Symmetric Gibbs Measures

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 7, July 1997, Pages 2775–2811 S 0002-9947(97)01934-X

SYMMETRIC GIBBS MEASURES

KARL PETERSEN AND KLAUS SCHMIDT

Abstract. We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures—a version of de Finetti’s theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, ‘canonical’ Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

1. Introduction It has long been known that many Gibbs measures on (topologically mixing) subshifts of finite type (SFT’s) are not only ergodic but are, in fact, K (they satisfy the Kolmogorov 0-1-law in that they have trivial (one-sided) tail fields). In fact the two-sided tail field is also trivial; another way to state this is to say that they are ergodic for the homoclinic or Gibbs relation (the countable equivalence relation RA in which two sequences are equivalent if and only if they disagree in only finitely many coordinates, i.e. if and only if they are in the same orbit under the action of the group Γ of finite coordinate changes). We prove that these Gibbs measures, which include the Markov measures fully supported on the subshift, are also nonsingular and ergodic for many cocycle-generated subrelations of the homoclinic relation, including the symmetric relation SA in which two sequences are equivalent if and only if one can be obtained from the other by a permutation of finitely many coordinates. On a one-sided SFT this orbit relation for the action of the group Π of permutations of finitely many coordinates is the orbit relation of the adic transformation, and ergodicity of this equivalence relation is the same as ergodicity of the adic transformation.

Received by the editors August 17, 1995 and, in revised form, August 20, 1996. 1991 Mathematics Subject Classification. Primary 28D05, 60G09; Secondary 58F03, 60J05, 60K35, 82B05. Key words and phrases. Gibbs measure, subshift of finite type, cocycle, Borel equivalence relation, exchangeability, adic transformation, tail field, interval splitting, Kolmogorov property, ratio limit theorem, Markov chain. First author supported in part by NSF Grant DMS-9203489.

c 1997 American Mathematical Society

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Our interest in these matters arose from the study of the dynamics of adic transformations, which were defined by A.M. Vershik (see [55]) as a family of models in which the cutting and stacking constructions of ergodic theory are organized in a way that makes them conveniently accessible to combinatorial analysis, and which has certain universality properties (such as containing a uniquely ergodic version of every ergodic, measure-preserving transformation on a Lebesgue space [54]). ‘Transversal flows’ had been investigated by Sinai [48], Kubo [28] and Kowada [25, 26], and also by S. Ito [23], who proved ergodicity of the adic transformation on an SFT for its measure of maximal entropy, a particular case of our Theo- rem 3.3. Adic transformations for actions of amenable groups were constructed by Lodkin and Vershik [34]. For the Pascal adic transformation (first defined by Vershik [53]), i.e. the adic transformation on the full shift, the ergodicity of Gibbs measures (including Bernoulli and Markov measures) is closely related to classical investigations of exchangeability and the triviality of remote sigma-fields. Given a (usually finite-valued) stochastic process X0,X1,... with (usually shift-invariant) distribution µ, we have the sigma-algebra of shift-invariant sets,thetail field + I = n 0 (Xn,Xn+1,...), and the exchangeable sigma-algebra consisting of F∞ ≥ B E all sets invariant under the group Π of permutations of N that move only finitely many coordinates.T (As sub-sigma-algebras of the product sigma-algebra of countably many copies of a fixed measure algebra, + , and the inclusions can I⊂F∞⊂E be proper. In the two-sided case there are also − , the sigma-algebra generated + F∞ by and − , and the two-sided tail = n 1 (Xi : i n).) Already for theF case∞ of theF∞ full shift, invariance and ergodicityF∞ under≥ B the symmetric| |≥ subrelation + + + T Sn of the Gibbs relation Rn of Σn leads to interesting results. A. M. Vershik [private communication] observed that the nonatomic, ergodic, invariant probability measures for the Pascal adic transformation on the 2-shift are in one-to-one + correspondence with the Bernoulli measures on Σ2 . By using dyadic expansions to regard this Pascal adic transformation as an infinite interval-exchange map on the unit interval [0, 1], one may recognize it as the map studied by Arnold [3] and proved by Hajian, Ito, and Kakutani [17] to be ergodic (with respect to Lebesgue measure) and later used by Kakutani [24] to prove the uniform distribution of his interval splitting procedure. (The connection was also noted in [49]. See [41] for more about interval splitting.) The Hewitt-Savage 0-1-law [19] says that for i.i.d. random variables is trivial, i.e. consists only of sets of measure 0 or 1 E+ (Bernoulli measures are Sn -ergodic). This was extended to Markov measures by Blackwell and Freedman [4]. There are analogous statements for the two-sided case ...X 1,X0,X1,X2,..., where there are past, future, and two-sided tail fields to deal with.− It took some time to sort out the relationships among the various tail fields [16, 21, 22, 32, 36, 37, 39, 38], to understand the representation of measures as mixtures in terms of the theory of Choquet simplices and ergodic decompositions [8, 7, 9, 42, 44, 56], and to investigate exchangeability in more general contexts [8, 7, 13, 20, 30, 42, 51]. For a survey of exchangeability, see [2]. Diaconis and Freedman [7] gave a necessary and sufficient condition for a measure to be a mixture of Markov measures, in terms of ‘partial exchangeability’—invariance under the subgroup of Π that preserves transition counts (as well as symbol counts). They also gave a general theory of ‘sufficient statistics’, describing how to present the (in some sense) most general symmetric measure as a mixture of extremal ones [8]. Several workers in statistical mechanics considered ‘canonical’ or ‘microcanonical’ Gibbs states, in the construction of which symbol counts, or the values of some

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other ‘energy function’, are fixed in finite regions. The point was to find the extremal measures in at least some classes of examples and thereby obtain a mixture- representation theorem for the most general such measures. Georgii [13, 14] found that certain Markov measures qualify. Similar results were obtained by Lauritzen [30], Höglund [20], and Thompson [51]. We prove (Theorem 3.3) that many Gibbs measures, including all mixing Markov measures, are ergodic under certain subrelations of RA which include the equivalence relation SA that corresponds to permutation of finitely many coordinates (or, equivalently, eventual equality of accumulated symbol counts). As a corollary, (letting k = 2 for simplicity) taking φ(x)=ψ(x)=cx0 in Theorem 3.3, we find measures µφ that are ergodic and invariant under the adic on the SFT ΣA, i.e. symmetric or exchangeable, when sequences are constrained to lie in the SFT ΣA (see Examples 5.1, 5.3, and 5.4). Similar results hold when ψ is a function of only finitely many coordinates of x. Using a well-known formula (see [40, p. 22]), one can recover explicit expressions for these Gibbs states (cf. [13, 14, 20, 30, 51]). As + a consequence of the ergodicity of Markov measures for SA we obtain some point- wise ratio limit theorems for adic transformations whose direct combinatorial proofs appear to be quite difficult (Section 7.4). Further, we identify all the probability measures on SFT’s that are invariant simultaneously for the shift and the symmetric relation SA (Theorem 6.2). More generally, we prove ergodicity of certain Gibbs ψ measures under subrelations SA of RA defined by cocycles generated by functions ψ :ΣA ,where is a ‘nearly abelian’ countable discrete group—see 3.13 and 7−→ G G Theorem 3.3; the symmetric relation SA is of this kind for a particular ψ.The dynamical and abstract approach provides a common framework for the diverse questions and results scattered through the literature and also leads to some interesting problems, mentioned in the final section of the paper, for example whether these measures are weakly mixing and whether there are some implications for various kinds of interval splitting or for the statistical mechanics of materials. The authors thank C. Ji, U. Krengel, and A. M. Vershik for their suggestions that contributed significantly to the progress of this work, and the referee for several improvements in the presentation.

2. Background In this section we review the elements that we will need from the general theory of Borel equivalence relations, deﬁne adic transformations, and recall what is known about ergodic invariant measures for the Pascal adic transformation on the 2-shift.

2.1. Borel equivalence relations. First we establish the basic terminology and notation concerning Borel equivalence relations that will be needed later (cf. [10, 11, 47]). Let (X, ) be a standard Borel space, and let R X X be a discrete Borel equivalenceB relation, i.e. a Borel subset which is an⊂ equivalence× relation, and for which in addition the equivalence class

R(x)= x0 X:(x, x0) R { ∈ ∈ } of every point x X is countable. Under this hypothesis the saturation ∈ R(B)= R(x) x B [∈

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of every Borel set B is again a Borel set. The full group [R]ofRis the group of all Borel automorphisms∈B W of X with Wx R(x) for every x X.Thereexists a countable subgroup G [R] such that ∈ ∈ ⊂ Gx = gx : g G = R(x) { ∈ } for every x X. A sigma-finite measure µ on is quasi-invariant under R if µ(R(B)) = 0∈ for every B with µ(B) = 0; itB is ergodic if, in addition, either ∈B µ(R(B)) = 0 or µ(X r R(B)) = 0 for every B . Let µ be a probability measure on which is∈B quasi-invariant under R.Thenµis also quasi-invariant under every W B[R]. In particular, if G [R] is a countable subgroup with Gx = R(x) for every x∈ X, then one can patch⊂ together the Radon- ∈ Nikodým derivatives dµg/dµ, g G, and define a Borel map ρµ : R (0, ) R with the following properties: ∈ 7−→ ∞ ⊂

(1) ρµ(Wx,x)=(dµW /dµ)(x) µ-a.e., for every W [R], ∈ (2) ρµ(x, x0)ρµ(x0,x00)=ρµ(x, x00) whenever (x, x0), (x, x00) R. ∈ The map ρµ is the Radon-Nikod´ym derivative of µ under R,andµis (R-)invariant if ρµ(x, x0) = 1 for all (x, x0) R r ((X N) (N X)), where N is a µ-null set. ∈ × ∪ × ∈B

2.2. The main examples. In order to describe a class of adic transformations of particular interest to us we assume that V =(Vk,k 0) is a sequence of ﬁnite, ≥ nonempty sets and put XV = k 0 Vk, with the product of the discrete topologies. ≥ Two elements x =(xk),x0 =(x0)inX are comparable (in symbols: x x0)if Q k V ∼ xk=x0 for all suﬃciently large k 0. The set k ≥ (2.1) R = (x, x0) X X : x x0 V { ∈ V × V ∼ } is a Borel set and an equivalence relation, and each equivalence class

(2.2) R (x)= x0 X :x0 x V { ∈ V ∼ } of RV is countable. In other words, RV is a discrete, standard equivalence relation in the sense of [10]; it is called the homoclinic equivalence relation or Gibbs relation of X . More generally, if Y X is a nonempty closed set, we denote by V ⊂ V (2.3) RY = R (Y Y ) V ∩ × the Gibbs relation of Y , and observe that RY (y)=R (y) Y and RY (B)= V ∩ RV(B) Yfor every y Y and B Y . The following conditions on Y are easily seen to be∩ equivalent: ∈ ⊂

(T1) the equivalence relation RY on Y is topologically transitive, i.e. there exists a y Y whose equivalence class RY (y)isdenseinY, ∈ (T2) for every pair 1, 2 of nonempty open subsets of Y there exists a point O O y Y with RY (y) i =∅for i =1,2. ∈ ∩O 6 If Y fulﬁlls (either of) these conditions we shall simply say that Y satisﬁes condition (T). Now assume that each of the sets Vk is totally ordered with an order

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+ minimal element x−; furthermore, if x X ,x=x , then there exists a unique ∈ V 6 smallest element succ(x) x0 R (x):x x0 , called the successor of x.Put ∈{ ∈ V ≺ } + C =R (x−) R (x ). V V ∪ V The restriction of the map x succ(x)toXVrCV is a homeomorphism of 7→ XV r CV; in order to extend this homeomorphism to a Borel automorphism of the entire space XV we set

succ(x)ifx XVrCV, T <(x)= ∈ V x if x C , ( ∈ V < and call TV = TV the adic transformation of XV determined by the orders < of the sets V. In contrast to the space XV, a nonempty, closed subset Y XV may have many minimal and maximal elements with respect to the partial order⊂ ; however, if y Y is not maximal, then there still exists a unique smallest element≺ succ(y) ∈ + ∈ y0 R (y) Y : y y0 .IfRY =R (Y Y), and if Y − and Y are the sets { ∈ V ∩ ≺ } V∩ × of minimal and maximal elements of Y and CY = RY (Y −) RY (Y −), then the ∪ map y succ(y) again induces a Borel automorphism TY : Y Y ,givenby 7→ 7−→ succ(y)ify YrCY, TYy= ∈ y if y CY , ( ∈ which is called the adic transformation of Y .Notethat n (2.4) TY(y):n Z = y0 Y :y0 y { ∈ } { ∈ ∼ } for every y Y r CY . ∈ We could have tried to define the adic transformation TY a little more elegantly on the set CY of exceptional points (cf. [15]). However, the definition of TY on CY will not really matter, since we shall only consider nonatomic measures and adic transformations TY on closed, nonempty subsets Y XV which satisfy the following condition: ⊂ + (M) the sets Y − and Y of minimal and maximal elements of Y in the partial order are both countable. ≺ Example 2.1. (The Pascal adic transformation (Vershik [53])) Let n 2, and let ≥ (n) (0) (n 1) n (0) (n 1) V = (j ,...,j − ) N : j + +j − = k k { ∈ ··· } (n) (n) (n) for every k 0. We put V =(Vk ,k 0), furnish each Vk with the reverse lexicographic≥ order, and set ≥ (2.5) (n) (0) (n 1) (n) Y = y =(yk) X (n) :yk =(y ,...,y − ) V and { ∈ V k k ∈ k y(i) y(i) for every k 0andi=0,...,n 1 . k ≤ k+1 ≥ − } The equivalence relation RY (n) and the adic transformation Tn = TY (n) are called the n-dimensional Pascal Gibbs relation and n-dimensional Pascal adic transformation on the space Y (n). It is easy to see that Y (n) satisfies the conditions (T) and (M). We picture Y (n) as a graded graph. At level k there are as vertices the elements (n) (n) (n) of V . There is a connection from yk V to yk+1 V if and only if k ∈ k ∈ k+1

