Some properties of cellular automata with equicontinuity points

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1 Some properties of cellular automata with equicotiuity poits Fraçois Blachard, Pierre Tisseur To cite this versio: Fraçois Blachard, Pierre Tisseur. Some properties of cellular automata with equicotiuity poits. Aales de l Istitut Heri Poicaré (B Probabilités et Statistiques, Istitute Heri Poicaré, 2000, 36, pp <hal > HAL Id: hal Submitted o 6 Dec 2003 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Some properties of cellular automata with equicotiuity poits. F. BLANCHARD ad P. TISSEUR ccsd (versio 1 : 6 Dec 2003 Abstract We ivestigate topological ad ergodic properties of cellular automata havig equicotiuity poits. I this class surjectivity o a trasitive SFT implies existece of a dese set of periodic poits. Our mai result is that uder the actio of such a automato ay shift ergodic measure coverges i Cesàro mea, assumig equicotiuity poits have measure 1; the limit measure is described by a formula ad some of the properties of its topological support are give. 1 Itroductio. Compared with their topological dyamics, the ergodic theory of cellular automata is still i its ifacy. Oe of the mai reasos is that few ivariat measures are kow, if ay, for ay give cellular automato. I this article for a family of CA defied by a topological property we give a rather simple costructio of measures that are ivariat both for the shift ad the automato; we also show that whe a automato belogig to this family is oto, there is a dese set of periodic poits for its actio. The property of havig equicotiuity poits was first cosidered for cellular automata by Gilma [6] i relatio with Wolfram s empirical classificatio [14]. Afterwards this property was used by Kůrka [8] as a basic elemet of his topological classificatio of CA accordig to their local behaviour. He distiguishes four classes: equicotiuous automata (E1; those that have equicotiuity poits without beig equicotiuous (E2; ad two other classes of CA, all of them sesitive to iitial coditios. I [9] Lid fids the exact Cesàro mea limit of the images of Beroulli measures by a simple additive cellular automato. His result was geeralized i [12] ad later exteded to Markov measures uder the actio of a larger class of additive cellular automata, all of them without equicotiuity poits, by Ferrari, Maass, Martiez ad Ney [5]. Boyle ad Kitches [1] also proved that periodic poits Istitut de Mathématiques de Lumiy - UPR 9016 du CNRS 1

3 are dese for left or right closig automata; this class has oly a small overlap with E2. These results cocer CA actig o the full set of cofiguratios A Z. Our settig is slightly larger: we cosider a cellular automatof actig o a subshift X A Z, that is, F(X X, ad the equicotiuity poits we cosider are those of the dyamical system (X, F. The article is devised as follows. After the Itroductio, Sectio 2 is devoted to geeral defiitios, i particular that of blockig words, which is essetial for CA havig equicotiuity poits. Sectio 3 cotais the results. The first oe is topological but we prove it with the help of Poicaré s recurrece theorem: assumig that the automato F acts surjectively o a trasitive subshift of fiite type X ad has equicotiuity poits, X cotais a dese set of F periodic poits. Our mai result is purely ergodic: if a blockig word (X, F has positive measure for a shift ergodic measure µ, the the images of µ uder the powers of F coverge i Cesàro mea. The limit µ c is of course F ad σ ivariat; it is give by a formula, which evertheless leaves ope several iterestig questios. The we examie properties of the topological support of µ c, ad give several examples of cellular automata to which our results ca be applied; we fiish with some ope questios. 2 Defiitios ad backgroud. 2.1 Dyamical systems, measures, cellular automata. A topological dyamical system (X, T cosists of a compact metric space X ad a cotiuous self map T. A poit x X is said to be a equicotiuity poit, or to be Lyapuov stable, if for ay ǫ > 0, there exists η > 0 such that if d(x, y η oe has d(t i (x, T i (y ǫ for ay iteger i > 0. Whe all poits of (X, T are equicotiuity poits (X, T is said to be equicotiuous: sice X is compact a equicotiuous system is uiformly equicotiuous. The dyamical systems i this article are all defied o a symbolic space. Let A be a fiite alphabet. Defie A to be the set of all fiite cocateatios of letters of A, called words; the legth of the word u is deoted by u. A laguage L is a subset of A. The set A Z of bi ifiite sequeces o A is edowed with the product topology, associated with the distace d(x, y = 2 i where i = mi{ j x(j y(j}; the shift σ: σ(x = (x i+1 i Z is a homeomorphism o A Z. Give a word u ad a iteger t, the clope set [u] t = {x A Z : x t = u 1...;x t+ u 1 = u u } is called a cylider set. Whe x A Z ad p q are two itegers, put x(p, q = x p... x q. A sequece (S i i N is said to be ultimately periodic if there exist two atural itegers p ad p such that S p +kp+i = S p +i for k, i N. The dyamical system (A Z, σ is called the full shift. A subshift X is a closed shift ivariat subset of A Z. A trasitive subshift S is oe such that for u, v 2

