site percolation, Z. Wahrsch. Verw. Gebiete 61 (1982), 75{81. [6] V.F.R. Jones and K. Schmidt, Asymptotically invariant sequences and approximate niteness, Amer. J. Math. 109 (1987), 91{114. [7] W. Krieger, On the nitary isomorphisms of Markov shifts that have nite expected coding time, Z. Wahrsch. Verw. Gebiete 65 (1983), 323{328. [8] A. Livshitz, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 6 (1972), 1278{1301. [9] K. Petersen and K. Schmidt, Symmetric Gibbs measures, preprint (1995). [10] K. Schmidt, Cocycles on ergodic transformation groups, MacMillan (India), Delhi, [11] K. Schmidt, Asymptotically invariant sequences and an action of SL(2; Z) on the 2-sphere, Israel J. Math. 37 (1980), 193{208. [12] K. Schmidt, The cohomology of higher-dimensional shifts of nite type, Pacic J. Math. 170 (1995), 237{270. Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria, and International Erwin Schrodinger Institute for Mathematical Physics, Pasteurgasse 6, A-1090 Vienna, Austria address: klaus.schmidt@univie.ac.at