Evolution of probability measures by cellular automata on algebraic topological Markov chains

In this paper we review some recent results on the evolution of probability measures under cellular automata acting on a fullshift. In particular we discuss the crucial role of the attractiveness of maximal measures. We enlarge the context of the results of a previous study of topological Markov chains that are Abelian groups; the shift map is an automorphism of this group. This is carried out by studying the dynamics of Markov measures by a particular additive cellular automata. Many of these topics were within the focus of Francisco Varela’s mathematical interests. Key terms: Topological Markov chains, cellular automata, maximal measure, entropy, additive cellular automata. Corresponding Author: Servet Martínez e-mail: smartine@dim.uchile.cl 1991 Mathematics Subject Classification . Primary: 54H20; Secondary: 37B20. 1 1 4 MAASS & MARTINEZ Biol Res 36, 2003, 113-118 a maximal excited state in equilibrium. It is easy to verify that the above iteration admits several limit behaviors. To illustrate, consider in the fullshift {0,1} Z the mod 2 CA given by (Fx)i = xi + xi+1, where addition is modulo 2. Then: ( Fx)i = xi + xi+2, and ∑ − = + − = 1 2


INTRODUCTION
Let A be a finite set of symbols and M = (M a,b : a,b ∈ A) be a {0,1}-valued matrix indexed by the alphabet A .We consider the set of allowed two-sided sequences of symbols in A or configurations, X M = {x = (x 1 ) i∈ z ∩ i ∈ Z, M xixi + 1 =1} The matrix M specifies a set of constraints of the configurations, M a,b =0 means that the path from a to b is forbidden, and on the contrary, when M a,b =1 it is allowed.In the event that all entries of M are equal to one, the set Χ Μ , is called fullshift and denoted by A Z .
A cellular automaton (CA) defined on Χ Μ is a map F : X M → X M such that for x ∈ X M and i∈ Z, (Fx) i = f (x i-r ...x i+r ) where r∈ N is the radius and f : A 2r+1 → A is a given local rule (see Figure 1).When the local rule does not depend on left coordinates χ i-r ,...,χ i-i , we say it is a one-sided CA.In this case the local rule is defined by f : A r+1 → A and (Fx) i = f (x i-r ...x i+r ) .
Among CA we distinguish the shift map σ M : X M → X M such that (σ M x) i = x i+1 .This map codes time in the sense that x i represents the state of system at time i.Observe that σ M is one-sided, and it is induced by the local rule with r = 1, f : A 2 ˝ A, f (x 0 x 1 ) = x 1 .. It is evident that When X M is a fullshift, we use the notation σ A instead of σ M .
Cellular automata have served as models of systems which present self organization, and the study of their dynamics has been the object of several papers over the last three decades.One approach to the study of such behavior is to consider a 'natural' probability measure µ of full support in Χ Μ (which represents a realistic amount of initial conditions) and the sequence of iterated measures (F n µ : n ∈ N), where (F n µ ) (B) = µ (F n (B)) for every measurable set B. The behavior of such a sequence has been studied in particular contexts in recent years: (Lind, 1984;Mass and Martínez, 1998;Ferrari et al, 2000;Pivato andYassawir, 2001, 2002).One of the primary objectives of this study is to link the limit behavior of this sequence with probability measures of maximal entropy, in other words, the way in which the CA goes to a maximal excited state in equilibrium.
It is easy to verify that the above iteration admits several limit behaviors.To illustrate, consider in the fullshift {0,1} Z the mod 2 CA given by (Fx) i = x i + x i+1 , where addition is modulo 2. Then: (F 2n x) i = x i + x i+2 , and , so for any Bernoulli measure µ the limits of (F n µ : n ∈N) are different along these subsequences.
In order to study the equilibrium, a natural point of view is to consider the convergence of the temporal mean for a class of translation invariant probability measures µ on Χ Μ .In fact, any limit for this sequence is also invariant for the CA.The main existing results concern fullshifts.This study began with Lind's pioneer (1984) work for the mod 2 CA and initial Bernoulli measures, where it is proven that the temporal mean converges to the uniform Bernoulli measure, which is the maximal entropy measure for F and σ {0,1} .The same result was shown for larger classes of group cellular automata and more general initial probability measures in (Mass and Martínez, 1998;Ferrari et al, 2000).Ferrari et al (2000) used regeneration theory of stochastic processes with finite state space.Lind's point of view is that of harmonic analysis: the cellular automaton is viewed as an endomorphism of the product group and the uniform Bernoulli measure is its Haar measure.This approach was followed in Pivato andYassawir (2001 and2002), where the authors proved the same kind of results for the class of probability measures they called harmonically mixing, and the family of additive cellular automata verifying a diffusive condition.The previous results motivated us to study those sensitive cellular automata which can be decomposed as the product of an additive cellular automaton with one having an equi-continuous direction (Host et al, 2002).
Let us provide some notation and background.