4 L(S there is w L(S such that uwv L(S; it is strogly mixig if for ay larger tha some 0 (u, v oe ca fid a word w of legth with the same property. To ay subshift X there correspods a uique laguage L(X: it is the set of all words that are foud as blocks of coordiates of a poit of X. Give ay subshift X the laguage L(X has two geeral properties: for u L(X, ay sequece of cosecutive letters of u is also i L(X; ad for ay word v i L(X there are letters a ad b i A such that avb L(X. A subshift of fiite type X is defied by forbiddig a fiite family of words E: the L(X is the smallest laguage havig the two properties above ad such that o word u L(X is of the form u = vew with e E. Trasitive subshifts of fiite type have a dese set of periodic poits. Whe a probability measure µ o A Z is shift ivariat, its topological support S(µ is closed ivariat, hece a subshift. O every trasitive subshift of fiite type oe defies a particular measure λ with support X called the Parry measure; the Parry measure of the full shift is the uiform measure. A sequece (µ N of probability measures o a compact set K is said to coverge vaguely to a limit µ if the sequece K fdµ teds to fdµ for ay K cotiuous fuctio f : K. O a subshift X a sequece (µ N of shift ivariat measures coverges vaguely if ad oly if for ay word u L(X the sequece (µ ([u] 0 N coverges. I this article we call cellular automato(ca for short a cotiuous map F : X X defied o a subshift X A Z ad commutig with the shift σ; we also call cellular automatothe dyamical system (X, F. The Curtis Hedlud Lydo theorem [4] states that for every cellular automato(x, F there is a iteger r, called the radius of F, ad a block map f : A 2r+1 L(X A such that oe has F(x i = f(x i r,..., x i,...,x i+r. If X is a trasitive subshift of fiite type, the automato F acts surjectively o X if ad oly if the Parry measure λ is F ivariat [3]. The set W(X, F = lim F i (X is called the limit set of the cellular automato(x, F; of course whe F is surjective W(X, F = X. 2.2 Blockig words ad equicotiuous poits. Defiitio: Let F be a cellular automatowith radius r actig o the subshift X. A word B A 2k+1 is called a blockig word for (X, F if there is a ifiite sequece of words v, v = 2i + 1 r such that for ay x A Z with x( k, k = B oe has F (x( i, i = v for Z. I the defiitio above we do ot assume X to be σ trasitive, but this coditio appears ecessary for most of our proofs. Remark that if B is a blockig word ad x( k, k = B, the F (x(, i does ot deped o x(k, +, ad reversely sice 2i + 1 r. A occurrece of a blockig word i a cofiguratio x completely discoects coordiates to its right ad left for the actio of the automato; hece the ame. 3