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(i) (i) (i) (i) yk yk+1 for all i =0,...,n 1. In fact then yk = yk+1 for all i except one, ≤ i0 i0 − i = i0, for which yk = yk+1 1. We think of the edge from yk to yk+1 as labeled − (j) with the symbol i0 0,...,n 1 ;denotethisi0 by φk(y). Then yk is equal to the number of appearances∈{ of the− symbol} j on the edges of the path y from level 0 to level k. This allows us to deﬁne a continuous bijection

(n) + N Φ: Y Σn =(Z/nZ) 7−→ (n) by setting, for every y =(yk) Y , ∈ Φ(y)k = φk(y). (n) (n) (n) If we put W =(Wk ,k 0) with Wk = Z/nZ for every k 0, and if we (n) ≥ + ≥ furnish each Wk with its natural order, then Σn = XW(n) , and we define the + (n) + partial order 0 on Σ = X (n) as above. It is clear that the map Φ: Y Σ ≺ n W 7−→ n is order-preserving, i.e. that Φ(y) 0 Φ(y0) if and only if y y0. However, if + ≺ + ≺ Rn = RW(n) is the Gibbs relation on Σn ,then + + (2.6) (Φ Φ)(RY (n) )=Sn (Rn. × + + + Indeed, S is the subrelation of R in which two points (z,z0) R are equivalent n n ∈ n if and only if the coordinates of z0 differ from those of z by a finite permutation. There exists a unique Borel automorphism T of Σ+ satisfying Σ∗n n

T ∗+ Φ(y)=Φ TY(n) Σn · · (n) for every y Y .ThemapT∗+ satisfies ∈ Σn k + (T ∗+ ) (x):k Z =Sn(x) { Σn ∈ } for all but countably many x Σ+. It is important to note the distinction between ∈ n the map T ∗+ just defined and the adic transformation T + arising from the partial Σn Σn + order 0 on Σn = XW(n) , the familiar adding machine or odometer, most of whose ≺ + orbits are equal to the equivalence classes of the Gibbs relation Rn ;inthecase n=2,T (0p1q10 ...)=1q0p01 ... for each p, q 0. We shall abuse terminology Σ∗+ 2 ≥ to some extent and call the transformation T ∗+ the adic transformation on the full Σn + shift Σn . Example 2.2. (The adic transformation of an SFT ) Define V(n) =(V(n),k 0) k ≥ and XV(n) for n 1 as in Example 2.1, and let A =(A(i, j), 0 i, j n 1) be an irreducible, aperiodic≥ transition matrix with entries in 0, 1 .Wedenoteby≤ ≤ − { } + + Σ = x=(xk) Σn :A(xk,xk+1) = 1 for every k N A { ∈ ∈ } the one-sided SFT defined by A,andwe (2.7) R+=R+ (Σ+ Σ+) A n ∩ A × A + for the Gibbs relation of ΣA.Put 1 + (2.8) YA=Φ− (Σ ) X (n) . A ⊂ V Then YA satisfies the conditions (T) and (M) above (since any maximal path in the associated graph, when traced backwards, must always follow the allowed edge with the largest possible label, and hence the labelling has to be eventually periodic).

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+ + + Let S R be the equivalence relation in which two points (z,z0) R A ⊂ A ∈ A are equivalent if there is K such that zk = zk0 for k K while for k

RY = RY (B):B Y , (2.11) BY { ∈B } TY = B Y :TYB=B BY { ∈B }

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of RY -saturated and TY -invariant Borel sets in Y coincide (mod µ). We denote by n Y the sigma-field generated by all cylinder sets of the form T ⊂B [vn,...,vn+k]n = y =(yi) Y :yn =vn,...,yn+k = vn+k , and define the tail { ∈ } sigma-field of Y as Y = n 0 n. T ≥ T RY Lemma 2.4. For every closed,T nonempty subset Y XV, Y = Y .Inpartic- ular, if µ is a nonatomic probability measure on Y which⊂ isT quasi-invariantB under RY (or, if Y satisfies the hypothesis (M) under the adic transformation TY ),then

RY TY Y = = (mod µ). T BY BY Proof. For every m 0weset ≥ (m) (2.12) R = (y,y0) Y Y : yk = y0 for all k m Y { ∈ × k ≥ } (n) (n) R (n) R Y RY Y and note that RY = n 0 RY and hence Y = n 0 .As n= Y for ≥ B ≥ B T B RY every n 0, we conclude that Y = . ≥ S T BY T The m-weight wm(y)ofanelementy Y X ,m 0, is defined as ∈ ⊂ V ≥ (m) (2.13) wm(y)= R (y) | Y | where F denotes the cardinality of a set F . Similarly, if C Y is a Borel set and m 0,| then| ⊂ ≥ (m) (2.14) wm(C, y)= R (y) C. | Y ∩ | (This corresponds to the dimension of a vertex in a Bratteli diagram, the number of paths into a vertex in a graph as in Example 2.3, or the height of a column in an (n) ergodic-theoretic cutting and stacking construction.) For example, if Y XV(n) ⊂ (n) is defined as in Example 2.1, then the m-weight wm(y)ofanelementy=(yk) Y (0) (n 1) (n) ∈ is given as follows: if ym =(ym ,...,ym − ) Vm ,then ∈ m! wm(y)= (0) (n 1) . ym ! ym − ! ··· Theorem 2.5 (Vershik [52]). Let Y X be a closed, nonempty subset, and let ⊂ V µ be a nonatomic probability measure on Y which is invariant and ergodic under B RY . For every Borel set B Y , ⊂ w (B,y) µ(B) = lim n for µ-a.e. y Y, n w (y) →∞ n ∈ where the weights wn( ) and wn(B, ) are defined in (2.13) and (2.14). · · 1 Proof. If f L (Y, Y ,µ), then ∈ B (n) RY 1 Eµ(f n)(y)=Eµ(f Y )(y)= f(z) |T |B R(n) Y z R(n) | | ∈XY for µ-a.e. y Y . The reverse martingale theorem and Lemma 2.4 imply that ∈ 1 RY lim f(z)=Eµ(f Y )(y)=Eµ(f )(y) n (n) |T |BY →∞ R (y) (n) Y z R (y) | | ∈ XY µ-a.e., and by setting f equal to the indicator function 1B of B we have proved the theorem.

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Remark 2.6. If µ is invariant but not necessarily ergodic, then

wn(B,y) Eµ(1B )(y) µ-a.e. wn(y) → |T for every Borel set B Y . ⊂ Theorem 2.7 (Hajian-Ito-Kakutani [17]). Let Y (2) be as in (2.5), and denote by (2) + N + Φ: Y Σ2 = 0, 1 the map described in Example 2.1, so that S2 = 7−→ { } + (Φ Φ)(RY (2) ) is the equivalence relation on Σ2 in which two points are equivalent× if and only if their coordinates diﬀer by a ﬁnite permutation. For any α with N 0 <α<1we set να(0) = α, να(1) = 1 α,writeµα =να for the corresponding + − Bernoulli measure on Σ2 ,andputµ¯α =µα Φ. Then the following equivalent statements are true: ·

(1)µ ¯α is invariant and ergodic under the Pascal adic transformation T2,

(2)µ ¯α is invariant and ergodic under RY2 ,

(3) µα is invariant and ergodic under the adic transformation T ∗+ of the full shift Σ2 + + Σ2 (or, equivalently, under the equivalence relation S2 ). Proof. The equivalence of (1), (2), and (3) is obvious; we prove (2). We write C(m, n) for the binomial coeﬃcient m = m!/(n!(m n)!). For each cylinder set n − C =[v0,...,vm]= y Y2 :yi =vi for i =0,...,m Y2 ending in a coordinate (0) (1) {(1)∈ (0) (1) }⊂ vm =(jm ,jm ) Vm with jm + jm = m,andforeveryy=(yk) Y2,n>m (0) (1)∈ (i) (i) ∈ and yn =(yn ,yn ) with jn jm ,i=0,1, ≥ w (C, y) C(n m, y(0) j(0)) (n m)!y(0)!y(1)! n = n m = n n − (0)− (0) −(0) (1) (1) wn(y) C(n, yn ) (yn jm )!(yn jm )!n! − − (0) (0) (0) (1) (1) (1) (yn jm +1) yn (yn jm +1) yn = − ··· · − ··· (n m +1) n − ··· (0) (1) (0) j(0) (1) j(1) j 1 j 1 y m y m m − j m − j = n n 1 1 n n · − (0) · − (1) j=1 yn j=1 yn Y Y m 1 1 − j − 1 · − n j=1 Y j(0) (j(1) ) α m (1 α) m → − forµ ¯α-a.e. y Y2 as n . In particular,µ ¯α(C) only depends on the last ∈ →∞ coordinate vm of C, which guarantees the invariance ofµ ¯α under RY2 ,and¯µα is ergodic because limn wn(C, )/wn( ) is constantµ ¯α-a.e. (cf. Remark 2.6). →∞ · · Remark 2.8. Since the measuresµ ¯α,α (0, 1), in Theorem 2.7 are obtained from + ∈ the Bernoulli measures µα on Σ2 via the map Φ, they are called the Bernoulli + measures on Y2. If we also consider the degenerate Bernoulli measures µ0,µ1 on Σ2 deﬁned as above, but with α 0,1 , then the measuresµ ¯0, µ¯1 are again invariant ∈{ } and trivially ergodic under RY2 : each of them is concentrated on an equivalence

class of RY2 consisting of a single point. Theorem 2.9 (de Finetti and Vershik). The only nonatomic probability measures which are invariant and ergodic for the Pascal adic transformation T2 are the measures µ¯α,α (0, 1), described in Theorem 2.7. ∈

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Proof. Let C =[v0,...,vm]= y Y2 :yi =vi for i =0,...,m be a cylinder set (0) (1) (2) { ∈ (0) (1) (2) } (i) (i) with vm =(jm ,jm ) Vm ,letvm+1 =(j ,j ) V with j jm for ∈ m+1 m+1 ∈ m+1 m+1 ≥ i =0,1, and put C0 =[v0,...,vm+1] Y2 (we are using the same notation as in Example 2.1 and Theorem 2.7). If µ is⊂ a nonatomic, ergodic, invariant probability measure for T2, then Theorems 2.5 and 2.7 imply that µ(C) w (C, y)/w (y) = lim n n µ(C ) n w (C ,y)/w (y) 0 →∞ n 0 n (0) (0) (0) C(n m, yn vm ) yn = lim − − =1 lim . n (0) (0) n C(n m 1,yn v ) − n →∞ − − − m+1 →∞ By Theorem 2.5 this limit exists µ-a.e. and is µ-a.e. equal to a constant α (0, 1), ∈ so we conclude that µ =¯µα. Remark 2.10. To make the connection between Theorem 2.9 and de Finetti’s theorem more explicit, let n 2, and letµ ¯ be a nonatomic, ergodic, invariant probability ≥ (n) 1 measure for the adic transformation Tn on Y (cf. Example 2.1). Then µ =¯µΦ− + is invariant under the relation Sn of ﬁnite coordinate changes, and de Finetti’s the- + orem shows that µ is a Bernoulli measure on Σn . More generally, every nonatomic invariant probability measure for Tn is a mixture of such Bernoulli measures. + If ΣA is an arbitrary, irreducible, aperiodic SFT, then the explicit determination + of all SA -invariant and ergodic probability measures is not so obvious. In Section 6 we will answer this question for shift-invariant measures. In some special cases, de Finetti’s theorem already gives a complete answer.

11 Theorem 2.11. Let A =(10). Then the only nonatomic probability measures on + + + ΣA Σ2 which are ergodic and invariant under SA (or, equivalently, under the adic⊂ transformation T ) are the Markov measures µ ,α (0, 1),whereµ has Σ∗+ α0 α0 A ∈ α 1 α transition matrix − and initial distribution p(0) = α, p(1) = 1 α (cf. (3.1)). 10 − + N + Proof. We deﬁne a continuous, bijective map Ψ: ΣA a, b = Σ2 by replacing 7−→+ { } ∼ each occurrence of the string 10 in an element x ΣA with a b and the remaining 0’s with a’s. Although the map Ψ is not shift-commuting,∈ it satisﬁes the condition + + + (Ψ Ψ)(SA )=S2; in particular, Ψ sends the set of nonatomic SA -invariant × + + and ergodic probability measures on ΣA bijectively onto the set of nonatomic R2 - + invariant, ergodic probability measures on Σ2 . The latter measures are characterized by de Finetti’s theorem as the Bernoulli + measures on Σ2 ,andthemeasuresµα0,α (0, 1), are just their images under 1 ∈ Ψ− .

