DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 9,Number6,November2003 pp. 1423–1446

UNIFORM BERNOULLI IN DYNAMICS OF PERMUTATIVE CELLULAR AUTOMATA WITH ALGEBRAIC LOCAL RULES

Bernard Host

Equipe d’Analyse et de Math´ematiques Appliqu´ees Universit´e de Marne la Vall´ee 5 Boulevard Descartes, Champs sur Marne 77454 Marne la Vall´ee Cedex, France Alejandro Maass, Servet Mart´ınez

Departamento de Ingenier´ıa Matem´atica and Centro de Modelamiento Matem´atico UMR 2071 UCHILE–CNRS Universidad de Chile Casilla 170–3, correo 3 Santiago, Chile

Abstract. In this paper we study the role of uniform Bernoulli measure in the dynamics of cellular automata of algebraic origin. First we show a representation result for classes of permutative cellular automata: those with associative type local rule are the product of a group cellular automaton with a translation map, and if they satisfy a scaling condition, they are the product of an affine cellular automaton (the alphabet is an Abelian group) with a translation map. For cellular automata of this type with an Abelian factor group, and starting from a translation with complete connections and summable decay, it is shown that the Ces`aro mean of the iteration of this measure by the cellular automaton converges to the product of the uniform Bernoulli measure with a shift invariant measure. Finally, the following characterization is shown for affine cellular automaton whose alphabet is a group of prime order: the uniform Bernoulli measure is the unique invariant probability measure which has positive entropy for the automaton, and is either ergodic for the shift or ergodic for the Z2-action induced by the shift and the automaton, together with a condition on the rational eigenvalues of the automaton.

1. Introduction. Our purpose is to show the role of uniform Bernoulli measure for a class of right permutative cellular automata whose local rules verify some algebraic conditions. In this direction we study the convergence of the Ces`aro mean of the dynamics of some translation invariant probability measures by such automata. This is done in the same spirit of the pioneering work of Lind [15] for the mod 2 automaton, where it is proven that, starting from any Bernoulli measure these Ces`aro means converge to the uniform Bernoulli measure. The same result was shown for larger classes of affine cellular automata and more general initial

1991 Mathematics Subject Classification. Primary: 54H20; Secondary: 37B20 . Key words and phrases. permutative cellular automata, uniform Bernoulli measure, chains with complete connections.

1423 1424 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ probability measures in [18] and [7]. The tool developed in [7] is regeneration theory of stochastic processes with finite state space. The point of view of Lind is that of harmonic analysis: the cellular automaton is viewed as an endomorphism of the product group and the uniform Bernoulli measure is the . This approach was followed recently in [20] and [21] where the authors prove the same kind of results for the class of probability mea- sures they called harmonically mixing and for the family of affine cellular automata verifying a diffusive condition. On the other hand, in the non-expansive case, in [2] it was proven that the limit of the Ces`aro mean exists for any shift ergodic measure of full topological support and any cellular automaton having equicontinuity points. All the previous results motivated us to study those sensitive cellular automata which can be decomposed as the product of two cellular automata; one an affine cellular automaton, the other a cellular automaton having an equicontinuous di- rection. From an ergodic point of view –and when spaces are endowed with the appropiate measures– this gives a decomposition into a product of disjoint dynam- ical systems [5]. Then, under suitable conditions on the initial measure and the cellular automaton, the weak limits of Ces`aro means can be decomposed into the product of its marginal measures: a uniform Bernoulli measure with a shift invari- ant one. In particular we consider initial measures which are chain connected and have summable decay,asin[7]. These ideas are carried out for two classes of right permutative cellular automata: those with ψ-associative local rules and those satisfying a N-scaling condition. We point out that these classes of permutative cellular automata introduced here are rich enough to describe the idea of the product representation we develop, but these classes are far from being exhaustive. The algebraic representation of these automata is studied in Section 3 where we distinguish between group cellular automata and affine cellular automata, a crucial difference is that for affine cellular automata the finite group defining the alphabet is Abelian. We prove ψ-associative (N-scaling) cellular automata are, up to a one- block map conjugacy, the direct product of a group (affine) cellular automaton with a translation map of a full shift. This is done in Theorem 3.6 (Theorem 3.8). We point out that for N-scaling bipermutative cellular automata representation results can be found in [19], which, from a purely algebraic point of view are intimately related with the classification of quasi-groups (see [4] or [3] for general results). Since probability measures obtained as weak limits of Ces`aro means are invariant for both the shift map and the automaton, then it is natural to identify these invariant measures. For affine cellular automata this problem has been called the “three dot problem” in reference to the addition cellular automaton defined on {0, 1}Z. This question is related to actual research of rigidity of Zd-actions in different contexts: for the geometric aspects see for example [11], [12], [13], [22], [9], [8], [25]; in a symbolic dynamics context, closer to our work, quote [24]; and in an algebraic context see [23]. In Section 4, Theorems 4.1 and 4.2, we prove for affine cellular automata whose underlying alphabets are of prime cardinality, that the uniform Bernoulli measure is the unique probability measure such that: (a) has positive entropy for the automaton and (b) the shift map verifies some conditions: either it is simple ergodic for the shift, or, it is ergodic for the Z2-action together with a condition on invariant sigma algebras for powers of the shift. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1425

Section 5 is mainly devoted to the study of the convergence of Ces`aro means of the iteration of classes of probability measures by product cellular automata. We also give some general criteria to identify invariant measures for the action of the shift and an affine-translation cellular automaton. First in subsection 5.2 we supply new classes of measures that are harmonically mixing. In Theorem 5.4 we prove that probability measures with complete con- nections and summable decay defined on a fullshift whose alphabet is an Abelian group, are harmonically mixing and thus, the Ces`aro mean of the iteration of this measures by affine cellular automata converges to the uniform Bernoulli measure, see Theorem 5.6. Subsection 5.3 is devoted to the study of the product of two cellular automata, one of them having an equicontinuous direction. In Theorems 5.8 and 5.9, we prove, under some ergodicity conditions, that probability measures invariant for both the shift map and the cellular automaton, are the product of the marginal measures. In Theorem 5.12 we consider affine-translation cellular automaton and an initial probability measure with complete connections and summable decay. We prove that, the Ces`aro mean converges to the product of the uniform Bernoulli measure with a shift invariant one. In particular, these results are applied in Corollaries 5.13 and 5.14, to right permutative cellular automata with ψ-associative local rules and those verifying the N-scaling condition. In the next section we provide definitions and introduce the classes of group- translation and affine-translation cellular automata.

2. Definitions. For a finite set A, called the alphabet, denote by A∗ the set of finite sequences or words w = w0...wn−1 with letters in A.By|w| we mean the ∗ Z Z length of w ∈ A . The shift map σ : A → A is defined by (σx)i = xi+1 for Z x =(xj)j∈Z ∈ A , i ∈ Z. If we need to distinguish a shift map according to its Z ∗ alphabet we denote it by σA.Forx ∈ A and i ≤ j in Z or x = x0...xn ∈ A and i ≤ j in {0, ..., n}, we denote by x[i, j]=xi...xj the finite word in x between coordinates i and j.Givenw ∈ A∗ and i ∈ Z, the cylinder set starting in coordinate Z i with word w is [w]i = {x ∈ A : x[i, i + |w|−1] = w}. A cellular automaton (c.a.)(AZ, f) is a map f : AZ → AZ defined by

Z (fx)i = f(xi+l(f), ..., xi+r(f)),x∈ A ,i∈ Z, where f : Ar(f)−l(f)+1 → A is a local rule, or (r(f)−l(f)+1)-block map. The integers l(f)andr(f), l(f) ≤ r(f), are the left and right radius respectively and if it is clear from the context we will write l and r respectively. If l ≥ 0, we say that f is one- sided andweassumel = 0. We will also use f to indicate the action of the local rule ∗ on words of length greater or equal to r − l + 1. That is, for w = w0...wn ∈ A with |w|≥r − l + 1, we put f(w)=f(w0, ..., wr−l)f(w1, ..., wr−l+1)...f(wn−r+l, ..., wn). Given M ∈ Z, f ◦σM is a c.a. with the same local rule as f but l(f ◦σM )=l(f)+M and r(f ◦σM )=r(f)+M. Analogously we define c.a. acting on the set of one-sided infinite sequences AN. The c.a. (AZ, f) is said to be right permutative if for every w ∈ Ar−l the map f(w, ·):A → A defined by f(w, ·)(a)=f(wa) is one-to-one. This implies that r−l r−l r−l   for every w ∈ A the map fw : A → A given by fw(w )=f(ww ) is also one-to-one. Analogously we define left permutative and bipermutative c.a. Clearly the shift map is not left permutative. 1426 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