5 The two followig results are essetially due to Kůrka [8]. Propositio 2.1 Ay equicotiuity poit of a cellular automato(x, F has a occurrece of a blockig word. Coversely if there exist blockig words, ay poit with ifiitely may occurreces of a blockig word to the left ad right is a equicotiuity poit; if moreover X is trasitive for σ equicotiuity poits are dese i X. Proof: Let x be a equicotiuity poit of (X, F; applyig the defiitio of equicotiuity poits to cellular automata, there is a iteger k such that if d(x, y < 2 k, for ay oe has F x( r, r = F y( r, r, so that B = x( k, k is a blockig word. Coversely let V (X be the set of all poits with ifiitely may occurreces of blockig words to the left ad right; whe X is trasitive V (X is o empty, eve dese. Let x V (X. To ay give ε > 0 oe associates a iteger t such that 2 t < ε. There exist a real umber η ad itegers t < k such that 2 k < η ad the words x( k, t ad x(t, k cotai a occurrece of B each. For every poit y belogig to the cylider set [x( k, k] k oe has F i (x( t, t = F i (y( t, t; sice ε is chose arbitrarily oe cocludes that x is a equicotiuity poit. For a equicotiuous cellular automato, there is a atural iteger k such that all words of L(X with legth 2k + 1 are blockig words; thus Propositio 2.2 The cellular automato(x, F is equicotiuous if ad oly if there are two itegers p ad p such that for x X the sequece (F (x N is ultimately periodic with period p ad preperiod p. 3 Results. 3.1 Dese periodic poits. Propositio 3.1 Let X be a trasitive subshift of fiite type ad suppose that the cellular automato(x, F has a equicotiuity poit. The F is oto if ad oly if it possesses a dese set of periodic poits. Proof: Let F act surjectively o the trasitive subshift of fiite type X ad suppose it has equicotiuity poits. For ay word v L(X we costruct a σ periodic poit ū X such that ū(k, v 1+k = v for some iteger k, which is also F periodic; this establishes the desity of F periodic poits i X. Fix v L(X. By Propositio 2.1 F has a blockig word B L(X; as X is trasitive ad has a dese set of σ periodic poits, there is a word u = Bwvw L(X such that ū X, where ū is the periodic poit costructed o u ad such that a occurrece of u starts at 0. The cylider set C = [ub] 0 cotais ū, ad λ(c > 0 if λ is the Parry measure of X. Sice λ is F ivariat we apply the Poicaré recurrece theorem: there is m > 0 such that λ(c F m C > 0; i particular there are a poit x X ad q = ( ub 1 such that x(0, q = F m (x(0, q = ub. 4

6 But B is a blockig word. All the coordiates of ū coicide with those of x o the segmet [0, q], ad sice there is a occurrece of B at the begiig of this segmet ad oe at the ed, for ay > 0 oe has F x(i, q i = F ū(i, q i, where i < 1 2 B is as i the defiitio of blockig words. We have thus show that F x ad F ū coicide o a segmet of legth q 2i q B = u, which is greater tha or equal to the commo σ period of ū ad F m ū: therefore F m ū = ū. The coverse is straightforward. We have proved this topological result ergodically. There should be a purely combiatorial proof. The followig simple cosequece is kow but seems to be owhere i writte form. Corollary 3.1 A cellular automato(x, F is equicotious ad surjective if ad oly if there is p > 0 such that ay x X is periodic of period p. Proof: By Propositio 2.2, F beig equicotiuous, there is a iteger p such that for ay x A Z the sequece (F p + (x N is periodic with period p; the ay periodic poit has period p. By Propositio 3.1 the set of periodic poits is dese; as i the proof of this propositio oe idetifies the block of coordiates F (x( k, k with the correspodig block of a periodic poit with period p for every, ad oe reaches the coclusio by lettig k go to ifiity. 3.2 Cesàro mea covergece of measures. We start with a easy result o equicotiuous CA. If µ is a measure o A Z ad M a Borel set, deote by µ (M = 1 µ ( F i (M its Cesàro mea of order with respect to F. Propositio 3.2 Let (X, F be a equicotious cellular automatowith period p ad preperiod p, ad let µ be a shift ergodic measure with support X. The µ coverges vaguely i Cesàro mea to the measure µ c = 1 p 1 µ F (i+p. p Proof: It suffices to show that for u L(X the sequece (µ ([u] 0 N coverges to the right limit. By Propositio 2.2 there are p ad p that for ay poit x, ay pair of itegers ad i oe has F p +i+p (x = F p +i (x. Thus if u L(X ad > p oe has µ ([u] 0 = 1 p 1 µ ( F i ([u] µ ( F i ([u] 0. i=p 5