3. Gibbs measures and subrelations of Gibbs equivalence relations on two-sided shift spaces As we saw in Theorems 2.5, 2.7 and 2.9, the ergodicity of Bernoulli measures + under the equivalence relation Rn on the full n-shift has dynamical implications for the n-dimensional Pascal Gibbs relation RY (n) and the n-dimensional Pascal adic (n) transformation T ∗+ on Y . In this section we generalize our discussion of ergod- Σn icity of Bernoulli measures under the Pascal adic transformation by showing that certain Gibbs measures on SFT’s are ergodic under a natural class of subrelations of the Gibbs relation of the subshift. However, since the discussion of ergodicity of

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these subrelations is in some sense more natural on two-sided shift spaces, we ﬁrst discuss equivalence relations on two-sided SFT’s before turning to one-sided SFT’s in the next section. Assume that A =(A(i, j), 0 i, j n 1) is an irreducible, aperiodic n n transition matrix with entries in≤0, 1 ,denoteby≤ − × { } ΣA = x=(xk) Σn =(Z/nZ)Z : { ∈ A(xk,xk+1) = 1 for every k Z ∈ } the two-sided SFT deﬁned by A, and we write σ = σA for the shift

σ(x)k = xk+1 and

RA = (x, x0) ΣA ΣA : xk = xk0 for only ﬁnitely many k Z { ∈ × 6 ∈ } for the Gibbs relation on ΣA. A one-step Markov measure µP on ΣA is determined by a stochastic matrix P =(P(i, j), 0 i, j n 1) with P (i, j) > 0 if and only if A(i, j)=1(sucha matrix is said to≤ be compatible≤ − with A): ifp ¯ =(¯p(0),...,p¯(n 1)) is the unique probability vector withpP ¯ =¯p, then the measure of every cylinder− set

C =[v0,...,vl]m = x=(xk) ΣA :xm+i =vi for i =0,...,l { ∈ } is given by

(3.1) µP (C)=¯p(v0)P(v0,v1) P(vl 1,vl). ··· − Any such Markov measure µP is easily seen to be quasi-invariant under the Gibbs relation RA: its Radon-Nikod´ym derivative is given by

∞ P (x ,x ) (3.2) ρ (x, x )= k k+1 µP 0 P (x ,x ) k= k0 k0+1 Y−∞ for every (x, x0) RA (note that the inﬁnite product in (3.2) consists mostly of 1’s). ∈ More generally, let φ:ΣA R be a continuous function and put, for every k 0, 7−→ ≥ ω0(φ)=max φ(x) φ(x0) : x, x0 ΣA , {| − | ∈ } ωk(φ)=max φ(x) φ(x0) : xl = x0 for l

(3.3) ω(φ)= ωk(φ)< . ∞ k 0 X≥ We denote by M1(ΣA) the set of Borel probability measures on ΣA, furnished with the weak∗ topology. A measure µ M1(ΣA)isaGibbs measure of a map ∈ φ:ΣA R with summable variation if 7−→ k k (3.4) log ρµ(x, x0)= (φ σ (x) φ σ (x0)) · − · k X∈Z for every (x, x0) RA r (N N), where N ΣA is a µ-null set; the set of all such ∈ φ× ⊂ measures is denoted by M1 (ΣA). By deﬁnition, any Gibbs measure of φ is quasi- invariant under RA. The Markov measure µP in (3.1) is a Gibbs measure of the map φ(x) = log P (x0,x1). Conversely, if the function φ depends only on the coordinates

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φ x0,x1, then there is only one Gibbs measure for φ, and it is Markov: M1 (ΣA)= µP for some Markov matrix P compatible with A (see [40]). The following {generalization} of this fact is part of the lore of the theory of Gibbs measures; see, for example, [5, 31, 43, 57]. Theorem 3.1. Let A =(A(i, j), 0 i, j n 1) be an irreducible and aperiodic ≤ ≤ − 0-1-matrix, ΣA Σn the associated shift of finite type, and RA the Gibbs equiva- ⊂ lence relation of ΣA described above. If φ:ΣA R is a function with summable 7−→ φ variation, then there exists a unique Gibbs measure µφ M (ΣA). Moreover, µφ ∈ 1 is ergodic under RA, and invariant and K under the shift σ = σA. φ Proof. First we show that M (ΣA) is nonempty. For every K 0weset 1 ≥ (K) RA = (x, x0) ΣA ΣA : xk = x0 for k K . { ∈ × k | |≥ } We claim that the set (K) M(φ, K)= µ M1(ΣA):µ satisfies (3.4) for every (x, x0) RA { ∈ ∈ } is nonempty for every K 0. Indeed, fix L>1 such that every entry of AL is positive, take K L, and≥ group the cylinder sets determined by central (2K +1)- blocks into n2 classes≥ according to their first and last entries. We pick a member from each class and define µ on it arbitrarily, for example as a Markov measure consistent with the transition matrix A. Now we can use (3.4) to carry µ over to each of the other cylinder sets in the same class. Thus if C is one of our chosen cylinder sets and γ is the finite coordinate change that carries C to C0 by changing its central (2K +1)-block (of course the endpoints of the block do not change), then m m the summable variation property of φ shows that m [φ σ (x) φ σ (γx)] is ∈Z · − · a continuous function of x C,soforE C0 we can define ∈ ⊂ P µ(E)= exp [φ σmγx) φ σm(x)] dµ(x). 1 · − · γ− E m Z X∈Z Since each M(φ, K) is also convex, compact, and nonincreasing in K, it follows that φ M1 (ΣA)= M(φ, K) = ∅. 6 K 0 \≥ φ Next we show that M1 (ΣA) consists of a single measure which is ergodic under RA; for this we require a little bit of notation and a lemma. Since the matrix A is irreducible and aperiodic, there exists an integer L 1 such that AL(i, j) > 0 2 ≥ r+1 for every (i, j) (Z/nZ) .Astrings=s0...sr in 0,...,n 1 is allowed if ∈ { − } A(sj ,sj+1) = 1 for all j =0,...,r 1. If s0 = s0 ...s0 is a second allowed string, − r0 then s and s can be concatenated if sr = s , and the string ss = s0 ...srs ...s 0 0 0 10 r00 is the concatenation of s and s0. Finally, if s = s0 ...sr is an allowed string and m Z,then ∈ [s]m= x ΣA:xm=s0,...,xm+r =sr . { ∈ } φ Lemma 3.2. Suppose that φ:ΣA R has summable variation and µ M1 (ΣA). Given M 0 and disjoint cylinder7−→ sets ∈ ≥ C =[i M...iM] M = x ΣA :xl =il for l M , − − { ∈ | |≤ } D=[j M ...jM] M = x ΣA :xl =jl for l M , − − { ∈ | |≤ } there is a homeomorphism V [RA] and a cylinder set C0 C such that ∈ ⊂

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(1) V 2 = identity, (2) D0 = V (C0) D, ⊂ 2L (3) µ(C0) µ(C)n− , ≥ 4L 16ω(φ) 8Lω (φ) (4) µ(D0) n− e− − 0 µ(D), ≥ (5) log(dµV/dµ)(x) log(dµV/dµ)(x0) < 8ω(φ) for all x, x0 C0. | − | ∈ Proof. For every (i, j) 0,...n 1 2 we denote by ∈{ − } L+1 S(i, j) (Z/nZ) ⊂ the set of allowed strings of the form s = is1 ...sL 1j, and we write c = i M ...iM − − and d = j M ...jM for the strings defining the cylinders C and D.If(i, j) 2 − ∈ (Z/nZ) , s S(i, i M ), s0 S(iM ,j), t S(i, j M )andt0 S(jM,j), we consider ∈ − ∈ ∈ − ∈ the concatenations scs0,tdt0 of length 2L +2M+ 1 and denote by [scs0] M L and − − [tdt0] M L the corresponding cylinder sets. For every x [scs0] M L we define − − ∈ − − x0 [tdt0] M L by ∈ − − xk if k M+L, | |≥ tl if k = L M + l, l =0,...,L, xk0 = − − jk if k M,  | |≥ tl0 if k = M + l, l =0,...,L,  and note that the map x x0 is a homeomorphism of [scs0] L M onto [tdt0] L M .  − − − − → t,t0 As the cylinders C, D are disjoint, this allows us to define an element V [RA] s,s0 by setting ∈

x if x ΣA r ([scs0] L M [tdt0] L M ), − − − − t,t0 ∈ ∪ Vs,s (x)= x0 if x [scs0] L M , 0  ∈ − −  t,t0 1 (Vs,s )− (x)ifx [tdt0] L M . 0 ∈ − − k k k k From the deﬁnition of ω(φ), comparing φ(σ x)toφ(σ x0)(andφ(σ y)toφ(σ y0)) k k k k when k M+L,andφ(σ x)toφ(σ y)(andφ(σ x0)toφ(σ y0)) when k

t,t0 t,t0 (3.5) max log ρµ(Vs,s (x),x) log ρµ(Vs,s (y),y) <8ω(φ). x,y [scs ] | 0 − 0 | ∈ 0 In order to check the dependence of (3.5) on s, s0,t,t0 we ﬁx arbitrary points x¯ C, y¯ D and calculate that ∈ ∈ (3.6) M+L t,t0 k k log ρµ(V (x),x) (φσ (¯y) φσ (¯x)) < 8ω(φ)+4Lω0(φ) s,s0 − − k= M L −X− 2 for every (i, j) (Z/nZ) , s S(i, i M ), s0 S(iM ,j), t S(i, j M ), t0 S(jM ,j) ∈ ∈ − ∈ ∈ − ∈ and x [scs0] L M . In other words, we have found a constant ∈ − − L+M ξ(D, C)= φσk(¯y) φσk(¯x) − k= L M −X− such that the logarithm of the Radon-Nikod´ym derivative of each of the maps V t,t0 s,s0 is within 8ω(φ)+4Lω0(φ) of this constant. In particular,

8ω(φ) 4Lω (φ)ξ(D,C) µ([tdt0] L M ) 8ω(φ)+4Lω (φ)ξ(D,C) (3.7) e− − 0 < − −

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2 for every (i, j) (Z/nZ) , s S(i, i M ), s0 S(iM ,j), t S(i, j M )andt0 ∈ ∈ − 2L ∈ ∈ − ∈ S(jM,j). Since µ([scs0] L M ) µ(C)n− foratleastonechoiceofs, s0,we − − ≥ conclude that, for any suitable choice of t, t0,

8ω(φ) 4Lω0(φ)ξ(D,C) µ(D) µ([tdt0] L M ) µ([scs0] L M )e− − ≥ − − ≥ − − 2L 8ω(φ) 4Lω (φ)ξ(D,C) µ(C)n− e− − 0 . ≥ Similarly we see that, for suitable choices of s, s0,t,t0,

2L 2L 8ω(φ)+4Lω0(φ)ξ(D,C) µ(D) µ([tdt0] L M )n µ([scs0] L M )n e ≤ − − ≤ − − µ(C)n2Le8ω(φ)+4Lω0(φ)ξ(D,C), ≤ so that

2L 8ω(φ) 4Lω (φ)ξ(D,C) µ(D) 2L 8ω(φ)+4Lω (φ)ξ(D,C) (3.8) n− e− − 0 n e 0 . ≤ µ(C) ≤ 2 We choose (i, j) (Z/nZ) and s S(i, i M ),s0 S(iM,j) such that (3) holds − ∈ ∈ ∈ t,t0 for C0 =[scs0] L M ,chooset S(i, j M ),t0 S(jM,j), and set V = Vs,s . − − ∈ − ∈ 0 Conditions (1)–(2) follow from the deﬁnition of V with

D0 =[tdt0] L M , − − (4) is a consequence of (3.7)–(3.8), and (5) is (3.5).

φ Completion of the proof of Theorem 3.1. Every probability measure in M1 (ΣA) has its Radon-Nikod´ym derivative under RA given by (3.4). In particular, by the φ chain rule any two distinct RA-ergodic elements of M1 (ΣA) have to be mutually φ singular. Thus if M1 (ΣA) contains more than one probability measure, then it must therefore also contain a measure which is not RA-ergodic. Choose a Borel set B = RA(B) ΣA with 0 <µ(B)<1, and let ⊂ ε =1/(100n4Le24ω(φ)+8Lω0(φ)), where L 0ischosensothateveryentryofAL is positive. ≥ Since µ(B)µ(ΣA r B) > 0, there exist an integer M 0 and cylinder sets ≥ C =[i M,...,iM] M, D =[j M,...,jM] M in ΣA such that − − − − µ(B C) > (1 ε)µ(C),µ((ΣA r B) D) > (1 ε)µ(D); ∩ − ∩ − we apply Lemma 3.2 to ﬁnd cylinder sets C0 C and D0 D, satisfying the conditions (1)–(5) of that lemma. According to⊂ (3), ⊂ 2L µ(C0 B) >µ(C0) εµ(C) µ(C0)(1 εn ), ∩ − ≥ − and (5) guarantees that

8ω(φ) 2L µ(V (C0 B)) >µ(D0)e− (1 εn ) ∩ − 4L 24ω(φ) 8Lω (φ) 2L µ(D)n− e− − 0 (1 εn ) ≥ − 99 = µ(D) 100n4Le24ω(φ)+8Lω0(φ) >εµ(D)>µ(B D), ∩ which shows that B cannot be invariant under V . This contradiction proves that φ M1 (ΣA) consists of a single RA-ergodic measure, as claimed.