Every c.a. is the composition of a one-sided c.a. with a power of the shift. As we are interested mostly in shift invariant measures, we consider in the sequel only one-sided c.a. We restrict also to c.a. of radius 1. A canonical way to reduce to this case is by considering the higher power shift construction: that is, take C = Ar and define f bl : CZ → CZ by f bl = πbl ◦ f ◦ (πbl)−1, where πbl : AZ → CZ is the well bl bl defined one-to-one onto map (π x)k = x[kr, (k +1)r − 1]. The map π defines a between f and f bl - that is, it is a continuous bijection and f bl ◦ πbl = πbl ◦ f (see [16])- and the map f bl is of radius 1. Since a main ingredient in our study is the set of starting measures, which will be only shift invariant without a suitable structure with respect to the c.a.,we will consider one-block conjugacies of the canonical recoding of the c.a.,thatis, local rules which are simple renamings of letters. We denote by sA a one-block Z conjugacy from A onto itself, so l(sA)=r(sA) = 0 and its local rule sA : A → A Z is one-to-one. The c.a. (A , sA ◦ σ) will be called a translation c.a., its local rule is given by ((sA ◦ σ)x)i = sA(xi+1). Let (AZ, f)beac.a. and µ be a translation invariant probability measure on AZ. n n −n As usual, f µ denotes the measure given by f µ(B)=µ(f B)forB a . In MN 1 N−1 n this paper we study the convergence of the Ces`aro mean µ (f)= N n=0 f µ.If this limit exists as N →∞, we denote it by Mµ(f). Notice that in the study of the N convergence of Mµ (f) we only need to consider one-sided c.a. In fact, if l(f) < 0we −l(f) N −l(f) N replace f by f ◦σ which is a one-sided c.a. and verifies Mµ (f ◦σ )=Mµ (f). Z Z A main role will be played by the uniform product measure λA of A , where λA is the equidistributed probability measure in A. Let us introduce two classes of c.a. For (K, ·) a finite group, the cellular automa- Z ton (K , g) with local rule given by g((a1,a2)) = a1 · a2 is called a group c.a. Now, let (K, +) be a finite Abelian group, θ, η two commuting automorphisms of K and c ∈ K. The cellular automaton (KZ, g) with local rule given by

g((a1,a2)) = θ(a1)+η(a2)+c is called an affine c.a. If c = 0 we call it a linear c.a. Observe that (KZ, +) is a Z compact group with + defined componentwise and the measure λK is the associated Haar measure. It is characterized as the unique probability measure µ such that ∈ Z  µ(χ)= KZ χdµ = 0 for every non-trivial character χ K ,thatisforχ =1.

3. Representation of some classes of permutative c.a. We consider here one- sided right permutative c.a., and we restrict to the case where the radius r is 1; we have already explained how to reduce to this case.

Z Z 3.1. Maximal bipermutative part. Let (K , gK ) and (B , gB)bec.a. The Z product c.a. ((K × B) , gK × gB) is defined by gK × gB(x, y)=(gK (x), gB(y)). Z Z Z Z We denote by πK :(K × B) → K and πB :(K × B) → B the associated projections, under which gK and gB are the natural factors of gK × gB. Z Z In the special case where (K , gK ) is an affine (resp. group) c.a. and (B , gB) is a translation c.a., the product will be called an affine-translation c.a. (resp. Z a group-translation c.a.). More generally, when (K , gK ) is bipermutative and Z (B , gB) is a translation c.a., the product will be called a bipermutative-translation c.a. Every bipermutative-translation c.a. gK × gB is clearly right permutative, but (unless B is a singleton set) gK × gB is not left permutative. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1427

Remark 3.1. If g = gK ×sB ◦σB is an affine-translation c.a. (or a group-translation c.a.) with K not being a singleton set, the affine c.a. gK is bipermutative and it represents the maximal bipermutative part of g up to one-block factors. That is, if π is a one-block factor of g onto a bipermutative c.a. h : CZ → CZ, then π can be factorized through gK . On the other hand, if K is a singleton set, the action of g is a translation and the c.a. g is completely right permutative, i.e., up to one-block conjugacy there is no bipermutative part. Proof. We show this only for affine-translation c.a., the other case is similar. Con- sider (a, b) ∈ K × B with π((a, b)) = d and put d = π(g((a, b), (a, b))) = π((θ(a)+   η(a)+c, sB(b))). Clearly, for any b ∈ B, g((a, b), (a, b)) = g((a, b ), (a, b)). Then, since h is a factor of g, h(d, d)=d and h(π((a, b)),d)=d.Buth is bipermutative, so π((a, b)) = d. This proves that {(a,b) ∈ K × B : π((a,b)) = d} = K(d) × B for some K(d) ⊆ K, which implies the desired result.

It would be interesting to find simple necessary and sufficient conditions for a c.a. to be isomorphic to a group-translation or an affine-translation c.a. by a one-block map conjugacy. We consider below two classes of c.a. with this property: those with ψ-associative local rules and those satisfying a N-scaling condition. We point out that these classes of right permutative c.a. are rich enough to exhibit various representations of this kind, but to our view they are far from being exhaustive.

3.2. Necessary conditions. Here we establish necessary conditions for a c.a. to be bipermutative-translation, and we provide a method for recovering its bipermu- tative part in this case. Let (AZ, f) be a one-sided right permutative c.a., with r(f) = 1. We define an operation • on A by a•b = f(a, b); this operation is right cancelable since f is right permutative. We define also an equivalence relation ∼ on A by a ∼ b ⇔ fa = fb. We writea ˜ the equivalence class of a ∈ A,andA˜ = A/ ∼ the quotient space. We use generally Greek letters for elements of A˜.Forα ∈ A˜, we define the bijection fα : A → A to be equal to fa for an arbitrary element a of α.Asfa is a bijection for every a, fα is also a bijection for every α. We say that ∼ is compatible with • if,

For every a, b, b ∈ A, b ∼ b ⇒ a • b ∼ a • b . (3.1) It follows immediately that, For a, a,b,b ∈ A, a ∼ a,b∼ b ⇒ a • b ∼ a • b . In this case, the operation • on A induces an operation ˜• on A˜, witha ˜ ˜• ˜b = a• b for every a, b ∈ A. We can define a map f˜ : A˜2 → A˜ by f˜(˜a,˜b)=f(a, b)=˜a ˜• ˜b for every a, b ∈ A. We write (A˜Z,˜f)thec.a. with local rule f˜. It is a factor of (AZ, f), associated to the one-block map which associates to each a ∈ A its equivalence classa ˜. For α ∈ A˜, the map f˜α : A˜ → A˜ defined as usual by f˜α(β)=f˜(α, β) satisfies f˜α(˜b)=fα(b)=α ˜• ˜b for every b ∈ A.Asfα : A → A is onto, f˜α : A˜ → A˜ is onto also, and it follows that it is a bijection. This means that the c.a. (A˜Z,˜f) is right permutative. 1428 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

Consider the case where the original c.a. (AZ, f) is a bipermutative-translation Z     c.a. ((K ×B) , gK ×gB). For (k, b), (k ,b) ∈ K ×B,wehave(k, b) ∼ (k ,b)ifand only if k = k. Property (3.1) is clearly satisfied, and the factor c.a. (A˜Z,˜f)isequal Z to (K , gK ), up to a one-block conjugacy. This means that, when looking for c.a. which are conjugate to bipermutative-translation c.a. (by one-block conjugacies), we can restrict ourselves to those which satisfy Property (3.1), and such that (A˜Z,˜f) is bipermutative. This last condition can be written:

For every a, a,b∈ A, a • b ∼ a • b ⇒ a ∼ a . (3.2) Properties (3.1) and (3.2) alone are clearly not sufficient to ensure that the given c.a. is conjugate to a bipermutative-translation one, and we shall restrict ourselves to some particular cases. But we begin with a simple preliminary remark. Lemma 3.2. If Properties (3.1) and (3.2) are satisfied, then all equivalence classes α ∈ A˜ have the same number of elements. Proof. Let α, β ∈ A˜. By permutativity of ˜f, there exists γ ∈ A˜ with γ ˜• α = β. The map fγ : A → A is one-to-one, and maps the class α into the class β,thus |α|≤|β|. Exchanging the role played by α and β we get the converse inequality, and our lemma is proved.