7 The first term teds to 0; usig periodicity oe gets µ ([u] 0 = 1 p p 1 ( µ F (i+p ([u] k. Defiitio: Let F be a cellular automatoactig o the subshift X. A probability measure µ o X is said to be equicotiuous for (X, F if the set of equicotiuity poits of (X, F has measure 1. Lemma 3.1 Let (X, F be a cellular automatoad µ be a measure o X, ergodic for σ. The the two followig properties are equivalet: (1 there exists a blockig word B such that µ([b] 0 > 0; (2 µ is equicotiuous for (X, F. Proof: (1 (2: sice µ is σ ergodic ad µ([b] 0 > 0, almost every poit cotais ifiitely may occurreces of B to the left ad right, so it is a equicotiuity poit by Propositio 2.1. (2 (1: agai by Propositio 2.1, every equicotiuity poit cotais a occurrece of a blockig word; sice the family of blockig words is at most coutable, there is a blockig word B such that µ([b] 0 > 0. Defiitio: Give a word B, which we shall always suppose to be a blockig word for (X, F, let R(k, m, B be the set of all poits of X havig at least oe occurrece of B betwee the coordiates m k ad k, ad aother oe betwee the coordiates k ad m+k. Wheever there is o ambiguity o B we deote it by R(k, m. Theorem 3.1 Let (X, F be a cellular automatoad µ be a shift ergodic, equicotiuous measure o X. The µ coverges vaguely i Cesàro mea uder F. The limit µ c is F ad σ ivariat, ad for every word u L(X oe has µ c ([u] 0 = lim m p(k,m 1 p(k, m ( µ R(k, m F (i+ p(k,m ([u] 0. Proof: It is sufficiet to show that for ay word w L(X, w = 2k + 1, the sequece (µ ([w] k N coverges. By Lemma 3.1 there is a blockig word B for (X, F with µ([b] 0 > 0. The limit of the icreasig sequece of sets (R(k, m m N is the set of all poits havig at least two occurreces of B, oe to the left of k ad the other to the right of k. Sice µ is σ ergodic the set V (B of poits havig ifiitely may occurreces 6

8 of B to the right ad left has measure 1. Thus lim m µ(r(k, m = 1 ad for ay iteger k, ay word w L(X A 2k+1 oe has µ ([w] k = lim µ(f i ([w] k R(k, m. m We prove that µ ([w] k by usig the twofold covergece of the double sequece ( µ(f i ([u] k R(k, m m, N. Ideed sice the iterval [0, 1] i which µ takes its values is compact, if ( µ(f i ([u] k R(k, m m, N coverges simply as ad uiformly i as m, the two limits commute ad oe obtais the desired covergece: lim lim m µ(f i ([u] k R(k, m = lim µ ([w] k. Let us show first that the sequece coverges as for fixed m. Let x ad y belog to R(k, m: by the defiitio of R(k, m there are blockig words to the left of their k th coordiate ad to the right of their k th coordiate, so that if y is such that u = y( m k, m + k = x( m k, m + k, the for ay iteger i oe has F i (x( k, k = F i (y( k, k. I particular if ū is the periodic poit with period 2m + 2k + 1 such that ū( m k, m + k = u, the sequece (F (ū N = (F (x( k, k N is ultimately periodic. Let p(x, k, m be its period ad p (x, k, m be its preperiod. Deote by p(k, m the least commo multiple of the values of p(x, k, m for x R(k, m ad by p (k, m the correspodig iteger for p (x, k, m. Let w be a word of legth 2(k + m + 1 such that [w] k m R(k, m. For ay x [w] k m ad i, j N oe has F p (k,m+j+ip(k,m (x( k, k = F p (k,m+j (x( k, k; thus for ay word u of legth 2k + 1 oe has R(k, m F (ip(k,m+j+p (k,m ([u] k = R(k, m F (p (k,m+j ([u] k. A argumet similar to that of the proof of Propositio 3.2 shows that lim µ ( F i ([u] k R(k, m = p(k,m 1 ( µ F (i+ p(k,m ([u] k R(k, m p(k, m which is what we wat. Now let us prove that the sequece ( µ(r(k, m F i ([u] k m N (1 coverges uiformly i whe m. 7