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φ Finally, M1 (ΣA) is clearly shift-invariant, so the unique measure µφ it contains is shift-invariant. Ergodicity of µφ under RA is equivalent to triviality of the two- sided tail ﬁeld, which implies triviality of the two one-sided tail ﬁelds, which in turn is equivalent to the K property.

For the remainder of this section we fix a map φ:ΣA R with summable 7−→ variation and write µφ for the unique Gibbs measure of φ. According to Theorem 3.1, µφ is ergodic under the Gibbs equivalence relation RA; we shall now prove that µφ is also ergodic under a class of shift-invariant subrelations of RA which was discussed in [47]. Let be a countable, discrete group with finite conjugacy classes (i.e. with 1G hgh− : h < for every g ), and let ψ :ΣA be a continuous map. |{ ∈G}| ∞ ∈G 7−→ G Following [47] we set, for every (x, x0) RA, ∈ (3.9) ψ K K 1 + (x, x0) = lim ψ(x) ψ(σ (x)) (ψ(x0) ψ(σ (x0)))− , J K ··· · ··· →∞ ψ K 1 1 K 1 (x, x0) = lim (ψ(σ− (x)) ψ(σ− (x)))− ψ(σ− (x0)) ψ(σ− (x0)) − K J →∞ ··· · ··· n n (since ψ(σ (x)=ψ(σ x0)fornoutside a finite interval, these limits exist). The ψ maps : RA are cocycles of RA, i.e. J± 7−→ G ψ ψ ψ (3.10) (x, x0) (x0,x00)= (x, x00) J± J± J± whenever (x, x0), (x, x00) RA.Ifh:ΣA is a continuous map, then ∈ 7−→ G 1 (3.11) ψ0(x)=h(x)ψ(x)h(σ(x))− ,

ψ0 and if : RA are the cocycles deﬁned as in (3.9), but with ψ0 replacing ψ, then J± 7−→ G

ψ0 ψ 1 (3.12) (x, x0)=h(x) (x, x0)h(x0)− J± J± ψ ψ0 for every (x, x0) RA (this means that and are cohomologous cocycles of ∈ J± J± RA). The set ψ ψ ψ (3.13) SA = (x, x0) RA : + (x, x0)= (x, x0) { ∈ J J− } ψ is a subrelation of RA, and (3.12) shows that SA is unaﬀected if ψ is replaced by ψ0 in (3.13). We shall prove the following theorem.

Theorem 3.3. Let φ :ΣA R be a function with summable variation, and let be a countable, discrete group7−→ with finite conjugacy classes. For every continuousG map ψ :ΣA the Gibbs measure µφ is ergodic under the equivalence relation ψ 7−→ G SA defined in (3.13). The proof of Theorem 3.3 requires some preparation. Since is discrete there ex- G ist integers K, K0 0 such that ψ(x) only depends on the coordinates x K ,...,xK ≥ − 0 of every x ΣA, and by applying a standard recoding argument (going to a ∈ higher-block presentation of ΣA) we may assume without loss in generality that L K =0,K0 =1.WefixL 1 such that every entry of A is positive and find al- (i) (i) ≥(i) lowed strings s =0s1 ...sL 1i for every i 0,...,n 1 . Define a continuous − ∈{ − } map h¯ :ΣA by setting 7−→ G ¯ (x0) (x0) (3.14) h(x)=ψ(0,s1 ) ψ(sL 1,x0) ··· −

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for every x =(xk) ΣA, put ∈ 1 (3.15) ψ¯(x)=h¯(x)ψ(x)h¯(σ(x))−

ψ¯ for every x ΣA, and denote by the cocycles deﬁned by (3.9) with ψ¯ replacing ψ. ∈ J±

ψ¯ Lemma 3.4. Let ∆ = (x, x0):(x, x0) RA . ± {J± ∈ } (1) For every (x, x0) RA there exist allowed strings s = s0 ...sm, s0 =s0 ...s0 , ∈ 0 m t = t0 ...tm , t = t ...t such that s0 = sm = s = s = t0 = tm = t = 0 0 0 m0 0 0 m0 0 0 t0 =0and m0 ψ¯ 1 + (x, x0)=ψ(s0,s1) ψ(sm 1,sm)(ψ(s0,s10 ) ψ(sm0 1,sm0 ))− , J ··· − ··· − ψ¯ 1 (x, x0)=(ψ(t0,t1) ψ(tm 1,tm))− ψ(t0,t10 ) ψ(tm0 1,tm0 ). J− ··· − ··· − (2) For every m 1 and all allowed strings s = s0 ...sm, s0 = s0 ...s0 with ≥ 0 m s0 = sm = s0 = sm0 =0, 1 ψ(s0,s1) ψ(sm 1,sm)(ψ(s0,s10 ) ψ(sm0 1,sm0 ))− ∆+, ··· − ··· − ∈ 1 (ψ(s0,s1) ψ(sm 1,sm))− ψ(s0,s10 ) ψ(sm0 1,sm0 ) ∆ . ··· − ··· − ∈ − (3) ∆ is a subgroup of . ± G (4) For every g ∆ and every B Σ with µφ(B) > 0 there exist elements ± ∈ ± ∈B A (x, x0), (y,y0) RA (B B) with xk = xk0 for all k 0, yk = yk0 for all k 0,and ∈ ∩ × ≤ ≥ ψ¯ ψ¯ g+= +(x, x0),g = (y,y0). J − J− In fact we can accomplish this with x = y, x0 = y0. ψ¯ Proof. We shall prove the conditions (1)–(4) for ∆+ and + ; the proofs for ∆ ¯ J − and ψ are completely analogous and will be omitted. J− Condition (1) follows from (3.12): if (x, x0) RA satisfy xk = xk0 for k K L, say, then ∈ | |≥ − ψ¯ ¯ ¯ ¯ 1 ¯ 1 + (x, x0)=ψ(x0,x1) ψ(xK 1,xK)ψ(xK0 1,xK0 )− ψ(x0x10 )− J ··· − − ··· (x0) (x0) =ψ(0,s1 ) ψ(sL 1,x0)ψ(x0,x1) ψ(xK 1,xK) ··· − ··· − 1 1 (x0) 1 (x0) 1 ψ(xK0 1,xK0 )− ψ(x0x10 )− ψ(sL 1,x0)− ψ(0,s1 )− . · − ··· − ··· Choosing a string B such that xK B0 is allowed and setting

(x0) (x0) (x0) (x0) s =0s1 ...sL 1x0 ...xKB0,s0=0s1 ...sL 1x0 ...xK0 B0 − − ψ¯ shows that (x, x0) is of the required form. J+ In order to prove (4) and in the process (2) we assume that s = s0 ...sm, s0 =s0 ...sm0 are allowed strings with s0 = s0 = sm = sm0 =0,set 1 g=ψ(s0,s1) ψ(sm 1,sm)(ψ(s0,s10 ) ψ(sm0 1,sm0 ))− ··· − ··· − and denote by 1 (g)= h :hgh− = g C { ∈G } the commutant of g.As has finite conjugacy classes, the quotient space = G H / (g) is finite, and we define a homeomorphismσ ¯ :ΣA ΣA by setting G C ×H 7−→ ×H 1 σ¯(x, h (g)) = (σ(x), ψ¯(x)− h (g)) C C

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for every x ΣA and h .Wedenotebyνthe normalized counting measure on ∈ ∈G and setµ ¯ = µφ ν.IfB Σ with µφ(B) > 0, then the mean ergodic theorem, H × ∈B A applied to the indicator function of B0 = B (g) ΣA , yields that ×{C }⊂ ×H l 1 1 − k σ¯ 2 lim 1B0 σ¯ = Eµ¯(1B0 Σ )inL(¯µ), l l · |B A×H →∞ k=0 X σ¯ where Σ denotes the family ofσ ¯-invariant Borel sets in ΣA . Hence B A×H ×H l 1 1 − k k 1 lim µφ(B σ− (B) x ΣA :ψ¯(x) ψ¯(σ − (x)) (g) ) l l ∩ ∩{ ∈ ··· ∈C } →∞ k=0 X l 1 1 − k = lim µ¯(B0 σ¯− (B0)) |H| · l l ∩ →∞ k=0 2X 2 µ¯(B0) =µφ(B) / . ≥|H|· |H| In particular, there exist inﬁnitely many l 0 with ≥ l l 1 µφ(B σ− (B) x ΣA :ψ¯(x) ψ¯(σ − (x)) (g) ) (3.16) ∩ ∩{ ∈ ··· ∈C } 2 µφ(B) /2 . ≥ |H| Put C =[s]0,D=[s0]0 and deﬁne W [RA]by ∈

(...,x 1,s0,...,sm0 ,xm+1,...)ifx=(xk) C, − ∈ Wx = (...,x 1,s0,...,sm,xm+1,...)ifx D,  − ∈ x if x ΣA r (C D). ∈ ∪  Suppose thatB ΣA,B [0]0 and µφ(B) > 0, and choose an r 1and ∈B ⊂ ≥ an allowed string t = t0 ...trm such that trm =0andthesetB¯=B [t]0 has ∩ positive measure. We denote bys ¯ =¯s0...s¯rm the r-fold concatenation of s and deﬁne W 0 [RA]by ∈ (...,x 1,s¯0,...,s¯rm,xrm+1,...)ifx=(xk) [t]0, − ∈ W0x= (...,x 1,t0,...,trm,xrm+1,...)ifx[¯s]0,  − ∈ x if x ΣA r ([t]0 [¯s]0). ∈ ∪ Then, since  l l lim µφ(B¯ σ− Wσ (B¯)) = µφ(B¯), l ∩ →∞ we have

l l l lim µφ(B¯ σ− (B¯) σ− Wσ (B¯)) l ∩ ∩ →∞ l l l = lim µφ(B¯ σ− (B¯) σ− W 0σ (B¯)) l ∩ ∩ →∞ l l l 2 = lim µφ(B¯ σ− (B¯) σ− WW0σ (B¯)) = µ(B¯) , l ∩ ∩ →∞ and (3.16) guarantees that

l l l l l µφ(B¯ σ− (B¯) σ− W 0σ (B¯) σ− WW0σ (B¯) (3.17) ∩ ∩ ∩ l 1 x ΣA :ψ¯(x) ψ¯(σ − (x)) (g) )>0 ∩{ ∈ ··· ∈C }

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for inﬁnitely many l 0. In particular, if x =(xk) lies in the set occurring in (3.17), then the points≥

x =(...,x 1,x0,...,xl,t1,...,trm,xrm+1,...), − x0 =(...,x 1,x0,...,xl,s¯1,...,s¯rm,xrm+1,...), −

x00 =(...,x 1,x0,...,xl,s10 ,...,sm0 ,s¯m+1,...,s¯rm,xrm+1,...) − all lie in B¯ B,and ⊂ l 1 l 1 h=ψ¯(x) ψ¯(σ − (x)) = ψ¯(x0) ψ¯(σ − (x0)) ··· ··· l 1 = ψ¯(x00) ψ¯(σ − (x00)) (g). ··· ∈C Furthermore, (x0,x00) RA (B B), and ∈ ∩ × ψ¯ 1 (x0,x00)=hgh− = g, J+ which proves (4) for any Borel set B [0]0 with positive measure. ⊂ If B Σ is an arbitrary Borel set with positive measure, then there exists an ∈B A allowed string t0 = t0 p ...tp0 with tp0 = t0 p =0andµφ(B [t0] p)>0, since µφ is − − ∩ − p K and hence mixing of every order. Put B0 = B [t0] t and B00 = σ (B0) [0]p, ∩ − ⊂ and apply the preceding part of this proof to find, for every g ∆+,anelement ψ¯ ∈ (y,y0) RA (B00 B00) with yk = yk0 for all k 0and +(y,y0)=g. From the ∈ ∩ ψ¯ × ≤ J ψ¯ p p definition of + it is clear that there exists an h ∆+ with + (σ− (y),σ− (y0)) = ψ¯ J1 ∈ J h (y,y0)h− for every (y,y0) RA (B00 B00). Note that h does not depend on g. J+ ∈ ∩ × We conclude that there exists, for every g ∆+, an element (x, x0) RA (B0 B0) ψ¯ 1 ∈ ∈ ∩ × with (x, x0)=hgh− , yielding what is claimed in (4). J+ For the proof of (3) we assume that g,h ∆+ and apply (1) to find allowed ∈ strings t = t0 ...tm , t =t ...t , such that 0 0 0 m0 0 1 h = ψ(t0,t1) ψ(tm 1,tm)(ψ(t0,t10 ) ψ(tm0 1,tm0 ))− . ··· − ··· − According to (4) there exists an element (x, x0) RA such that xk = xk0 for all ψ¯ ∈ k 0andg= (x, x0). We choose m 0 such that xk = x0 whenever k m, ≤ J+ ≥ k ≥ put C =[x0,...,xm]0, and apply(3.16) to find a point y =(yk) Cand an integer l>msuch that σl(y) C and ∈ ∈ l 1 g0 = ψ¯(y) ψ¯(σ − (y)) (h). ··· ∈C Put

y0 =(...,y 1,x0,...xm,ym+1,...,yl 1,t0,...,tm ,zl+m +1,...), − − 0 0 y00 =(...,y 1,x0,...xm0 ,ym+1,...,yl 1,t0,...,tm0 ,zl+m +1,...), − − 0 0

where the coordinates zl+m0+1,zl+m0+2,... are arbitrary (but, of course, allowed), ψ¯ and obtain that (y0,y00) RA and (y,y0)=gh. ∈ J+ ψ ψ Lemma 3.5. Let =( +, ): RA ∆=∆+ ∆ be the cocycle J J J− 7−→ × − ψ ψ (x, x0)=( +(x, x0), (x, x0)), J J J− and let R be the equivalence relation on X¯ =ΣA ∆deﬁned by × R = ((x, g, h), (x0,g0,h0)) : (x, x0) RA, { ∈ ψ ψ g = + (x, x0)g0,h= (x, x0)h0 . J J− }

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If ν is the counting measure on ∆ and λ = µφ ν,thenλis quasi-invariant and × ergodic under R.