3.3. c.a. with ψ-associative local rules. Definition 3.3. Let (AZ, f) be a one-sided right permutative c.a. We say that f has ψ-associative local rule for some ψ : A → A if ∀a, b, c ∈ A, ψ((a • b) • c)=a • (b • c) . (3.3) The motivation of this definition is the following. Let (K, ·) be a finite group Z and (K ,gK ) be the group c.a. given by the local rule gK (x, y)=x · y for x, y ∈ K. Z Let (B , gB) be a translation c.a., where gB = sB ◦ σB. Then the product c.a. Z ((K × B) , gK × gB)isψ-associative, for the map ψ : K × B → K × B given by ψ(x, b)=(x, sB(b)) for x ∈ K and b ∈ B. We shall show below that every ψ-associative one-sided right permutative c.a. is of this type; but we begin by some remarks. Remark 3.4. Let (AZ, f) be a one-sided right permutative c.a. Then it is ψ-associative for some ψ if and only if: ∀a, b, c, a,b,c ∈ A :(a • b) • c =(a • b) • c ⇔ a • (b • c)=a • (b • c). Remark 3.5. Assume that f has ψ-associative local rule. Then the map ψ is unique and it is a bijection. Proof. By right permutativity, for every d ∈ A there exist a, b, c ∈ A such that d =(a • b) • c. Therefore, the map ψ satisfying (3.3) is unique. By permutativity again, for every d ∈ A there exist a, b, c ∈ A with d = a • (b • c); it follows that the map ψ : A → A satisfying (3.3) is onto, and thus it is a bijection.

Theorem 3.6. Let (AZ, f) be a one-sided right permutative c.a. with ψ-associative local rule. Then it is conjugate through a one-block map to a group–translation c.a. Proof. Step 1. We show now that f satisfies Properties (3.1) and (3.2). This means that the c.a. ˜f is well defined and bipermutative. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1429

Let a, b, b ∈ A with b ∼ b. For every c ∈ A we have     (a • b) • c = ψ−1 a • (b • c) = ψ−1 a • (b • c) =(a • b) • c thus a • b ∼ a • b. This proves (3.1). We take a ∼ a and b ∈ A. There exists c ∈ A with a • c = a • c, and there exists d ∈ A with c = b • d.Wehavea • (b • d) = a • (b • d), thus (a • b) • d =( a • b) • d, thus a • b ∼ a • b. Property (3.2) is proved. Step 2. We show here that (A,˜ ˜• ) is a group. To show (A,˜ ˜• ) is a group, it suffices for us to show that ˜• is associative, or, equivalently, that, for all a ∈ A, ψ(a) ∼ a. Let a ∈ A. By right permutativity of f, there exists b ∈ A with a • b = a;by left permutativity of ˜f, there exists c ∈ A withc ˜˜• a˜ =˜a,thatis(c • a) ∼ a.By right permutativity of f again there exists b ∈ A with (c • a) • b = a.Wehave (˜c ˜• a˜) ˜• ˜b =˜a =˜a ˜• ˜b =(˜c ˜• a˜) ˜• ˜b, and, by right permutativity of ˜f, ˜b = ˜b,thus b ∼ b. By using Property (3.1), we get   ψ(a)=ψ (c • a) • b = c • (a • b) ∼ c • (a • b) ∼ c • a ∼ a. Therefore, for every a, b, c ∈ A,wehave(a • b) • c ∼ a • (b • c), and (˜a ˜• ˜b) ˜• c˜ = a˜ ˜• (˜b ˜• c˜): the operation ˜• on A˜ is associative. This operation is also left and right cancelable by permutativity; as A˜ is finite, it follows easily that (A,˜ ˜• ) is a group. Step 3. Let be the identity element of the group (A,˜ ˜• ). Clearly (viewed as a subset of A) is stable under the operation •. We recall that the bijection f : A → A is defined as equal to fe for an arbitrary e ∈ . In other words, e • a = f(a)for every e ∈ and every a ∈ A. We have that

the map f is equal to ψ. (3.4)

In fact, for every a ∈ A there exists b ∈ A with f(b)=a. Now, for an arbitrary e ∈ we have e • e ∈ thus (e • e) • b = f(b)=a and

ψ(a)=ψ((e • e) • b)=e • (e • b)=e • f(b)=e • a = f(a) .

Step 4. By right permutativity, for every a ∈ A there exists an unique ea ∈ A with a • ea = a.Wehave˜a ˜• ea =˜a,thusea = and ea ∈ . We show

the restriction of the map a → ea to each class α ∈ A˜ is a bijection onto . (3.5)

Let a, b ∈ A with a ∼ b and a = b.Wehavea • eb = b • eb = b and a • ea = a, thus ea = eb. Thus the restriction of the map a → ea to each equivalence class α ∈ A˜ is one-to-one. As the classes α and have the same cardinal by Lemma 3.2, this map is a bijection and we conclude the proof of (3.5). Finally we prove that

for every a, b ∈ A, ea•b = ψ(eb). (3.6) Indeed, for a, b ∈ A we have,     (a • b) • ea•b = a • b = a • (b • eb)=ψ (a • b) • eb = ψ (a • b) • ea•b • eb =(a • b) • (ea•b • eb)=(a • b) • f(eb)=(a • b) • ψ(eb) where in the last equality we use (3.4). Step 5: end of the proof. 1430 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

We have now all the tools needed to prove Theorem 3.6. Let B = and sB : B → Z Z B be the restriction of bijection ψ to , and let sB : B → B be the corresponding Z one-block map; B is endowed with the translation c.a. gB = sB ◦ σB. Z Z Let u : A → A˜ × B be given by u(a)=(˜a, ea), and u : A → (A˜ × B) the corresponding one-block map. By (3.5), u is bijective, thus u is bijective too. It follows immediately from (3.6) and the definitions that u is a conjugacy from (AZ, f) Z onto ((A˜ × B) ,˜f × gB).

3.4. N-scaling right permutative c.a. Definition 3.7. Let (AZ, f) be a one-sided right permutative c.a. and N ≥ 2. We say that f satisfies the N-scaling property, or is N-scaling if N Z (f x)0 = x0 • xN for every x ∈ A . (3.7) In this Section, we prove: Theorem 3.8. Let (AZ, f) be a one-sided right permutative N-scaling c.a. Then it is conjugate to an affine–translation c.a. through a one-block map. Moreover, the associated group is isomorphic to (Z/pZ)s for some prime number p and some integer s. The following property of bipermutative N-scaling c.a. was established in [4] (Theorems 2.2.1 and 2.2.2): Theorem 3.9. [4] Let (AZ, f) be a bipermutative N-scaling c.a. Then A can be endowed with an operation +, such that (A, +) is an Abelian group, and the local rule is given by f(a, b)=θ(a)+η(b)+c,whereθ, η are commuting automorphisms of (A, +) and c ∈ A is a constant. In the next result we obtain additional properties in the bipermutative case. We keep the hypotheses and notation of Theorem 3.9, in particular, (A, +) is an Abelian group, and the local rule is given by f(a, b)=θ(a)+η(b)+c, where θ, η are commuting automorphisms of (A, +) and c ∈ A is a constant. Let e be the identity element of A. Lemma 3.10. Let (AZ, f) be a bipermutative N-scaling c.a. with |A| > 1.Then there exists a prime number p such that: 1. N = pt for some integer t ≥ 1; 2. (A, +) is isomorphic to ((Z/pZ)s, +) for some integer s ≥ 1; 3. there exists d ∈ A with d • d = d. Proof. By a direct computation we find that, for every x ∈ AZ,

N N −1 N N N−i i i (f x) = θ ◦ η (xi)+ (θ + η) (c). (3.8) 0 i i=0 i=0 Z N Let us take x ∈ A such that x[0,N]=e...e. From (3.7), (f x)0 = c. Therefore from (3.8) we get

N −1 N −1 (θ + η)i(c)=c or equivalently (θ + η)i(c)=e ; (3.9) i=0 i=1 UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1431 hence, for every x ∈ AZ it holds N N N−i i θ(x )+η(xN )= θ ◦ η (xi). 0 i i=0 Finally, by evaluating this last equality in points x ∈ AZ such that x[0,N]= e...eae...e for some a ∈ A we deduce

θN−1 =idandηN−1 =id, (3.10) and N a = e for every a ∈ A and every i ∈{1,...,N − 1} . (3.11) i    N ≤ { } 1) If gcd i :1 i