ALPHABET, WORDS, SEQUENCES
A finite set of symbols A is called alphabet, and by A we mean the set of finite sequences or words w = w 0 ...w n-1 with letters in A and w denotes the length of w ∈ A + .Given x= (x j ) j∈AE ∈ A Z and i ≤ j in Z, or x = x 0 ...x n ∈ Α + and i ≤ j in {0,...,n}, we denote by x[i,j] = x i ...x j the finite word in x between coordinates i and j.
Let X M ={x=(x j ) j∈AE ∈A Z :∩j∈AE,M xj, xj+1 =1} for M a {0,1}-valued matrix indexed by A This is the topological Markov chain induced by M .
For any n∈ N we define the set of words of The set of all words or language of Χ Μ is L(X M ) = ∪ n∈ι L n (X M ).Given w∈L (X M ) and i∈ N, the cylinder set starting in coordinate i with word w is Cylinder sets are a fundamental base for the topology of Χ Μ , which is metrizable and compact.
Let F : X M → X M be a CA.We can apply its local rule f (of radius r) to words.Given w = w 0 ...
This allows us to define by induction the N-

CLASSES OF CA
A CA F : X M → X M is said to be right permutative if for every w n ∈ L r2 (X M ) the map f (w,.) defined by f (w,.)(a) = f (wa), when w n wa∈ L r2+1 (X M ), is one-to-one.Analogously we define left permutative and bipermutative CA.
Assume (Χ Μ ,+) is a group.We say that F is additive if for every x,y ∈ XL M F (x+y) = F (x) + F (y).By Kitchens' results (1987), any topological Markov chain that is a group and such that the shift σ M is an automorphism, is equivalent to one induced by a group (L 1 (X M ),+) in such a way that x+y = (x i + y i ) i∈Ζ .Therefore the local rule of an additive CA F satisfies f (w+w) = f (w)+f (w') for w,w' ∈ L r2+1 (X M ).A particular class of CA additive is that whose local rules ƒ can be written f (w 0 w 1 ) = w 0 +w 1 for every w 0 w 1 ∈ L 2 (X M ), which is equivalent to F (x) = x + σ M(x) These CA are right permutative.
Another class of right permutative CA are additive-translation CA.They are defined on a product space by ) a oneto-one local rule.This class of CA appear naturally as the representation of right permutative CA whose local rules verify some algebraic conditions in [Host et al, 2002].

Probabilistic background
Relevant shift invariant probability measures are Bernoulli and Markov measures.Let us describe them in A z.It is enough to give its value in the set of cylinder sets.
Let π = (π a :a ∈ A) be a probability vector; the Bernoulli measure µ π is given by µ π ( a 0 ,...,a n Under this measure all coordinates are independent, so there is no memory.
Markov measures have memory one, and they are defined by a stochastic matrix P and its stationary probability vector π and we denote it by µ π,P It is given by where Let us introduce a relevant class of infinite memory probability measures.Denote N * = N\{0} and -N * = {-i:i ∈ N * }.Let µ be a shift invariant probability measure on A Z For w ∈ A -Í* let µ w be the conditional measure on A N given w that is for any m ≥ 0 and a 0 ,...,a m ∈ A, The measure µ has summable decay if . This is a uniform continuity condition on µ w as a function of W Entropy One of the fundamental invariants in ergodic theory is entropy, both topological and measure-theoretical.Here we introduce them in the context of cellular automata.Let F : X M → X M be a CA and µ a F-invariant probability measure defined on measurable sets of Χ Μ ,, that is, µ(C) = µ(F -1 (C)) for every cylinder set C Let α n = {[w] -n : w ∈ L 2n+1 (X M )} be the partition of centered cylinders of length 2n + 1 in Χ Μ .The dynamics of this partition α n under the transformation F is given by the sequence of partitions (α n (F,N)N ∈ Í), where The entropy of F with respect to µ is There is a topological counterpart.The topological entropy of F is The variational principle links both notions of entropy in the following way: is said to be of maximal entropy.These measures carry the maximal amount of information of the system.
Maximal entropy measures for topological Markov chains Χ Μ such that M m > 0 for some m ∈ Í can be explicitly described in terms of algebraic characteristics of the matrix.Let u,v be left and right eigenvectors of M associated to the eigenvalue λ of maximum modulus, that is, uM = λu, Mv = λv.By Perron-Frobenius theorem λ > 0 and also u > 0, v > 0. Assume