Proof. The quasi-invariance of λ under R is obvious. For every (g,h) ∆ we deﬁne a homeomorphism T : X¯ X¯ by setting ∈ (g,h) 7−→

T(g,h)(x, g0,h0)=(x, g0g,h0h)

for every (x, g0,h0) X¯. For every Borel set B X¯ with λ(B) > 0and(g,h) ∆ we have ∈ ⊂ ∈

T(g,h)(R(B)) = R(T(g,h)(B)). Lemma 3.4(4) is easily seen to imply that every R-saturated set is invariant under the transformations T , (g,h) ∆, and the ergodicity of RA guarantees that (g,h) ∈ λ(X¯ rR(B)) = 0 for every Borel set B X¯ with λ(B) > 0 (cf. e.g. [58, 59, 45]). ⊂ Proof of Theorem 3.3. We use the notation of Lemmas 3.4–3.5 and denote by 1 1 G the identity element in . Deﬁne a map η :∆ by η(g,h)=gh− and set G 7−→ G Ξ=η(∆). We denote by R0 the equivalence relation on X¯ 0 =ΣA Ξgivenby × ψ ψ 0 1 R = ((x, g), (x0,g0)) : (x, x0) R,g= +(x, x0)g0 (x, x0)− { ∈ J J− } ¯ ¯ ¯ ¯ 1 and denote by ξ : X X0 the map ξ(x, g, h)=(x, gh− ). If ν0 is the count- 7−→ ¯ ¯ ¯ ing measure on Ξ and λ0 = µ ν0,thenξ:(X,µ¯) (X0,λ0) is a nonsingu- × 7−→ lar map which sends R-equivalence classes to R0-equivalence classes, and Lemma

3.5 implies that λ0 is ergodic under R0. In particular, the equivalence relation R0 ((ΣA 1 ) (ΣA 1 )) induced by R0 on ΣA 1 is ergodic. Since ∩ ×{ G} × ×{ G} ψ ψ ×{ G} 0 0 ((x, 1 ), (x0, 1 )) R if and only if (x, x0) SA = SA , we have proved Theorem 3.3. G G ∈ ∈

ψ Remark 3.6. As a special case of the fact that the equivalence relation SA in (3.13) 1 is unaﬀected if ψ is replaced by a function of the form ψ0 = hψ(h σ)− in (3.11), ψ ψ · ψ we obtain that SA is shift-invariant: (σ(x),σ(x0)) SA if and only if (x, x0) SA. ψ ∈ ∈ These properties of SA are particularly transparent if is abelian. Deﬁne a map ψ G : RA by J 7−→ G (3.18)

ψ ∞ k k 1 ψ ψ 1 (x, x0)= (ψ(σ (x)ψ(σ (x0)− )= + (x, x0) (x, x0)− J J J− k= Y−∞ ψ for every (x, x0) RA.Then is a cocycle, i.e. satisﬁes the equation (3.10) with ∈ J ψ replacing ψ,and J J± ψ ψ S = (x, x0) RA : (x, x0)=0 . A { ∈ J }

ψ ψ0 As = whenever ψ,ψ0:ΣA are related via the equation (3.11), J J 7−→ G ψ ψ0 ψ we obtain once again that SA = SA . However, the individual cocycles are J± obviously changed if we replace ψ by ψ0.

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4. Gibbs measures and subrelations of Gibbs equivalence relations on one-sided shift spaces As in the preceding section we assume that A =(A(i, j), 0 i, j n 1) is an irreducible, aperiodic transition matrix with entries in 0, 1≤. We≤ define− the + + { } + one-sided SFT ΣA and the Gibbs relation RA as in Example 2.2 and write σ = σA for the one-sided shift σ(x)k = xk+1 + + on Σ .Ifφ:Σ R is a continuous map we define ωk(φ),k 0, as in the A A 7−→ ≥ two-sided case and say that φ has summable variation if k 0 ωk(φ) < .A + ≥ ∞ probability measure µ on ΣA is a Gibbs measure of a continuous map φ:ΣA R + P 7−→ if µ is quasi-invariant under RA and k k (4.1) log ρµ(x, x0)= (φσ (x) φσ (x0)) − k 0 X≥ + for every (x, x0) RA (cf. (3.4); in the one-sided case this definition makes sense even if φ does not∈ have summable variation). The same argument as in Theorem 3.1 φ + + shows that the set M1 (ΣA) M1(ΣA) of Gibbs measures of φ is nonempty, weak∗- ⊂ + closed, and convex. Furthermore, since every function φ:Σ R with summable A 7−→ variation can be viewed as a function on ΣA which again has summable variation, φ + M1 (ΣA) contains a single measure µφ which is ergodic under the Gibbs relation + RA. In contrast to the two-sided case the measure µφ need not be shift-invariant, although Theorem 3.1 guarantees that it is equivalent to a shift-invariant and exact + probability measureµ ¯φ on ΣA. For example, if P =(P(i, j), 0 i, j n 1) is a stochastic matrix which is compatible with A in the sense of Section≤ ≤ 3, and− ifp ¯ is the unique probability vector satisfyingpP ¯ =¯pand p(i)=1/n for i =0,...,n 1, then −

(4.2) µlog P (C)=p(v0)P(v0,v1) P(vl 1,vl) ··· − foreverycylinderset + C=[v0,...,vl]m = x=(xk) Σ :xm+i =vi for i =0,...,l , { ∈ A } whereas the shift-invariant measureµ ¯log P ,givenby

(4.3) µ¯log P (C)=¯p(v0)P(v0,v1) P(vk 1,xk), ··· − is the Gibbs measure of the function φ(x) = log P (x0,x1)+log¯p(x0) logp ¯(x1) (cf. [40]). [The reason for the difference between the one- and two-sided− cases is the following. If φ is changed to a cohomologous function, the Radon-Nikodým derivative of the two-sided measure under the finite coordinate changes is unaffected, but this is no longer true in the one-sided case. The uniqueness in the one-sided case is proved exactly as in the two-sided case; however, since the Radon-Nikodým derivative changes on the one-sided shift if φ is changed by a cohomologous function, the one-sided measure changes as well. The condition of shift-invariance in the one-sided case singles out a particular element in the cohomology class of φ.] For easier reference we summarize this discussion in a theorem. Theorem 4.1. Let A =(A(i, j), 0 i, j n 1) be an irreducible, aperiodic 0-1- + ≤ ≤ − + matrix, ΣA the associated one-sided shift of finite type, and RA the Gibbs relation + + of Σ .Ifφ:Σ R is a function with summable variation, then there exists A A 7−→ a unique Gibbs measure µφ of φ which is ergodic under RA.Furthermore,µφis

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equivalent to a probability measure µ¯φ which is invariant, ergodic and exact under + the shift σ on ΣA.

Remark 4.2. According to [57], the measureµ ¯φ is actually Bernoulli for the shift σ. We encounter a similar phenomenon when dealing with cocycle-generated sub- + + relations of the Gibbs relation RA on the one-sided SFT ΣA. Fix a function + φ:ΣA R with summable variation, denote by µφ the Gibbs measure of φ, 7−→ + and consider a continuous map ψ :ΣA ,where is a countable, discrete 7−→ G G ψ group with finite conjugacy classes. We define a cocycle + by the first equation in (3.9), and put J ψ+ + ψ (4.4) SA = (x, x0) RA : + (x, x0)=1 , { ∈ J G} where 1 is the identity element in . In contrast to the situation described in G G ψ+ Remark 3.6, the equivalence relation SA is changed if ψ is replaced by a function 1 + ¯ + ψ0 = hψ(h σ)− ,whereh:ΣA is continuous. In particular, if h:ΣA is · 7−→¯ G + 7−→ G the function defined in (3.14) and ψ :ΣA is given by (3.15), then essentially the same proof as that of Theorem 3.3 yields7−→the G following result. + Theorem 4.3. Let φ :ΣA R be a function with summable variation, and let be a countable, discrete group7−→ with finite conjugacy classes. For every continuousG + + map ψ :ΣA there exists a continuous map h:ΣA such that the Gibbs 7−→ G ψ07−→+ G measure µφ is ergodic under the equivalence relation SA defined by (4.4) with 1 ψ0 = hψ(h σ)− · replacing ψ. ψ+ ψ+ The measure µφ is ergodic under the original relation SA if and only if SA is topologically transitive. Proof. The proof of Theorem 3.3 shows that the first assertion is a consequence ψ+ of Lemma 3.5. The topological transitivity of SA is obviously necessary for the ψ+ ergodicity of µφ under SA , since every open subset of ΣA has positive µφ-measure. In order to see that topological transitivity is also sufficient for ergodicity we observe + that there exists a finite partition 1,..., m of ΣA into closed and open subsets O O+ ψ0+ on each of which the continuous map h:ΣA is constant, and that SA ψ+ 7−→ G ∩ ( i i)=SA ( i i)fori=1,...,m (cf. (3.12)). The ergodicity of µφ O ×Oψ0+ ∩ O ×O under SA guarantees that the restriction of µ to i is ergodic under the relation ψ+ O ψ+ ψ+ SA ( i i), and the topological transitivity of SA implies that SA ( i)isa ∩ O ×O + O dense, open subset of ΣA for every i =1,...,m (note that there exists a countable + ψ+ group G of homeomorphisms of ΣA with SA = Gx = gx : g G for every ψ+ { ∈ } x ΣA). In particular, SA ( 1) meets each i in a set of positive measure, so ∈ ψ+ O O that SA is ergodic.

Remark 4.4. Assume for simplicity that ψ is a function of the two variables (x0,x1), and call two symbols a, b 0,...,n 1 equivalent if there exist allowed strings ∈{ − } s, t of equal length and a symbol i Z/nZ such that the strings asi and bti are ∈ ψ+ allowed and are permutations of each other. Then it is easy to see that SA is topologically transitive if and only if all symbols in Z/nZ are equivalent (cf. [7, 8]). ψ+ Thus ergodicity of µφ under SA depends only on A, and not on φ or ψ.

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Remark 4.5. The results in sections 3–4 are largely unaﬀected if we drop the as- sumption of aperiodicity of A.IfAis irreducible, but no longer aperiodic, then the alphabet 0,...,n 1 decomposes into periodic components C0,...,Cm 1,m 2, { − } − ≥ such that xi+k Cj+k (mod m) whenever x =(xk) ΣA,xi Cj,andk Z.If ∈ ∈ ∈ ∈ m Xj= x ΣA:x0 Cj ,j=0,...,m 1, then each Xj is invariant under σ , { ∈ ∈ } − and a standard recoding argument allows us to regard Xj as an irreducible and m aperiodic SFT with regard to the shift σ . By applying Theorem 3.1 to each Xj we see that there exists, for every function φ:ΣA R with summable variation, 7−→ and for every j =0,...,m 1, a unique Gibbs probability measure µ(j) for φ on − φ Xj. In particular, there is no longer a unique probability measure on ΣA satisfying (3.4). (0) The orbit-average of the Gibbs measure µφ , say, under σ is the unique shift- invariant Gibbs measure µφ of φ on ΣA (it coincides, of course, with the orbit- averages of the other measures µ(j),j =1,...,m 1). If we deﬁne the Gibbs φ − relation RA exactly as in the aperiodic case, then RA(Xj)=Xj, and the analogue (j) of Theorem 3.3 holds for each of the measures µφ on Xj. However, µφ will no longer ψ be SA-invariant for any ψ. Similarly one obtains the corresponding statements on one-sided SFT’s.

5. Examples Throughout this section we assume that A =(A(i, j), 0 i, j n 1) is an Z ≤ + ≤ − N irreducible and aperiodic 0-1-matrix, write ΣA (Z/nZ) and ΣA (Z/nZ) for ⊂ ⊂ + the two- and one-sided SFT’s deﬁned by A, and denote by RA and RA the Gibbs + relations on ΣA and ΣA.