N i = 0 = =0 modp pt 1 0 0 i≥1   t N and we get a contradiction, since p

Lemma 3.11. Any N-scaling c.a. satisfies Properties (3.1) and (3.2) and the me- diality property: (a • b) • (c • d)=(a • c) • (b • d) for every a, b, c, d ∈ A. The mediality property was studied in [19] to find algebraic representations of bipermutative c.a. 1432 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

Proof. (1) We show first (3.2). Let a, a,b satisfy a • b ∼ a • b and let c ∈ A. Z  Consider points x, y ∈ A such that x0 = a, x1 = b, xN = c, y0 = a and yn = xn 2 2 for every n = 0. Clearly, (f x)n =(f y)n for every n>0and 2  2 (f x)0 =(a • b) • (b • x2)=(a • b) • (b • x2)=(f y)0 . 2 2 N N Thus (f x)n =(f y)n for every n ≥ 0 and it follows that (f x)0 =(f y)0.Bythe N-scaling property we get a • c = a • c. As it holds for every c, we find a ∼ a. Z (2) We show here the mediality property. We choose x ∈ A with x0 = a, N N x1 = b, xN = c and xN+1 = d. By (3.7), (f x)0 = a • c and (f x)1 = b • d,thus N+1 (f x)0 =(a • c) • (b • d). On the other hand, (fx)0 = a • b,(fx)N = c • d,and N+1 by (3.7) again (f x)0 =(a • b) • (c • d). (3) We prove now (3.1). Let a, b, b ∈ A with b ∼ b. By permutativity, for every c ∈ A we can choose c,c ∈ A with c • c = c.Wehave (a • b) • c =(a • b) • (c • c)=(a • c) • (b • c) =(a • c) • (b • c)=(a • b) • (c • c)=(a • b) • c, then (a • b) ∼ (a • b).

Proof of Theorem 3.8. As the Properties (3.1) and (3.2) are satisfied, the c.a. (A˜Z,˜f) is well defined and bipermutative. Clearly, this c.a. is also N-scaling. By Theo- rem 3.9 and Lemma 3.10, it remains only to prove that (AZ, f) is conjugate to the product of (A˜Z,˜f) by some translation c.a. By Lemma 3.10 part (3), there exists ∈ A˜ with ˜• = . The map f maps the class to itself. Let ψ : → be the restriction of f to this class. By left permutativity of ˜f, for every a ∈ A there exists a unique δa ∈ A˜ such that δa ˜• a˜ = . We choose an arbitrary da in the class δa and write ea = da • a.  This element does not depend on the choice of da, because any d ∈ δa verifies  d • a = da • a.  We prove the map a → ea is one-to-one on each equivalence class. Let a, a ∈ A    be such that a ∼ a and a = a .As˜a =˜a , by uniqueness we get δa = δa .Wehave  ea = da • a = da • a = da • a = ea . This proves the desired property. For a, b ∈ A, by the mediality property established in Lemma 3.11 we have: (da • db) • (a • b)=(da • a) • (db • b)=ea • eb ∈ thus (δa ˜• δb) ˜• (a • b)= .

By uniqueness, δa ˜• δb = δa•b. Then,

ea•b = da•b • (a • b)=(da • db) • (a • b)=ea • eb = ψ(eb) .

We conclude as in the proof of Theorem 3.6. We put B = , sB = ψ and gB is the translation c.a. associated to this map. The map u : A → A˜ × B given by u(a)= Z Z (˜a, ea) induces a one-block conjugacy from (A , f)onto((A˜ × B) ,˜f × gB).

4. Invariant measures for affine c.a. on Zp with p prime. Let K = Zp with p a prime number, and g an affine c.a. on KZ. Since g commutes with the shift σ, thecouple(g,σ) defines a Z2 action on KZ.Letµ be a (g,σ)-invariant probability measure on KZ. In this section we will give a criteria based on ergodicity and Z entropy that ensures µ = λK . We recall that a (g,σ)-invariant probability measure is ergodic for the action (g,σ) if every Borel invariant set under g and σ has µ measure 0 or 1. Z −n Let B be the Borel σ-algebra of K , we put Bn = g (B)forn ∈ N.Let P be the cylinder partition of KZ corresponding to the coordinate 0. For p

  ∞    −n  hµ(P−, g)=Hµ P− g (P−) n=1   ∞ −n  = − log E(1A| g (P−)) dµ. A n A∈P− =1 Theorem 4.1. Let (KZ, g) be an affine c.a. with |K| = p, a prime number, and µ a (g,σ)-invariant probability measure on KZ.Ifµ is ergodic for σ and of positive Z entropy for g, then µ = λK . Ergodicity with respect to σ is an extremely strong assumption, but the natural assumption of ergodicity for the action (g,σ) is not sufficient to guarantee the result, and counterexamples can be built. The next result illuminates the role played by the periods, that is, the rational eigenvalues for the shift. Theorem 4.2. Let (KZ, g) be an affine c.a. with |K| = p, a prime number, and µ a (g,σ)-invariant probability measure on KZ. Assume that (i) µ is ergodic for the Z2-action (g,σ); (ii) µ has positive entropy for g; (iii) the sigma algebra of σ(p−1)p-invariant sets coincide with the sigma algebra of σ-invariant sets up to sets of µ-measure 0. Z Then µ = λK . A direct proof of Theorem 4.1 could be given without using all the machinery we introduce below for the proof of Theorem 4.2, but we prefer to have similar proofs for both results. Henceforth, µ is a probability measure on KZ invariant by (g,σ). We assume that µ is ergodic for the action (g,σ) and that hµ(g) > 0. 4.1. Entropy. We begin with a classical remark that holds on any one-sided biper- mutative c.a. of radius 1.

Lemma 4.3. hµ(g)=Hµ(P|B1). n Proof. Let >0. From permutativity, to know (g x)[− , ] for all n>0isthe − ∞ ∞ −n P −1 P∞ same as to know (gx)[ , ). This means that n=1 g ( −)=g ( −) where P∞ ∞ −j P − ∞ − = j=− σ ( ). Moreover, by permutativity again, for a given (gx)[ , ), −1 ∞  −1 ∞ x[− , ] is determined by x0. This fact means that g (P−)∨P− = g (P−)∨P. Thus,   ∞      −n  −1 ∞ hµ(P−, g)=Hµ P− g (P−) = Hµ P|g (P−) . n=1 The result follows by taking the limit as →∞.

4.2. The groups Dn. Since K = Zp with p a prime number, the compact group Z Z K is a vector space over the field K.Fory ∈ K , we write Ty for the translation x → x + y in KZ. To the affine c.a. g(x)=ax + bσ(x)+c, we associate the homogeneous c.a. ∈ Z ≥ n n g0(x)=ax + bσ(x), for any x K . Notice that for any n 0, g (x)=g0 (x)+cn where cn is a fixed constant in K. 1434 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

Z n Let us introduce the following σ-invariant subgroups of K : Dn = ker(g0 )for n ≥ 0, and D∞ = ∪n≥0Dn. n Claim. Every d ∈ Dn is periodic (for the shift) of period (p − 1)p .

Proof. The groups Dpn can be described explicitly:

n   p n pn p pn−i i Z g x = a b xk i = axk + bxk pn ,n≥ 0,x∈ K ,k∈ N, 0 k i + + i=0 because a, b are non-zero elements of Zp and Lucas lemma [17]. Z Thus Dpn is the set of d ∈ K such that adk + bdk+pn = 0 for all k ∈ N.It n follows that every d ∈ Dpn is periodic of period (p − 1)p . Moreover, the sequence of groups {Dn : n ∈ N} is increasing and thus for every n we have Dn ⊂ Dpn .

An element d of D1 is uniquely determined by its coordinate d0,thus|D1| = p, n and then D1 is isomorphic to Zp. By induction, |Dn| = p for every n,andan element d of Dn is uniquely determined by d[0,n− 1]. Thus Dn acts by translations Pn−1 ∈ { ∈Pn−1} Pn−1 on the partition 0 , that is for any d Dn, d + A : A 0 = 0 . Also, Z n two points x, y ∈ K have the same image by g if and only if y = Td(x) for some d ∈ Dn.Thustheσ-algebra Bn consists of Td-invariant sets for every d ∈ Dn,up to µ-null sets.

Lemma 4.4. The unique σ-invariant infinite subgroup of D∞ is D∞ itself. More- Z over, D∞ is dense in K .