∈ = λ
The probability vector π M = ((π M ) a : a∈A) given by (π M ) a = u a v a is the stationary vector for this matrix, that is, π M P M = π M .Then the associated stationary Markov measure

Temporal mean of iteration of measures by CA
Let F: X M → X M be a CA and µ be a shift invariant probability measure on Χ Μ .As usual, F n µ denotes the measure given by (F n µ (B) = µ (F -n µ) for B a measurable set.We will discuss the convergence in distribution of the temporal mean measure

CONVERGENCE OF TEMPORAL MEANS TO MAXIMAL ENTROPY MEASURE
Let F: X M → X M be a CA.In this section we will supply conditions ensuring the convergence of to a maximal entropy measure for σ M .This is shown for different kinds of translation invariant probability measures µ.In particular we will consider the case of stationary Markov probability measures on Χ Μ .

Convergence of temporal means for additive CA
Let (K, +) be a finite Abelian group and F: K Z → K Z be an additive CA.In a recent article, Pivato and Yassawir (2002) used a harmonic analysis framework to study the convergence of to the uniform Bernoulli measure of K 2 .For this purpose they introduce the notion of harmonically mixing measures.A shift invariant probability measure µ on K 2 is called harmonically mixing if for all ε > =, there is some m > 0 so that for every character χ ∈ KZ with rank (χ) > m it holds , where rank (χ) is the number of coordinates of dependence of χ.The statement (2.1) ) ( lim was shown to hold for any additive CA and any harmonically mixing measure in Pivato and Yassawir (2001) (a previous proof for K=Z p ,p a prime number, was shown in Pivato and Yassawir (2002)).The proof is based on the fact that for every non-trivial character χ ∈ KZ , there is some subset of density one, so that rank (χº F j ) goes to ∞ when j tends to ∞ inside D X .Moreover, Pivato and Yassawir (2002) also proved that Bernoulli measures and Markov measures whose transition matrices have strictly positive entries are harmonically mixing.
Our approach to (2.1) was probabilistic.In Ferrari et al (2000) we show that (2.1) holds for any measure with complete connections and summable decay memory for groups where #K is a power of a prime number.In addition, in Host et al (2002) we extend the result to any group by showing that measures with complete connections and summable decay are harmonically mixing.
In addition to these results, in an ergodic context, when #K = p a prime number, if all weak limits of are ergodic for the shift map and have positive entropy for F, we prove in Host et al ( 2002) that (2.1) holds.

Convergence of temporal means for additive-translation CA
In Host et al (2002) we consider more general classes of CA, those which admit a product representation as an additive-translation CA.They contain one-sided right permutative CA of radius 1 acting on a fullshift such that the local rules f : A 2 → A are associative, that is, there is a one-to-one map ψ: A → A such that ψ:[f(f(a,b),c)] =f(a,f(a,c)); or satisfy a Nscaling condition, that is, where f N is the N-th power of f.Therefore temporal means can be computed for classes of CA that a priori do not have any algebraic structure.A main point in this decomposition is that in one coordinate we have a group structure and in the other a translation CA for which temporal means are periodic.
Let F x G : (K x A) AE → (K x A) Z be an additive-translation CA and let µ be a shift invariant probability measure on (K x A) Z .Consider a weak limit . This measure has two natural projections in each coordinate, π K μ and π A μ , but a priori it is not true that μ = π K μ × π A μ .We prove in Host et al, ( 2002) that if π K μ = λ K Z and for some L ∈ AE the CA G oσ A L has zero entropy for π A μ , then μ = λ K Z × π A μ .A corollary of the previous result and the properties described in previous subsection is: for any shift invariant probability measure µ with complete connections and summable decay it holds has zero entropy because it is a CA of radius O.

Convergence of temporal means for a particular additive CA
In this section we announce a new result on convergence of temporal means to the maximal entropy measure, in the context of additive CA acting on an Abelian group (X M ,+).We will not give a proof of the result but we will illustrate it on a particular example constructed on (X M ,+) given in Kitchens (1987).
Since F is right permutative it can be shown that for any allowed word a 0 ,...,a m-1 ∈ L m (X M ) and n ≥ 1 there are two words b 0 ...b m , c 0 ...c m