Example 5.1. Invariant measures for equivalence relations.Letφ:ΣA[0] R 7−→ be a continuous map which takes only finitely many values, and let Rbe the countable subgroup generated by the values of φ, furnished with theG⊂ discrete topology. We put ψ = φ:ΣA and apply Theorem 3.3 to obtain that the 7−→ G Gibbs measure µφ is ergodic, and obviously invariant, under the equivalence relation ψ ψ S RA. Note that the full group [S ] satisfies A ⊂ A ψ [S ]= W [RA]:W preserves µφ . A { ∈ } One special case of this construction is obtained by taking a stochastic matrix P =(P(i, j), 0 i, j n 1) compatible with A and setting φ(x) = log P (x0,x1) ≤ ≤ − for every x ΣA. As noted in the discussion preceding Theorem 3.1, in this case ∈ µφ is the shift-invariant Markov probability measure determined by P . (So every Markov measure is a µφ in this way.) Another special case arises from φ = ψ = constant. Then µφ is the unique measure of maximal entropy on ΣA. In the one-sided case, it is equivalent to the + unique invariant measure for the stationary adic on ΣA. Example 5.2. Failure of ergodicity. In order to illustrate Theorem 4.1 and Remark + N + 4.4, consider the one-sided two-shift ΣA =(Z/2Z) and the map ψ :ΣA[0] Z defined by 7−→ ψ(x)=x0 x1 + − for every x =(xk) Σ .Then ∈ A ψ (x, x0)=x0 x0 J+ − 0

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+ ψ+ for every (x, x0) RA (cf. (3.9)), and the relation SA is obviously not topologically ∈ + transitive. In particular, if φ:ΣA R is a function with summable variation, 7−→ ψ+ then the Gibbs measure µφ is nonergodic under SA . Example 5.3. Ergodicity of Gibbs measures on one-sided SFT’s under finite per- + + mutations. Suppose that SA RA is the equivalence relation in which two points + ⊂ x =(xk),x0 =(x0)inΣ are equivalent if and only if the coordinates xk,k 0, k A ≥ and xk0 ,k 0, are finite permutations of each other (cf. Examples 2.1–2.2). If + ≥ /n ψ0 :Σ ZZ Z is the map defined by A 7−→ (x0) ψ0(x)=e + (0) (n 1) for every x =(xk) Σ ,wheree =(1,0,...,0),...,e − =(0,...,0,1) are ∈ A the unit vectors in Z/nZ n, then it is clear that S+ = Sψ0+. Z ∼= Z A A If 11 A =(11), + then SA is obviously topologically transitive, since the strings 010 and 100 are allowed and permutations of each other (cf. Remark 4.4). However, if

110 A = 001 , 110 + then SA is not topologically transitive. Indeed, if s = s0s1 ...sm is any cycle (i.e. any allowed string with s0 = sm), then the numbers of 1’s and 2’s among the entries s0,...,sm 1 must be equal. In particular, if the symbols 1 and 2 were equivalent in the sense− of Remark 4.4, we could find allowed strings s and t of equal length and a symbol i 0,1,2 such that 1si and 2ti are allowed and permutations of each other. We∈{ may obviously} assume that i = 1; then the number of 1’s in 1s1 exceeds the number of 2’s by 1, whereas the number of 1’s and 2’s in 2t1isequal. This shows that 1s1and2t1 cannot be permutations of each other, so that 1 and ψ+ 2 are not equivalent and SA is not topologically transitive (cf. Theorem 4.3 and Remark 4.4). + If φ:ΣA R is a function with summable variation, then the Gibbs measure + 7−→ + + µφ on ΣA is ergodic under SA only if SA is topologically transitive (Theorem 4.3). All these statements have, of course, obvious translations into properties of the adic + transformations TA∗ on ΣA and TYA on YA (cf. Example 2.2). Example 5.4. Exchangeability and partial exchangeability. The preceding example is a special case of a general problem: given some group of symmetries of a one- or two-sided SFT, what are the Gibbs measures which are invariant (or symmetric) under this group of symmetries? The most elementary examples of such symmetry groups are the group of all finite permutations of the coordinates xk,k 0, and the ≥ group of all finite permutations of the transitions [xk,xk+1]ofthepointsx=(xk) in ΣA. We have met the first of these groups in the context of adic transformations in Examples 2.1–2.2 and in Section 2, and our analysis of them depends on the fact that they generate the orbits (= equivalence classes) of certain subrelations of RA + and RA defined by cocycles. In order to describe these cocycles we consider the 0,...,n 1 l+1 (i ,...,i ) ( /n )l+1 group l+1 = Z{ − } ,denotebye 0 l Z Z Z the unit vector G ∈

(i0,...,il) 1if(i0,...,il)=(j0,...,jl), e(j ,...,j ) = 0 l (0otherwise,

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and observe that the orbits of the group of all permutations of the l-fold transitions + [xk,...,xk+l],k 0, of the points in ΣA and ΣA are the equivalence classes of the ψl ≥ ψl+ relations SA and SA ,whereψl is deﬁned by

(x0,...,xl) ψl(x)=e l+1 ∈G + for every x in ΣA or ΣA. Theorem 3.3 shows, for a two-sided SFT ΣA, the invariance and ergodicity of every fully supported Markov measure under the group of all finite permutations of the l-fold transitions described above. Conversely, Theorem 6.2 implies that every shift-invariant probability measure on ΣA which is invariant and ergodic under the group of finite permutations of the l-fold transitions must be an l-step Markov measure (where we are referring to the Bernoulli measures as zero-step Markov measures). These results carry over to the one-sided case with the precaution described in Remarks 4.4 and 4.5 and Examples 5.1 and 5.3. The adic version of the statement of ergodicity and invariance of Markov measures under the groups of all finite permutations of the l-fold transitions is the Hewitt-Savage 0-1-law as extended to Markov chains by Diaconis and Freedman [7]. Example 5.5. Derangement-equivalent sequences. In all the examples so far the group has been abelian. An elementary example where is nonabelian is obtained G G by setting equal to the group Sn of permutations of the symbols Z/nZ,andby putting G

ψ(x)=(x0x1)

for every x =(xk) ΣA,where(ij) Sn is the transposition of i and j.SinceSn ∈ ∈ is finite, Theorem 3.3 implies that, for every function φ:ΣA R with summable 7−→ variation, the Gibbs measure µφ M1(ΣA) is ergodic under the relation S = ψ ∈ SA. In Examples 5.3 and 5.4, two sequences are equivalent if they eventually accumulate the same symbol counts (or perhaps the same transition counts). Here two sequences are equivalent if they eventually accumulate the same derangement effects, when their symbols act as permutations on a fixed set of n letters. The equivalence relation S defined here is not comparable with the relation ψ1 SA arising from the finite permutations of the transitions [xk,xk+1] appearing in Examples 5.3–5.4. However, since every finite Cartesian product of groups with finite conjugacy classes is again a group with finite conjugacy classes, Theorem 3.3 guarantees the following: if φ:ΣA R is a function with summable variation (i) (i) 7−→ and ψ :ΣA are continuous maps taking values in countable, discrete 7−→ G (i) groups with finite conjugacy classes ,i=1,...,l, then the Gibbs measure µφ G l ψ(i) is ergodic under the equivalence relation i=1 SA . In particular, µφ is ergodic under Sψ1 S, the relation that keeps track of both accumulated symbol accounts A T and accumulated∩ derangement effects. As a concrete example, consider the case where n =4and

1111 1111 A= 1111 , 1111 so that ΣA =Σ4 is the full 4-shift. The points

...x 10123020x7 ..., ...x 10230120x7 ... − −

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ψ1 are SA -equivalent, but S-inequivalent, whereas the points

...x 1010x3 ..., ...x 1000x3 ... − −

ψ1 are S-equivalent, but SA -inequivalent.

6. Symmetric measures on SFT’s We continue to let A =(A(i, j), 0 i, j n 1) be an irreducible and aperiodic + ≤ ≤ − 0-1-matrix, ΣA and ΣA the associated two- and one-sided SFT’s, and SA RA the equivalence relation generated by the ﬁnite coordinate permutations (cf.⊂ Example 2.2). As is well known, the only probability measure µ on ΣA which is invariant under the Gibbs relation RA is the measure of maximal entropy. However, if R RA ⊂ is a proper subrelation, the picture changes considerably. For example, if ΣA is a full shift, then de Finetti’s theorem states that the (shift-invariant) Bernoulli measures are precisely the SA-invariant measures. If A is arbitrary (but still irreducible and aperiodic), and if φ:ΣA R is a function depending on the single coordi- 7−→ nate x0, then the equation (3.4) shows that the Gibbs measure µφ M1(ΣA)is ∈ invariant under SA, and the ergodicity of µφ follows from Theorem 3.3. In Theo- rem 2.11 we saw that for the golden mean shift ΣA,everySA-invariant probability + measure on ΣA was the Gibbs measure of a function of a single variable. In this section we extend the statement to arbitrary SFT’s by showing that every shift- and SA-invariant probability measure on ΣA is the Gibbs measure for a potential function that depends on just one coordinate. Very closely related results have been obtained previously by Georgii [13] and Diaconis and Freedman [7]. By an obvious ψm extension to m-step SFT’s, every shift- and SA -invariant probability measure µ is the Gibbs measure of a function depending on the m + 1 variables x0,...,xm. + We write πN :ΣA ΣA for the projection onto the nonnegative coordinates and observe that, in the7−→ notation of Example 2.2,

n 1 − + + (6.1) S ([i]0 [i]0) S =(π π )(SA) R . A ∩ × ⊂ N× N ⊂ A i=0 [ Lemma 6.1. Let µ be a nonatomic, shift-invariant probability measure on ΣA + 1 which is invariant and ergodic under SA. Then the projection µ = µπ− of µ + N onto ΣA is quasi-invariant and ergodic under S =(πN πN)(SA).Furthermore,the + + × restriction of µ to the set [i]0 = x ΣA : x0 = i is invariant and ergodic under + { ∈ } + the relation S ([i]0 [i]0) for every i 0,...,n 1 with µ ([i]0) > 0. A ∩ × ∈{ − } Proof. The quasi-invariance and ergodicity of µ+ under S are obvious from (6.1) + + and the deﬁnition of µ . Suppose that µ ([i]0) > 0, and that there exist disjoint + Borel sets B1,B2 [i]0 with positive measure, each of which is saturated under SA ⊂ + ∩ ([i]0 [i]0). The ergodicity of µ under S implies that there exists a homeomorphism × + V [S] with µ (VB1 B2)>0. In particular we can ﬁnd a j 0,...,n 1 and ∈ (1) ∩ (1) (1) (2) (2) (2) ∈{ −(1)} allowed strings s = is1 ...sm 1j, s = is1 ...sm 1j, such that V ([s ]0)= (2) − − [s ]0 and the sets

(l) Cl = Bl [s ]0,l=1,2, ∩

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(l) (l) have positive measure. We put s0 = i, sm = j for l =1,2 and define, for every (p) p 0, V [RA]by ≥ ∈ (2) p s if σ (x) C1 and p k p + m, k p ∈ ≤ ≤ (V(p)x) = (1)− p k sk p if σ (x) C2 and p k p + m, − ∈ ≤ ≤p xk if kp+m, or σ (x) / (C1 C2). ∈ ∪ (p) + (0) (1) In other words, V checks for each x ΣA whether either of the strings s ,s occurs at the positions p,...,p+m; if one∈ of these strings occurs, then it replaces it by the other one, and if neither occurs, then the point is left unchanged. The measure µ+ is quasi-invariant under each of the maps V (p), and the shift-invariance of µ+ guarantees that the sequence of Radon-Nikodým derivatives (dµ+V (p)/dµ+,p ≥ 0) is uniformly integrable. By approximating Ci by closed and open sets and using uniform integrability we see that + (p) + lim µ (C )=µ (Cl) p l →∞ for l =1,2, where (p) (p) C = Cl V Cl. l ∩ The ergodicity of µ+ under S implies that µ+ is exact and hence mixing under + the shift σ on ΣA,sothat + (k) + + (k) lim µ (Cl [s ]p)=µ (Cl)µ ([s ]0) p →∞ ∩ for every k, l 0,1 . Hence ∈{ } + (p) (k) + + (k) lim µ (C [s ]p)=µ (Cl)µ ([s ]0) p l →∞ ∩ for every k, l 0,1 , and by setting ∈{ } (0) (p) (p) (2) (p) (1) V V x for x (C [i ]p) (C [i ]p), W (p)x = ∈ 1 ∩ ∪ 2 ∩ (x otherwise, + for every p>m, we have constructed a sequence of maps in [SA ] with + (p) + + (2) lim µ (C2 W C1)=µ (C2)µ ([s ]0) > 0. p →∞ ∩ + This violates the invariance of B1 and B2 under SA ([i]0 [i]0), and we conclude + +∩ × that the restriction of µ to [i]0 is ergodic under S ([i]0 [i]0). A ∩ × With this lemma at hand we can characterize the set of all shift-invariant measures on ΣA which are invariant and ergodic under SA. Any 1-step (irreducible aperiodic) SA-invariant SFT inside ΣA with a potential function on it depending on only a single coordinate determines such a measure, and we will show that they all arise in this way. In the following theorem we denote by ΣA ΣA an irreducible 0 ⊂ and aperiodic subshift defined by a 0-1-matrix A0 =(A0(i, j), 0 i, j n 1) with 2 ≤ ≤ − A0(i, j) A(i, j) for every (i, j) 0,...,n 1 .ThematrixA0may have some ≤ ∈{ − } zero rows and columns; however, the irreducibility of ΣA0 allows us to permute the alphabet 0,...,n 1 of ΣA, if necessary, and to assume that there exists an { − } integer n0 1,...,n 1 such that the first n0 rows and columns of A are nonzero ∈{ − } and the remaining ones zero, and to regard ΣA0 as an (irreducible and aperiodic) subshift of 0,...,n0 1 Z. Any probability measure µ on ΣA with µ(ΣA )=1will { − } 0 also be regarded as a probability measure on the SFT ΣA ΣA, and vice versa. 0 ⊂

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Theorem 6.2. Let µ be a nonatomic shift-invariant probability measure on ΣA which is invariant and ergodic under SA. Then there exist an irreducible, aperiodic and SA-invariant SFT ΣA ΣA and a function φ:ΣA Rdepending on the 0 ⊂ 0 → single coordinate x0 of every point x ΣA such that µ is the unique (Markov) ∈ 0 Gibbs measure of φ on ΣA0 . Conversely, if ΣA ΣA is an irreducible, aperiodic and SA-invariant SFT and 0 ⊂ φ:ΣA R is a function of a single coordinate, then the (Markov) Gibbs measure 0 7−→ µφ is invariant and ergodic under SA.