Proof. Let Γ be a σ-invariant infinite subgroup of D∞. Since g0 = a idKZ + bσ thenΓisg0-invariant: for every d ∈ Γ, g0(d)=ad + bσ(d) ∈ Γ, thus g0(Γ) ⊆ Γ. Fix n ≥ 0. As Dn is finite, there exists d ∈ Γ with d/∈ Dn.Letm be the smallest integer such that d ∈ Dm, which is clearly greater than n. The point m−n−1 g0 d belongs to Γ and Dn+1, but not to Dn. Hence

Γ ∩ (Dn+1 \ Dn) = ∅ . (4.12)

Now, we prove by induction that Γ contains Dn for every n ≥ 1, thus Γ = D∞. By (4.12), Γ contains a non-zero element of D1, and since D1 is isomorphic to Zp, we get Γ ⊃ D1. Assume that Γ ⊃ Dn for some n ≥ 1. Take d ∈ Γ ∩ (Dn+1 \ Dn) (it is possible   n by (4.12)). The group Γ spanned by Dn and d satisfies |Γ | > |Dn| = p and is a n+1  n+1 subgroup of Dn+1, thus its order divides |Dn+1| = p . Therefore, |Γ | = p ,  Γ = Dn+1, and we get that Γ ⊃ Dn+1. Z Finally, for x ∈ K , n>0andj ∈ Z, there exists a unique d ∈ Dn with Z d[j, j + n)=x[j, j + n), which implies that D∞ is dense in K .

4.3. The conditional measures µn,x. For n>0, let µn,x be a regular represen- tation of the conditional probability E(·|Bn)(x). This means that the Borel map Z Z x → µn,x from K to the set of probability measures on K is Bn-measurable, and Z that for every Borel subset A of K , E 1A |Bn (x)=µn,x(A) µ–a.e. As the σ-algebra Bn consists of Td-invariant sets for d ∈ Dn, it follows that, for µ-almost all x, µn,x = µn,x+d for every d ∈ Dn,andµn,x is concentrated on x + Dn = {x + d : d ∈ Dn}. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1435

Moreover, as σ commutes with g and preserves µ, σµn,x = µn,σx µ–a.e. For n ≥ 1 we write ζn,x = T−xµn,x. It is a probability measure concentrated on Dn and, for d ∈ Dn and µ-almost all x,

ζn,x+d = T−dζn,x and σζn,x = ζn,σx. (4.13)

(p−1)pn (p−1)pn −(p−1)pn For d ∈ Dn we have σ d = d,thusσ ζn,x = ζn,x ◦ σ = ζn,x (because ζn,x is concentrated on Dn), and from (4.13) we get that

(p−1)pn the map x → ζn,x is invariant by σ . (4.14)

For n>0andd ∈ Dn we write  Z En,d = {x ∈ K : ζn,x({d}) > 0},En = En,d .

d∈Dn\Dn−1

(p−1)pn By (4.14) the set En,d is invariant by σ , and by (4.13) the set En is invariant by σ. We write η(x)=ζ1,x({0})=µ1,x{x}. This is invariant by σ,and n−1 E1 = {x : η(x) < 1}. From the definitions it follows immediately that η(g (x)) = −n+1 µn,x(x+Dn−1)=ζn,x(Dn−1). Therefore En = {x : ζn,x(Dn−1) < 1} = g (E1). This function η is closely related to the entropy: fix A ∈P and consider x ∈ A. For 0 = d ∈ D1 we have d0 =0thus x + d/∈ A. It follows that µ1,x(A)= µ1,x({x})=η(x). We get  for any A ∈P and for any x ∈ A, η(x)=E 1A |B1)(x) , and it follows by Lemma 4.3 that 

hµ(g)= − log(η(x)) dµ(x) . KZ

As hµ(g) > 0, the set E1 where η(x) < 1 has a positive measure. As this set is invariant under σ, by the ergodicity of µ for (g,σ), we get that almost every x −n+1 belongs to En = g (E1) for infinitely many values of n. We have proved,

Lemma 4.5. Assume that µ is ergodic for the action (g,σ) and hµ(g) > 0.For µ–almost every x ∈ KZ and for infinitely many values of n>0 there exists d ∈ Dn \ Dn−1 such that x ∈ En,d.   ∈ ≥ Lemma 4.6. For d Dn, n 0, the measure Td 1En,d µ is absolutely continuous with respect to µ.

Z Proof. We want to prove that, for a Borel subset A of K and d ∈ Dn,

µ(A)=0=⇒ µ(T−dA ∩ En,d)=0.  As µ(A)= KZ µn,x(A) dµ(x), µn,x(A)=0forµ–almost every x. In particular, for µ–almost every x in T−dA,0=µn,x(A) ≥ µn,x({x + d}), because x + d ∈ A,thus x/∈ En,d, and the claim is proved.   c It can be checked that the measure Td 1En,d µ is singular with respect to µ. 1436 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

4.4. Proof of Theorem 4.1. Let µ be as in Theorem 4.1. Let ν be an ergodic component of µ for σp−1. This probability measure is invariant and ergodic for this transformation, and it is absolutely continuous with respect to µ. Claim. gp−1ν = σk(p−1)ν = ν. Proof of the claim. Each ergodic component δ of µ for σp−1 is equal to σjν for some j ∈{0, ..., p − 2}. Since gν is a probability measure invariant and ergodic for σp−1, and absolutely continuous with respect to gµ = µ, then it is one of these ergodic components, and thus equal to σkν for some k ∈{0, ..., p−2}. This finishes the proof.

For e ∈ D1 we write φ(x)=ζ1,x({e}). By (4.14), the function φ is invariant by σp−1, therefore, for each σp−1 ergodic component ν of µ, this function is equal ν- p−1 almost everywhere to some constant cν , or equivalently g ν-almost everywhere. p−1 This means that φ(g (x)) = cν = φ(x)forν-almost every x.Asitistruefor every ergodic component, φ gp−1(x) = φ(x)forµ-almost every x.

Claim. ζ1,x is invariant by Td for every d ∈ D1 and then it is the uniform measure on D1. ∈  Proof of the claim. Let d D1, d = 0, by Lemma 4.6, the measure Td(1E1,d µ)is absolutely continuous with respect to µ, then, p−1 p−1 φ(x + d)=φ(g (x + d)) = φ(g (x)) = φ(x)forµ-almost all x ∈ E1,d . Therefore,

ζ1,x({e})=ζ1,x+d({e})=ζ1,x({d + e})forµ–almost all x ∈ E1,d .

Since e ∈ D1 is arbitrary, for almost every x ∈ E1,d the measure ζ1,x is invariant by Td. Finally, d is a non-zero element of D1 and |D1| = p, a prime number, thus d spans D1. We conclude ζ1,x is the uniform measure on D1.

Therefore, for 0 = d ∈ D1, the measure µ1,x is equidistributed on x + D1 for µ- almost every x ∈ E1,d andthisistrueforµ-almost every x ∈ E1. Since hµ(g) > 0, E1 is invariant by σ and of positive measure, thus of measure 1, and µ1,x is the uniform measure almost everywhere. It follows that µ([α]0)=1/p for every α ∈ K, and that the partition P is independent of B1. Finally, from Lemma 4.3 and the Z Z fact that λK is the unique maximal measure we conclude that µ = λK . 4.5. Proof of Theorem 4.2. We begin with a simple remark about the third hypothesis of Theorem 4.2. It means that almost every ergodic component δ of µ for g,isergodicforg(p−1)p. n Remark 4.7. For every n ≥ 1, the σ-algebra of σ(p−1)p -invariant sets coincides with the σ-algebra of σ-invariant sets. Proof. It suffices to show that almost every ergodic component δ of µ for g is ergodic n for g(p−1)p for every n ≥ 1. The proof goes by induction. Let n ≥ 2 and assume that this property holds for n − 1, and it does not hold for n. There exist a function h of modulus one on KZ and a complex λ = 1 with n λ(p−1)p = 1, such that h(g(x)) = λh(x)forδ-almost every x.Wegethp(g(x)) = n−1 n λphp(x)andhp(g(p−1)p (x)) = λ(p−1)p hp(x)=hp(x). By ergodicity of δ for n−1 g(p−1)p the function hp is constant and λp = 1. It follows that h is invariant under g(p−1)p, thus constant. A contradiction follows. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1437

Z  We turn to the proof of Theorem 4.2. Let χ be a character of K with µ(χ)=  KZ χdµ= 0. We want to show that χ =1. As n →∞,      E χ |Bn → E χ | Bm µ–a.e. m≥1 and this limit is not identically 0 because its integral is equal to µ(χ). Thus we can choose n0 ∈ N such that the set     Z A = x ∈ K : E χ |Bn (x) =0forevery n>n0 satisfies µ(A) > 0. j Lemma 4.8. For every n>n0 and d ∈ Dn,ifµ(A∩En,d) > 0 then χ(σ d)=χ(d) for every j ∈ Z. Proof. For µ almost every x in A we have    0 = E χ |Bn (x)= χdµn,x = χ(x) ζn,x(χ). KZ

(p−1)pn By (4.14), the map x → ζn,x is invariant by σ , thus it is invariant by σ j by the remark above. By (4.13) it implies that, for every j ∈ Z, σ ζn,x = ζn,x for µ–almost every x.