Proof. First we construct an irreducible and aperiodic SFT ΣA ΣA with µ(ΣA ) 0 ⊂ 0 = 1, such that µ is the Gibbs measure of a function φ:ΣA0 R of only two coordinates. 7−→ Let Σ be a countably generated sigma-algebra, and let be a countable T⊂BA D generator of .Wewrite [x] :x ΣA for the space of atoms of ,where T { T ∈ } T [x] = x D D for every x ΣA, and denote by µxT : x ΣA aregular T ∈ ∈D ∈ { ∈ } decomposition of µ over , i.e. a family of Borel probability measures on ΣA with the followingT properties: T

(a) µx([x] ) = 1 for every x ΣA, T ∈ (b) the map x µxT (B) from ΣA to R is -measurable for every Borel set 7→ T B ΣA,and ⊂ µT(B)=Eµ(1B )(x) µ-a.e. x |T In view of (b) we may also assume without loss in generality that

(c) µxT = µxT for every x ΣA and x0 [x] . 0 ∈ ∈ T Let 0 = [0]0,...,[n 1]0 be the state (time-0) partition of ΣA, and let P k{ − } = k 0 σ− ( 0) ΣA be the sigma-algebra generated by all cylinder sets of A ≥ P ⊂B m+1 the form C =[i0,...,im]0,wherem 0and(i0,...,im) 0,...,n 1 .Then W 1 ≥ ∈{ − } −=σ− ( ) , and the atoms of and − are of the form A A ⊂A A A [x] = x0 ΣA : xk0 = xk for k 0 , A { ∈ ≥ } [x] = x0 ΣA : xk0 = xk for k 1 A− { ∈ ≥ } for every x ΣA. The conditional information function Iµ( −)isgivenby ∈ A|A − Iµ( −)(x)= log Eµ(1[x ] −)(x)= log µxA ([x] ) A|A − 0 |A − A (6.2) dµxA =log 1 (x) σ− ( ) dµx A

for every x ΣA (cf. (3.11) in [6]). The map ∈ Iµ( −) (6.3) J = e− A|A :ΣA [0, 1] R 7−→ ⊂ is obviously -measurable, and we claim that it is a function of the coordinates A x0,x1 of every point x ΣA. ∈ In order to prove this claim we assume that s = s0 ...sm and s0 = s0 ...sm0 are allowed strings with s0 = s0, s1 = s10 and sm = sm0 , and such that the entries in s and s0 are permutations of each other, and consider the map V [SA]which ∈ interchanges s and s0, i.e.

s0 if x [s]0 and 0 k m, k ∈ ≤ ≤ (Vx)k = sk if x [s0]0 and 0 k m,  ∈ ≤ ≤ xk if x/([s]0 [s0]0)ork>m, ∈ ∪ 

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for every x ΣA.ThenV( )= ,V( −)= −, and (6.2)–(6.3) show that ∈ A A A A J(x)=J(Vx) + for every x ΣA.The -measurability of J allows us to view it as a map J :Σ ∈ A A 7−→ R, and the argument in the preceding paragraph shows that this map is invariant + + under the subrelation of SA consisting of all (x, x0) SA with x0 = x0 and x1 = x10 . According to Lemma 6.1 this implies that J is constant∈ µ+-a.e. on each cylinder set + + + [i0,i1]0 Σ ,whereµ is the projection of µ onto Σ , and thus a function of the ⊂ A A coordinates x0,x1. This also proves our claim for the original map J :ΣA R. 7−→ Deﬁne a 0-1-matrix A0 =(A0(i, j), 0 i, j n 1) by setting (with the under- standing that 00 =0) ≤ ≤ −

0 + J(x0,x1) 0,1 if x [i, j]0 ΣA, A0(i, j)= ∈{ } ∈ ⊂ (0if[i, j]0 = ∅.

Clearly A0(i, j) A(i, j) for every (i, j) 0,...,n 1 . We claim that the SFT ≤ ∈{ − } ΣA ΣA deﬁned by A0 has the following properties: 0 ⊂ (d) ΣA0 is equal to the (closed) support of µ; (e) if R+ is the Gibbs relation of the one-sided SFT Σ+ ,then A0 A0 dµ+(x) J(σk(x)) = dµ+(y) J(σk(y)) k 0 Y≥ + for every (x, y) RA ; ∈ 0 + (f) if RA is the Gibbs relation of Σ ,then 0 A0 dµ+(x) J(σk(x)) = dµ+(y) J(σk(y)) k 0 Y≥ for every (x, y) RA ; ∈ 0 (g) SA(ΣA0 )=ΣA0. Indeed, (e) is an immediate consequence of the deﬁnition of J in terms of conditional measures, (f) follows from (e) and the shift-invariance of µ, (d) follows from the

fact that µ is the Gibbs measure of log J on ΣA0 , and (g) is a consequence of this as well as the SA-invariance of µ.Sinceµis shift-invariant and SA-ergodic, Remark 4.5 shows that the matrix A0 is aperiodic. + + Next we prove that the one-sided measure µ is equivalent to an SA -invariant measure ν+. Renumber the symbols 0,...,n 1ofA, if necessary, choose an integer n0 − ∈ 1,...,n 1 such that first n0 rows and columns of A0 are nonzero and the { − } remaining ones zero, and view S = 0,...,n0 1 as the alphabet of ΣA { − } 0 ⊂ ΣA SZ 0,...,n 1 Z.PutP(i, j)=J(x0,x1) whenever x =(xk) ΣA ∩ ⊂{ − } ∈ 0 satisfies x0 = i, x1 = j.Asµis invariant under SA, the conditions (f) and (g) above show that dµ(x) P (x ,x ) (6.4) = k k+1 =1 dµ(y) P (yk,yk+1) k Y∈Z + + whenever (x, y) SA0 = SA (ΣA0 ΣA0 ). We set SA = SA (ΣA0 ΣA0 )and + ∈ ∩ × + 0 ∩ × define the S -equivalence classes of symbols of Σ 0,...,n0 1 N as in Remark A0 A0 ⊂{ − } 4.4. For each such equivalence class C 0,...,n0 1 choose and fix a symbol ⊂{ − }

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+ iC C. Given any x ΣA , ﬁnd the equivalence class C with x0 C,choose ∈ + ∈ 0 ∈ y S (x) with y0 = iC,andset ∈ A0 dµ+(x) b(x)= . dµ+(y) Then (from (6.4)) b(x) is well-deﬁned, i.e. independent of the choice of y,andonly + depends on the coordinate x0 of x. For any (x, y) S , ∈ A0 dµ+(x) b(x) = . dµ+(y) b(y) + + Let ν be the unique probability measure on ΣA which is a constant multiple of 1 + + 0 b− µ .Thenν is Markov, + dν (x) b(y) P (xk,xk+1) + = dν (y) b(x) P (yk,yk+1) k 0 Y≥ for all (x, y) R+ ,andν+ is invariant under S+ (which is equivalent to saying A0 A0 + ∈ + + that ν , regarded as a probability measure on ΣA, is invariant under SA ). Finally we show that µ can also be written as the Gibbs measure of a function of a single coordinate on ΣA 0,...,n0 1 Z. 0 ⊂{ − } Regard b as a function on the alphabet 0,...,n0 1 of ΣA0 by setting b(i)=b(x) { ¯ − } 1 whenever x0 = i, i =0,...,n0 1, and put P(i, j)=b(i)− P(i, j)b(j) for every 2 − (i, j) 0,...,n0 1 with A0(i, j) = 1. We claim that P¯(i, j)=P¯(i, j0) whenever ∈{ − } A0(i, j)=A0(i, j0) = 1 (or, equivalently, whenever µ([ij]0)µ([ij0]0) > 0). As we have seen above, dν+(x) P¯(x ,x ) = k k+1 dν+(y) P¯(y ,y ) k 0 k k+1 ≥ + Y for every (x, y) RA .GivenA0-allowable 2-blocks ij and ij0, Lemma 6.1 implies ∈ 0 + + + + that there exists a pair (x, y) S ([ij]0 [ij0]0)=S (Σ Σ ) ([ij]0 [ij0]0). A0 A A0 A0 + + ∈ ∩ × ∩ × ∩ × As ν is SA -invariant, dν+(x) P¯(x ,x ) = k k+1 =1. dν+(y) P¯(y ,y ) k 0 k k+1 Y≥ + + + + However, the pair (σ(x),σ(y)) also lies in S = S (Σ Σ ), since x0 = y0, A0 A A0 A0 so that ∩ × dν+(σ(x)) P¯(x ,x ) = k k+1 =1. dν+(σ(y)) P¯(y ,y ) k 1 k k+1 Y≥ ¯ ¯ By comparing the last two equations we have established that P(i, j)=P(i, j0), which proves our claim. Let φ: 0,...,n0 1 Rbe the map satisfying { − }7−→ φ(x0) = log P¯(x0,x1)= Iµ(x0,x1) log b(x0) + log b(x1) − − for every x ΣA . From condition (f) above we know that ∈ 0 ρµ(x, y)= (φ(xk) φ(yk)) − k X∈Z

for every (x, y) RA0 . In other words, µ is the unique Gibbs measure of φ on ΣA0 . This completes∈ the proof of the ﬁrst part of the theorem. The converse assertion is obvious from (3.4), the deﬁnition of SA and Theorem 3.3.

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Remark 6.3. If µ is atomic, invariant and ergodic under SA,thenµhas to be concentrated on a ﬁxed point.

11 Example 6.4. Symmetric measures on the Golden Mean SFT. If A =(10), then + Theorem 2.11 describes all the SA -invariant probability measures on ΣA.Inin- terpreting this statement in the two-sided case one has to be a little careful (cf. (6.1)): the two points ...01010101010 ... of period two are both fixed under SA; therefore each of them carries an SA-invariant probability measure which is not shift-invariant. However, every nonatomic SA-invariant and ergodic probability measure on ΣA is also shift-invariant. Example 6.5. A nonatomic, symmetric but not shift-invariant measure on an SFT. If 110 A = 111 , 011 then the point ...000012222 ... is fixed under SA, but not under the shift, and carries an atomic, SA-invariant probability measure which is not shift-invariant, and whose orbit under the shift is infinite. In order to obtain a nonatomic example we set 11100 11100 A = 11111 00111 00111! and consider the unique SA-invariant probability measure on the set x =(xk) { ∈ ΣA :x0 =2 and x k 0,1 ,xk 3,4 for every k>0 . − ∈{ } ∈{ } }

Example 6.6. An irreducible, aperiodic, SA-invariant SFT ΣA0 ΣA which is not determined by its alphabet. Let ⊂ 111 110 A = 001 ,A0=001 . 100 100 Then ΣA0 has the same alphabet as ΣA and is irreducible, aperiodic and SA- invariant. This example shows that the irreducible, aperiodic and SA-invariant Z SFT’s ΣA0 ΣA which arise in Theorem 6.2 are not necessarily of the form ΣA S for some subset⊂ S 0,...,n 1 . ∩ ⊂{ − } 7. Questions and remarks

7.1. Weak mixing. As in Section 1 we let V =(Vk,k 0) be a sequence of ≥ nonempty, finite, totally ordered sets, put X = Vk, assume that Y X V k 0 ⊂ V is a nonempty, closed subset satisfying (M), write T ≥for the adic transformation QY of Y , and define the m-weights wm(y),y Y, by (2.13). In the case of a full shift ∈ with an adic-invariant measure, TY has an eigenfunction with eigenvalue ζ only if ζwm(y) 1 µ-a.e. How general is this statement? Conditions→ for existence of eigenfunctions in the stationary case were developed in [33] and [50]. We conjecture that even in the measure-preserving Bernoulli case w (y) on the full 2-shift the adic transformation TY is weakly mixing: ζ m 1 µ- → a.e. implies that ζ = 1. It is not difficult to see that TY is topologically weakly mixing. Also, because of the self-similar structure of Pascal’s triangle modulo a prime [29, 35], TY cannot have an eigenvalue (other than 1) that is a root of unity. If ζ is not a root of unity, perhaps ζwm(y) is even almost surely uniformly distributed in the unit circle, or at least dense. Conceivably, it is not convergent to 1 down every path inside Pascal’s triangle. Consider the skew product transformation T (z1,z2,z3,...)=(ζz1,z1z2,z2z3,...) on the infinite torus. For ζ not a root of

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unity, it is well known that this is uniquely ergodic, and so, as we look at any ﬁxed coordinate in the successive members of the orbit of any point, we see a uniformly distributed sequence. If we ﬂip a coin with probabilities α and 1 α of heads and tails, and each time the coin comes up heads we shift our view one− coordinate to the right, will we still see, with probability 1, a uniformly distributed, dense, or at least not convergent to 1 sequence?