As in the proof of Theorem 4.1, since Td(1En,d µ) is absolutely continuous with respect to µ, for almost every x ∈ En,d this property holds at x + d, that is j σ ζn,x+d = ζn,x+d and from (4.13) we get j j j T−σj dζn,x = T−σj dσ ζn,x = σ T−dζn,x = σ ζn,x+d = ζn,x+d = T−dζn,x .   j Thus Tσj d−d ζn,x = ζn,x and 1 − χ(σ d − d) ζn,x(χ)=0forµ–almost every x ∈ En,d.

We can now conclude the proof of Theorem 4.2. Let j  Γ={d ∈ D∞ : χ(d)=χ(σ d) for every j ∈ Z} and Γ = (idKZ − σ)Γ .

Γ is clearly a subgroup of D∞, invariant by σ. By Lemma 4.5, for infinitely many values of n>n0 there exists d ∈ Dn \ Dn−1 with µ(A ∩ En,d) > 0, and by Lemma 4.8 each of these d’s belong to Γ, thus Γ is infinite. But Γ is a σ-invariant subgroup of D∞, infinite because ker(idKZ − σ) is finite, therefore by Lemma 4.4 equal to D∞. Then χ(d) = 1 for every d ∈ D∞ and by density we conclude χ =1.

5. Convergence of Ces`aro means. In this section we consider cellular automata g which are products of an affine c.a. with a c.a. having an equicontinuous direction. N For a shift invariant measure µ we study conditions under which Mµ (g)converges to the product of the two marginal measures. We apply our results to the classes of right permutative c.a. studied in Section 3.

5.1. Probabilistic background. Denote N∗ = N \{0} and −N∗ = {−i : i ∈ N∗}. Z −N∗ Let µ be a shift invariant probability measure on A .Forw ∈ A let µw be the N conditional measure on A given w,thatisforanym ≥ 0anda0,...,am ∈ A,

µw{x0 = a0,...,xm = am} = µ{x0 = a0,...,xm = am | xi = wi,i<0}. 1438 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

∗ The measure µ can be disintegrated along the partition induced by A−N over AZ by [6], 

µ(·)= µw(·)dµ(w) . A−N∗ The measure µ is said to have complete connections if it satisfies −N∗ ∀a0, ..., am ∈ A, ∀w ∈ A , µw{x0 = a0, ..., xm = am} > 0. In this case define for m ≥ 0,     { }  ∗ µw x0 = a  −N γm =sup  − 1 : a ∈ A, v,w ∈ A ,vi = wi,i∈ [−m, −1] . µv{x0 = a}  ∞ ∞ The measure µ has summable decay if m=0 γm < . This is a uniform continuity condition on µw as a function of w. In particular, Bernoulli measures and Markov measures whose transition matrices have strictly positive entries belong to this class of measures. ∗ Let (Ti : i ∈ N ) be an increasing sequence of non-negative integer random variables. For every finite subset L of N let    ∗  N(L)= {i ∈ N : Ti ∈ L} . ∗ We say that (Ti : i ∈ N )isastationary renewal process with finite mean interre- newal time if

• (Ti − Ti−1 : i ≥ 2) are independent identically distributed with finite expec- tation, they are independent of T1 and P(T2 − T1 > 0) > 0. • For n ∈ N, P(T = n)= 1 P(T − T >n). 1 E(T2−T1) 2 1 The above conditions imply the stationary property: for every finite subset L of N and every a ∈ N the random variables N(L)andN(L + a) have the same distribution. The following fundamental construction of translation invariant probability mea- sures with complete connections and summable decay was stated in [7]. Proposition 5.1. Let µ be a shift invariant probability measure on AZ with com- plete connections and summable decay. ∗ There exists a stationary renewal process (Ti : i ∈ N ) with finite mean interrenewal −N∗ N time, and, for every w ∈ A , there exists a random sequence z =(zi : i ∈ N) ∈ A ∈ N∗ with distribution µw such that (zTi : i ) are i.i.d. uniformly distributed in A and independent of (zi : i ∈ N \{T1,T2, ...}).

We write Pw for the probability, when the random variables (zi : i ∈ N) are given the distribution µw. The probability measure P is the integral of Pw with respect ∗ to w ∈ A−N . Remark 5.2. From the construction of the renewal process in [7] we also get the following properties which are used in the sequel. ∗ 1. Given n, ∈ N ,1≤ k1 < ... < k ≤ n and j1, ..., jl ∈ N,   P ∈{ } w zi = ai,i 1, ..., n ; Tj1 = k1, ..., Tj = k 1   = Pw zi = ai,i∈{1, ..., n}\{k , ..., k}; Tj = k , ..., Tj = k |A| 1 1 1

for all a1, ..., an ∈ A. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1439   ∗ 2. For any n ∈ N and v ∈ A , Pw N({0,...,n− 1}) > 0 ∩ [v]n does not ∗ depend on w ∈ A−N ; 3. There exists a function ρ : N → R decreasing to zero such that P(N(L)= 0) ≤ ρ(|L|), for any finite subset L of N (Lemma 4.1, [7]). Proposition 5.3. Any translation invariant probability measure with complete con- nections and summable decay µ on AZ is mixing of all orders, that is, for all m ≥ 1, ∗ u0, ..., um ∈ A ,

−n1 −(n1+...+nm) lim µ([u0]0 ∩ σ [u1]0 ∩ ... ∩ σ [um]0)=µ([u0]0) · ... · µ([um]0). n1,...,nm→∞ Proof. We only prove the mixing property, the general case can be shown analo- gously. Let u, v ∈ A∗,and|u| = . ∗ Fix w ∈ A−N . We get that for n large enough   Pw [u]0 ∩ [v]n ∩{N({ , ..., n − 1} > 0})    = Pw [u]0) · Pwu N({0, ..., n − − 1}) > 0 ∩ [v]n−    = Pw [u]0) · P N({0, ..., n − − 1}) > 0 ∩ [v]n− by Remark 5.2 (2). Integrating with respect to w we get   P [u]0 ∩ [v]n ∩{N({ , ..., n − 1} > 0})    = P [u]0) · P N({0, ..., n − − 1}) > 0 ∩ [v]n− .

By using the Remark 5.2 (3), we obtain     µ([u]0 ∩ [v]n) − µ([u]0)µ([v]0) ≤ ρ(n − )(1+µ([u]0)) . Finally by taking the limit as n →∞we conclude that µ is mixing.

5.2. Convergence of Ces`aro means for affine c.a. Let (K, +) be a finite Abelian group and g be an affine c.a. Recently in [20] a framework based on har- N monic analysis is provided to study the convergence of Mµ (g) to the uniform Bernoulli measure. In this purpose therein it is introduced the notion of harmon- ically mixing measures and diffusive c.a. A shift invariant probability measure Z µ on K is called harmonically mixing if for all >0, there is some m>0so ∈ Z | | that, for all characters χ K with rank(χ) >mit holds KZ χdµ < ,where rank(χ) is the number of coordinates of dependence of χ.Thec.a. g is diffusive if Z for every non-trivial character χ ∈ K , there is some subset Dχ ⊂ N of density one j so that rank(χ ◦ g )goesto∞ when j tends to ∞ inside Dχ. The convergence to the uniform Bernoulli measure for K = Zp, p a prime number, was shown in [20] and it was generalized to any p ≥ 1 in [21]. In [20] it is also proved that Bernoulli measures and Markov measures whose transition matrices have strictly positive entries are harmonically mixing. Using the main construction of [7] stated in Proposition 5.1 and Remark 5.2 we show that measures with complete connections and summable decay are harmonically mixing. Theorem 5.4. Let (K, +) be a finite Abelian group. Any translation invariant probability measure on KZ with complete connections and summable decay is har- monically mixing. 1440 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

Proof. Let µ be a translation invariant probability measure on KZ with complete connections and summable decay. We keep the notation of Section 5.1. In particular ∗ −N∗ (Ti : i ∈ N ) is the renewal process induced by µ. Fix a past sequence w ∈ K . Z For a finite subset R of Z and x ∈ K we write xR for the sequence (xi : i ∈ R)in KR. Z  Let χ : K → T be a character, that is an element of KZ. There exist a finite  set R ⊂ Z and a sequence (χn : n ∈ Z)inK, with

Z χn =1forn/∈ R ; χn =1for n ∈ R and χ(x)= χn(xn) for every x ∈ K . n∈Z  We have to find an upper bound for | χdµ| depending only on |R|.Asµ is shift invariant we can assume that R ⊂ N.   R For any finite subset R of Z define χR : K → T by χR (y)= r∈R χr(yr). R Observe that χR (K ) is a subgroup of T and that we can identify χ with χR Z because χR(xR)=χ(x)forx ∈ K . R Denote Ξ = χR(K ) and define τ(R)=inf{i ∈ R : N({i})=1}, where inf ∅ = ∞.Wehave  χ(xR)dµw(x)= ξµw(χ(xR)=ξ) KZ ξ∈Ξ   = ξ Pw(χ(zR)=ξ,τ(R)=i)+ ξ Pw χ(zR)=ξ,τ(R)=∞ . i∈R ξ∈Ξ ξ∈Ξ

∈ \{} Ri ∈ Let i R and set Ri = R i ,soΞ=χRi (K )χi(K). For ξ Ξ we define − 1 ∈ ∈ Ri Vi(ξ)=χRi (ξχi(K)). A word y =(yr : r Ri) K belongs to Vi(ξ)ifand   only if there exists a ∈ K such that the word y obtained from y by putting yi = a satisfies χ(y)=ξ. − ∈ Ri ∈ 1 −1 For y K we put ξy = χRi (y) and for ξ Ξ we define Vi(ξ,y)=χi (ξξy ). Since Vi(ξ,y) is a coset of ker(χi), we get |Vi(ξ,y)| = | ker(χi)|. Therefore,

  ξ Pw χ(zR)=ξ,τ(R)=i ξ∈Ξ   = ξ Pw zr = yr,r∈ R; τ(R)=i

ξ∈Ξ y∈Vi(ξ) yi∈Vi(ξ,y) 1   = ξ Pw zr = yr,r∈ Ri; τ(R)=i |A| ξ∈Ξ y∈Vi(ξ) yi∈Vi(ξ,y)

|Vi(ξ,y)|   = ξ Pw zr = yr,r∈ Ri; τ(R)=i |A| ξ∈Ξ y∈Vi(ξ)

| ker(χi)|   = ξ Pw zr = yr,r∈ Ri; τ(R)=i |A| ξ∈Ξ y∈Vi(ξ) | |   ker(χi) P ∈ · = | | w zr = yr,r Ri; τ(R)=i ξ A R y∈K i {ξ∈Ξ:Vi(ξ,y)= ∅} where in the second equality we have used Proposition 5.1. Recall ξy = χRi (y). We have −1 −1 {ξ ∈ Ξ:Vi(ξ,y) = ∅} = {ξ ∈ Ξ:χi (ξξy ) = ∅} = ξyχi(K) . UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1441

Hence ξ = ξy ξ =0.

{ξ∈Ξ:Vi(ξ,y)= ∅} ξ∈χi(K) We conclude   ξ Pw χ(rR)=ξ, τ(R)=i =0. ξ∈Ξ Coming back to the integral we get,           χ(xR)dµw(x) = ξ Pw χ(xR)=ξ, τ(R)=∞ Z K ξ∈ Ξ   ≤|Ξ| Pw τ(R)=∞ ≤|K| ρ(|R|) . ∗   Since this inequality holds for any w ∈ K−N we have  χ(x) dµ(x) ≤|K| ρ(|R|). Since ρ(|R|) → 0as|R|→∞we conclude that µ is harmonically mixing. The following theorem can be shown by using the same arguments as those in the proof of Theorem 5.4. The proof is left to the reader. Theorem 5.5. Let (K, +) be a finite Abelian group, A a finite set, π : A → K a constant to one map and π : AZ → KZ the one-block map induced by π. If µ is a translation invariant probability measure on AZ with complete connec- tions and summable decay then πµ is harmonically mixing. From Theorem 5.4 we get the following result. Theorem 5.6. Let (KZ, g) be an affine c.a. If µ is a translation invariant proba- Z bility measure on K with complete connections and summable decay then Mµ(g) Z exists and is equal to λK . Proof. By Theorem 9 in [21] the c.a. g is diffusive. On the other hand, by The- orem 5.4 µ is harmonically mixing, therefore by Theorem 3 in [21] we get the result. When |K| is a power of a prime number Theorem 5.6 corresponds to Theorem 1.3 in [7]. Besides these results, when K = Zp for p a prime number, Theorem 4.1 and 4.2 can be used to prove that: Z Corollary 5.7. Let (K , g) be an affine c.a. with K = Zp for p a prime number. Let µ be a translation invariant probability measure on KZ. If every weak limit µ˜ N of (Mµ (g):N ∈ N) satisfies the hypotheses of either Theorem 4.1 or 4.2, then Z Mµ(g)=λK . Z 5.3. Convergence of Ces`aro means for affine-translation c.a. Let (B , gB) n be a c.a. Recall that gB is equicontinuous at some point if the family (gB : n ∈ N)of Z transformations of B is equicontinuous at this point and that gB is equicontinuous if it is equicontinuous at every point. Let us give an equivalent definition in terms of words. As usual we only consider ∗ one-sided c.a. of radius 1. A word u = u−k...u0...uk ∈ B is called an equicontinuous marker if for any x, y ∈ [u]−k n n x, y ∈ [u]−k =⇒ gB(x)0 = gB(y)0 for every n ∈ N .

The existence of an equicontinuous marker is equivalent to say that gB is equicon- tinuous at some point. If every point of the system contains an equicontinuous 1442 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ marker centered at the coordinate zero, then the system is equicontinuous. Given u ∈ B∗ we define: Z Bu is the set of y ∈ B such that u appears in y infinitely often to the right and to the left of the 0-th coordinate and for M,M0 ∈ N, M>M0,

{ ∈ ∃ ∈ − − − ∃ ∈ Bu,M,M0 = y Bu : j1 [ M, M0 1], j2 [M0 +1,M],

y[j1 −|u| +1,j1]=u, y[j2,j2 + |u|−1] = u}.

It is straightforward that (Bu,M,M0 : M>M0) is increasing for inclusion and that ∞ 

Bu = Bu,M,M0 . M0=1 M>M0 ∗ Z Let u = u−k...u0...uk ∈ B be an equicontinuous marker for (B , gB). Then, we deduce (see [14]) that for each M,M0 ∈ N, M>M0, there exists an integer ≥ preperiod T = TM0,M 0 and an integer period Γ = ΓM0,M > 0 such that, for any ∈ ∈ N ∈{ − } x Bu,M,M0 ,anym and any j 0, ..., Γ 1 , T +mΓ+j T +j gB (x)[−M0,M0]=gB (x)[−M0,M0] . (5.15) Z If in addition gB is onto, the uniform Bernoulli measure on B is invariant under gB. Therefore, we can use Poincar´e recurrence lemma to get that T = 0 (see [2]). If gB is equicontinuous and onto, then all points of the system are gB-periodics of the same period [14]. Z Let now µ be an ergodic measure for the shift of B and assume gB is onto. In [2] the authors prove that if µ([u]−k) > 0 (and thus, the shift invariant set Bu has full measure µ(Bu) = 1), then the Ces`aro mean Mµ(gB) exists, and for any clopen Z set C ⊆ B there is M0 ∈ N such that − N −1 ΓM 0,M 1 M 1 −n 1 −i ∩ µ(gB)(C) = lim µ(gB (C)) = lim µ(gB C Bu,M0,M ), N→∞N M→∞ ΓM ,M n=0 0 i=0 where the sequence in M defined by the terms in the second limit can be considered to be non-decreasing. Assume now that gB is equicontinuous. Let µ be a shift invariant measure on Z B , not necessarily ergodic. There exists an integer Γ0 > 0 such that for any