7.2. Super-K. Let us say that a ﬁnite-state process ( ,T), where T is a measure- preserving or nonsingular transformation on a probabilityP space (X, µ)and is a ﬁnite measurable partition of X, is (one- or two-sided) super-K if the associatedP (dependent, transient) random walk on 0,1,2,... |P|, which keeps track not only of which symbol appears (i.e., whichP×{ atom of is} entered) at each time n, but also how many times each symbol has appearedP up to time n,has(one-or two-sided, respectively) trivial tail. We have shown above that certain processes with Gibbs measures, including all mixing Markov processes, are two-sided super- K. Are there other natural examples, for example processes with the right uniform rate of mixing? Does every K-system have a super-K generator?

7.3. Invariant measures. In Section 6 we identified the shift- and SA-invariant measures on subshifts of finite type. More generally, if ψ is a continuous map on ΣA taking values in an arbitrary countable, discrete group with finite conjugacy ψ G classes, what can be said about the set of SA-invariant measures on ΣA?Itis ψ not difficult to check that, if ψ takesvaluesinafinite group, then the only SA- invariant measure is the measure of maximal entropy on ΣA (i.e. the unique RA- invariant measure). More generally, if is a countable, discrete group with finite G conjugacy classes and 0 a finite group, and if ψ :ΣA and ψ0 :ΣA 0 are G 7−→ G 7−→ G continuous maps, we denote by (ψ,ψ0): ΣA 0 the product map and leave it as an exercise to show that the set of nonatomic,7−→ G × shift-invariant G measures which ψ (ψ,ψ0) are invariant under SA coincides with the corresponding set for SA . 7.4. Asymptotic formulae. Let A =(A(i, j), 0 i, j n 1) be an irreducible and aperiodic 0-1-matrix, P =(P(i, j), 0 i, j n≤ 1)≤ a stochastic− matrix which is compatible with A,and¯pthe probability≤ vector≤ satisfying− pP ¯ =¯p. By (4.1), the shift-invariant Markov measureµ ¯P in (4.3) is quasi-invariant under the relation SA appearing in Example 2.2, and is equivalent to the Gibbs measure µlog P (cf. (4.2)). + /n We define ψ0 :Σ ZZ Z as in Examples 5.3 and 5.4, and set A 7−→ /n ψ0(0,x)=0 ZZ Z, ∈ m 1 − k ψ0(m, x)= ψ0(σ (x)), k=0 +X [[s]]m = x Σ : ψ0(m +1,x)=s { ∈ A } /n for every m 1, s ZZ Z and x ΣA. The following≥ proposition,∈ which∈ gives necessary and sufficient conditions for a (quasi-invariant) Markov measure to be ergodic for the symmetric equivalence + + relation SA (equivalently, the adic transformation T ∗+ ) on the SFT ΣA, should be ΣA compared with Theorem 2.5. Proposition 7.1. The following conditions are equivalent.

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(1) The measure µ¯P is ergodic under the adic transformation T ∗+ (or, equiva- ΣA + lently, under the relation SA ). (2) For every Borel set B Σ+, ⊂ A µ¯P (B [[ψ0(m +1,x)]]m [xm]m) lim ∩ ∩ m µ¯P ([[ψ0(m +1,x)]]m [xm]m) →∞ ∩ µ¯P (B [[ψ0(m +1,x)]]m) = lim ∩ =¯µP(B)¯µP-a.e.. m µ¯ ([[ψ (m +1,x)]] ) →∞ P 0 m (3) For every i 0,...n 1 , ∈{ − } µ¯P ([i]0 [[ψ0(m +1,x)]]m [xm]m) lim ∩ ∩ m µ¯P ([[ψ0(m, x)]]m [xm]m) →∞ ∩ µ¯P ([i]0 [[ψ0(m +1,x)]]m) = lim ∩ =¯p(i)¯µP-a.e. m µ¯ ([[ψ (m, x)]] ) →∞ P 0 m ( ,m) + ( ,m) Proof. Put R ∗ = (x, x ) S : x = x for k m ,m 0, write ∗ for Σ+ 0 A k k0 Σ+ A { ∈ ≥ } ≥ B A ( ,m) + the sigma-algebra of R ∗+ -saturated Borel subsets of ΣA,andset ΣA ( ,m) ∗+ = ∗ . Σ Σ+ B A B A m 0 \≥ ( ,m) For every m 0, R+ ∗ is the smallest sigma-algebra with respect to which the ≥ A maps ψ0(k, ),k m, are measurable. · ≥ As we saw in the proof of Theorem 4.3, there exists a ﬁnite partition 1,..., p + O O of ΣA into closed and open subsets such that the restriction ofµ ¯P to each i is ergodic under R ( ). From the choice of the map h¯ in (3.14) and RemarkO Σ∗ + i i A ∩ O ×O 4.4 it is furthermore clear that each of the sets i canbechosenas O i = [j]0, O j F [∈ i

where F1,...,Fp is the partition of the alphabet Z/nZ into R∗ + -equivalence classes ΣA of symbols, and that

Σ∗ + = 1,..., p (modµ ¯P ). B A {O O } If B Σ+ is a Borel set, then the martingale convergence theorem implies that ⊂ A ( ,m) lim Eµ¯P (1B ∗+ )=Eµ¯P(1B ∗ + ) m Σ ΣA →∞ |B A |B 1 + µ¯P -a.e. and in L (ΣA, Σ+ , µ¯P ). In particular, ifµ ¯P is ergodic, then B A µ¯P (B [[ψ0(m +1,x)]]m [xm]m) lim ∩ ∩ m µ¯P ([[ψ0(m +1,x)]]m [xm]m) →∞ ∩ 1 = lim m inf ty µ¯P ([[ψ0(m +1,x)]]m [xm]m) → ∩ ( ,m) E (1 ∗ ) 1 dµ¯ µ¯P B Σ+ [[ψ0(m+1,x)]]m [xm]m P · |B A · ∩ Z =¯µP(B)¯µP-a.e.,

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( ,m) 0,˙,sn 1 since [[s]] [i] ∗ for every s and i /n . The omission of m m Σ+ Z{ − } Z Z ∩ ∈B A ∈ ∈ the terms [xm]m is equivalent to replacing [[ψ0(m+1,x)]]m [xm]m =[[ψ0(m)]]m 1 ∩ − ∩ [[ψ0(m+1,x)]]m by [[ψ0(m+1,x)]]m and does not aﬀect the calculation. This shows that (1) implies (2) and hence (3). Conversely, ifµ ¯ is nonergodic, the description of the sigma-algebra shows P Σ∗ + B A that, for every i /n , E (1 )=0onΣ+ ,where Z Z µ¯P [i]0 Σ∗ + A r j ∈ |B A O + j = [i0]0i0 0,...,n 1 is S -equivalent to i . O { ∈{ − } A } [ ( ,m) It follows that there exists, for every suﬃciently large m,asetC ∗ of the Σ+ ∈B A form C =[[sm]]m [[sm+j]]m+j or, equivalently, of the form C =[i]m ∩ ··· ∩ ∩ [[sm+1]]m+1 [[sm+j]]m+j, with ∩···∩

µ¯P (1[i]0 C) 1 ( ,m) ∩ = 1 E (1 ∗ ) dµ¯ C µ¯P [i]0 Σ+ P µ¯ (C) µ¯ (C) · |B A P P Z µ¯ ([i] ) p¯(i) < P 0 = . 2 2

Since the measureµ ¯P is Markov,

µ¯P ([i]0 C) µ¯P ([i]0 [[sm]]m [i]m) ∩ = ∩ ∩ . µ¯P (C) µ¯P ([[sm]]m [i]m) ∩ From the two different ways of writing C it is clear that the terms [xm]m can be omitted in this calculation. We have proved that (3)—and hence (2)—cannot hold ifµ ¯P is nonergodic. Proposition 7.1 has combinatorial consequences whose direct proofs appear quite 11 + + difficult in all but the most elementary cases. Let A =(11), ΣA =Σ2, and let ab P = cd be a stochastic matrix which is compatible with A, i.e. which satisfies abcd > 0. We denote by µ =¯µP the shift-invariant probability measure defined in (3.1), wherep ¯ is the probability vector satisfyingpP ¯ =¯p.Sinceµ=¯µP is equivalent to the Gibbs measure µlog P , Theorem 4.3 and Remark 4.4 imply that µ + is ergodic under S2 ,sothatµmust satisfy the equivalent conditions (2) and (3) in Proposition 7.1. For i, j 0,1 ,m 1ands=(s0,s1) with s0 + s1 = m +1,let ∈{ } ≥ µ([i]0 [[(s]]m [j]m) Q (s)= ∩ ∩ . i,j p¯(i)

+ m0 If C =[i0,...,im0]0 Σ2 with j=0 ij = s10 , s0 = m0 +1 s10,ands0 =(s0,s10 ), then ⊂ − µ(C [[s]]Pm [j]m) ∩ ∩ = Qi j (s s0), µ(C) m0 − and Proposition 7.1 implies that

Qi xm (ψ0(m +1,x) s0) (7.1) lim m0 − =1 m Qj xm (ψ0(m +1,x) s00) →∞ m00 − + 2 for µ-a.e. x Σ2 ,andforanys0,s00 N . In other words, Proposition 7.1 is a statement about∈ ‘amnesia’ concerning the∈ initial coordinates of the points x Σ+. ∈ 2 In this special case the quantities Qij (s) can be determined explicitly (as sums

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of products of binomial coeﬃcients and other factors), but the veriﬁcation of the convergence (m) Qij (s s0) lim (m) − =1 m Qi j (s s ) →∞ 0 − 00 (m) (m) (m) (m) in (7.1) for suitable sequences (s =(s0 ,s1 ),m 1) with si for i =0,1 looks forbidding. ≥ →∞

7.5. Interval splitting. There should be some consequences for interval splitting of the ergodicity of Markov measures for the Pascal adic on the full shift. Perhaps if intervals are split in diﬀerent proportions depending on whether they arose as the left or right half of a previously-split interval, the resulting division points would still be uniformly distributed. Weak mixing of the Pascal adic might imply uniform distribution in a rectangle if two intervals are split simultaneously.

7.6. Dynamics. What are the joinings of the adic transformation on the full 2- shift with two diﬀerent Bernoulli measures? Are these adic systems disjoint, or at least not isomorphic? A very general problem is to describe the dynamics of the measures found in our above theorem, especially in case the equivalence relation ψ+ SA consists of the orbit relation for a single nonsingular transformation.

7.7. Possible applications. Subshifts of finite type are important for the design of actual communication systems, especially in magnetic recording. In biological or materials science applications (say polymer building) SFT’s might arise from something like the momentary disabling of a receptor: receipt of a certain symbol could make the receptor momentarily sensitive only to certain other symbols. If we want to model a signal recorded with constraints of the kind imposed by mem- bership in a subshift of finite type (for example to record efficiently on a disk that has already been used), in the absence of further information it might be reason- able to assume that the statistics of the signal are given by a measure with some of the symmetries discussed above. For example, as in de Finetti’s motivation for exchangeability (the distribution of repeated samples should be independent of the order in which they are drawn), perhaps it is natural to assume that cylinder sets in the SFT which map to one another under permutations of finitely many coordinates (or are symmetric in some other respect) have equal probabilities. For applications, the case of higher-dimensional actions needs serious development. The ψ+ general situations connected with SA -invariance (which can keep track of accumulated symbol or transition counts or derangement effects) might reflect a cost associated with sending or receiving certain signals, or a hysteresis or memory in materials. In image enhancement and pattern generation and recognition, symmetric Gibbs measures could account for the presence of texture, like bands or dapples [1, 12]. Knowing their dynamics, for example spectral properties, could help to filter out such background by using appropriate transforms, to code signals better by taking the structure into account, or to detect boundaries between regions with different textures. The presence of discrete spectrum might reflect subtle rhythms (periodic or aperiodic regularities different from spatial regularities, which are connected with the dynamics of the associated shift transformations). Weak mixing would be tantamount to the lack of any such almost periodic structure and might indicate resistance to filtration by Fourier methods.

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Department of Mathematics, CB 3250, Phillips Hall, University of North Carolina, Chapel Hill, North Carolina 27599 E-mail address: [email protected] Department of Mathematics, University of Vienna, Vienna, Austria E-mail address: [email protected]

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