M>M0 we can take ΓM0,M =Γ0 and

N −1 Γ 0−1 1 n 1 i Mµ(gB) = lim gBµ = gBµ. (5.16) N→∞ N Γ n=0 0 i=0

For simplicity in the sequel we will omit the subscript M0 from Bu,M,M0 , TM0,M and ΓM0,M , which will be clear from the context. If µ is a probability measure on Z a product space (K × B) denote µK = πK µ and µB = πBµ. Z Theorem 5.8. Let ((K × B) , gK × gB) be a product c.a. with gB onto and let µ˜ Z be a translation and gK × gB-invariant probability measure on (K × B) . Assume that, m (i) µ˜K is totally ergodic (that is, gK ergodic for any m ≥ 1), (ii) there is an equicontinuous marker u for gB such that µ˜B(Bu)=1. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1443

Then, µ˜ =˜µK × µ˜B. ∗ Z Proof. Let w = w−j...w0...wj ∈ B and let A be a Borel set of K .FixM0 >j. We prove thatµ ˜(A × [w]−j)=˜µK (A) · µ˜B([w]−j ). Sinceµ ˜ is gK × gB-invariant, for every N ≥ 1, N −1  × 1 ◦ n · ◦ n cN :=µ ˜(A [w]−j )= 1A gK (x) 1[w]−j gB(y) dµ˜(x, y). N K×B Z n=0 ( )

For M>M0 and N ≥ 1 we define N −1  1 ◦ n · ◦ n · cN,M := 1A gK (x) 1[w]−j gB(y) 1Bu,M (y) dµ˜(x, y). N K×B Z n=0 ( ) Then |cN − cN,M |≤1 − µ˜B(Bu,M ) uniformly in N,andcN,M → cN as M →∞. Let ΓM and TM be the period and preperiod defined above. Since gB is onto, we can assume TM = 0. Now write N = aN,M ΓM + j, where j ∈{0, ..., ΓM − 1}. Then for every n ∈{0, ..., N − 1},wehaven = mΓM + i for some m ∈{0, ..., aN,M } n i and i ∈{0, ..., ΓM − 1}. Hence, from (5.15), gB(y)[−M0,M0]=gB(y)[−M0,M0] n for all such n and y ∈ Bu,M , and, since j

cN,M =    a − Γ M −1 N,M 1 aN,M 1 ◦ mΓM +i ◦ i 1A gK (x) 1[w]−j gB(y)1Bu,M (y)dµ˜(x, y) N aN,M i=0 m=0 (K×B)Z

ΓM  + O . N Observe that ΓM /N → 0, aN,M /N → 1/ΓM and aN,M →∞as N →∞.By ΓM using the ergodicity ofµ ˜K with respect to gK and the fact thatµ ˜B(Bu)=1,we obtain Γ M −1 1 −i lim cN,M =˜µK (A) · µ˜B(Bu,M ∩ gB ([w]−j )) N→∞ ΓM i=0 Γ M −1     1 −i −i =˜µK (A) · µ˜B gB ([w]−j ) − µ˜B gB ([w]−j ) \ Bu,M ΓM i=0 Γ M −1   1 −i =˜µK (A) · µ˜B([w]−j ) − µ˜K (A) · µ˜B gB ([w]−j ) \ Bu,M ΓM i=0 and     lim cN,M − µ˜K (A) · µ˜B([w]−j ) ≤ 1 − µ˜B(Bu,M ) . N→∞ Taking the limit as M →∞, we deduce,

lim lim cN,M =˜µK (A) · µ˜B([w]−j ). M→∞ N→∞

Finally, since cN,M approaches cN uniformly in N we get that

µ˜(A × [w]−j) = lim lim cN,M = lim lim cN,M =˜µK (A) · µ˜B([w]−j ). N→∞ M→∞ M→∞ N→∞ 1444 BERNARD HOST, ALEJANDRO MAASS, SERVET MART´INEZ

Notice that if in the last theorem gB is equicontinuous, then hypothesis (ii) would not be necessary. In fact, every point of the system would be periodic with the same period, so the conclusion follows only by taking the limit of cN as N tends to ∞. Therefore we can state the following result. Z Theorem 5.9. Let ((K × B) , gK × gB) be a product c.a. with gB onto and equicontinuous. Let µ˜ be a translation and gK × gB-invariant probability measure Z on (K × B) .Ifµ˜K is totally ergodic, then, µ˜ =˜µK × µ˜B. Z Z Corollary 5.10. Let ((K × B) , gK × gB) be a product c.a. where (K , gK ) is an affine c.a. and µ a translation invariant probability measure on (K × B)Z with L complete connections and summable decay. Suppose gB ◦ σB is equicontinuous for ∈ Z M × Z ×M some L .Then, µ(gK gB) exists and it is equal to λK µB (gB). N Proof. Letµ ˜ be a weak limit of (Mµ (gK × gB):N ∈ N). L Remark that as gB ◦ σB is equicontinuous, from (5.16) we get that the limit M M measure µB (gB) exists, and necessarily,µ ˜B = µB (gB). M Z Now, by Theorem 5.6, µK = µK (gK )=λK . Since µ is translation invariant, M ◦ L M Z ◦ L we get µK (gK σK )= µK (gK )=λK . As gK σK is left or right permutative, Z L m λK is ergodic with respect to (gK ◦ σK ) for all m ≥ 1 [1]. Hence we conclude the result from Theorem 5.9. The proof of the next Proposition uses a disjointness argument for entropy. Z Z Proposition 5.11. Let ((K × B) , gK × gB) be a product c.a. where (K , gK ) is an affine c.a. and let µ be a translation invariant probability measure on (K × B)Z. N Z Consider µ˜ a weak limit of (Mµ (gK × gB):N ∈ N).Ifµ˜K = λK ,and,forsome L Z L ∈ Z, gB ◦ σB has zero entropy for µ˜B, then µ˜ = λK × µ˜B. L Proof. We assume that L ≥ 0, the case L ≤ 0 is similar. Then, gK ◦ σK is well defined on KN and it is right permutative. Also, sinceµ ˜ is translation invariant, its restriction to KN × BZ, which we will also denoteµ ˜, is well defined and it is × ◦ L translation and (gK gB) σ(K×B) invariant. Now, since µ is translation invariant, M ◦ L M Z N × Z × ◦ L µK (gK σK )= µK (gK )=λK . Therefore, (K B , gK gB σK×B, µ˜)isa N L N Z L joining of (K , gK ◦ σK ,λK ) and (B , gB ◦ σB, µ˜B). L N By [1] the restriction of gK ◦ σK to K is topologically conjugate to the shift of L N N L N n N (K ) by the map φ : K → (K ) ,(φx)n =(gK x)[0,L− 1], and φ projects λK into the uniform Bernoulli measure of (KL)N. N L N Z L Then (K , gK ◦ σK ,λK ) is a Bernoulli system. Since (B , gB ◦ σB, µ˜B)has zero entropy we conclude from [5] that these two systems are disjoint andµ ˜ = Z λK × µ˜B. Z Theorem 5.12. Let ((K × B) , gK × gB) be an affine-translation c.a. and µ a translation invariant probability measure on (K × B)Z with complete connections ∈ N Γ M and summable decay. Let Γ be such that sB = sB (hence, µB (gB) exists and Γ −1 1 i Z is equal to ν = s µB). Then, Mµ(gK × gB)=λ × ν. Γ B K i=0 −1 Proof. In this case gB ◦ σB = sB which is equicontinuous and has zero entropy. Hence the theorem follows either from Corollary 5.10 or Proposition 5.11. From Theorems 3.6, 3.8, 5.5 and 5.12, we deduce the following results on Ces`aro mean convergence for the classes of right permutative c.a. studied in Section 3. UNIFORM BERNOULLI MEASURE IN DYNAMICS OF CELLULAR AUTOMATA 1445

Corollary 5.13. Let (AZ, f) be a right permutative c.a. with ψ-associative local rule and assume that the group A˜r is Abelian. Let µ be a translation invariant probability measure on AZ with complete connections and summable decay, then Mµ(f) exists. Corollary 5.14. Let (AZ, f) be a one-sided right permutative c.a. that is N-scaling, N>1.Letµ be a translation invariant probability measure on AZ with complete connections and summable decay, then Mµ(f) exists. Acknowledgments. The authors acknowledge financial support from Programa Iniciativa Cientifica Milenio P01-005, FONDECYT 1980657 and program ECOS- Conicyt C99E10. The second author thanks the Equipe d’Analyse et de Math´e- matiques Appliqu´ees, Universit´e de Marne la Vall´ee, where part of this work was done. We also thank the referees for many valuable comments.

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