9 We already kow that for ay real umber ε > 0, for fixed k there is a iteger m 0 such that wheever m m 0 oe has µ(r(k, m 1 ε. Thus for ay iteger i ad m m 0 oe has µ((x R(k, m F i ([u] k ε, hece µ(f i ([u] k µ(r(k, m F i ([u] k ε. For ay iteger if m m 0 oe has µ(r(k, m F i ([u] k 1 µ(f i ([u] k ε = ε. Sice the two covergece coditios hold, we have proved that the two followig limits exist ad are the same: = lim µ c ([u] k = lim lim m lim m Equality (1 permits to coclude that µ c ([u] k = lim m µ ( R(k, m F i ([u] k µ ( R(k, m F i ([u] k = lim µ ([u] k. p(k,m 1 p(k, m ( µ R(k, m F (i+ p(k,m ([u] k. The ext corollary geeralizes Theorem 3.1 to a larger class of cellular automata. Its proof is straightforward. Corollary 3.2 Let (X, F, µ ad k Z be such that µ is σ ergodic ad equicotiuous for (X, F σ k ; the the coclusios of Theorem 3.1 hold. 3.3 The topological support of the measure µ c. Remark first that the topological support S(µ c is cotaied i W(F. Recall that R(k, m is the set of poits with at least oe occurrece of B betwee the coordiates k m ad m ad aother oe betwee the coordiates m ad m + k. We start with a techical lemma. Lemma 3.2 Let µ be a σ ergodic measure, equicotiuous for (X, F, ad let B be a blockig word such that µ([b] 0 > 0. For ay word u i A 2k+1 the sequece W m (u = p(k,m 1 µ(r(k, m F (i+p (k,m ([u] k p(k, m 8

10 is o decreasig. Proof: Let m 1 < m 2 be two atural itegers. The two sequeces (U i = ( µ(r(k, m2 F i ([u] k ad (V i = (µ ( R(k, m 1 F i ([u] k are ultimately periodic with preperiod ad period p (k, m 2 ad p(k, m 2 for the former, p (k, m 1 ad p(k, m 1 for the latter. Deote by p the greatest of the two itegers p (k, m 2 ad p (k, m 1, ad put p = p(k, m 2 p(k, m 1. The sequeces (V i ad (U i are ultimately periodic with preperiod p ad period p so that W m2 (u = 1 p p 1 µ (R(k, m 2 F (i+p ([u] k. Sice R(k, m 1 R(k, m 2 oe has W m2 (u 1 p p 1 µ (R(k, m 1 F (i+p ([u] k = p(k,m 1 1 µ (R(k, m 1 F (i+p (k,m 1 ([u] k = W m1 (u. p(k, m 1 Propositio 3.3 Suppose X is a trasitive subshift of fiite type, F is oto ad µ is equicotiuous for (X, F ad σ ergodic; the S(µ c S(µ. Proof: Choose a blockig word B such that µ([b] 0 > 0, ad y S(µ; sice µ([y( k, k] k > 0 for ay iteger k ad lim m µ(r(k, m = 1, there is a iteger m 0 such that wheever m m 0 oe has µ([y( k, k] k R(k, m > 0. For m N choose a poit x m i [y( k, k] k R(k, m. By Propositio 3.1 the set of F periodic poits is dese so there exists oe, y m, i the cylider set [x m ( k m, k + m] k m. The sequece (F (x m ( k, k does ot deped o the coordiates to the left of k m ad to the right of k + m; it is idetical to the periodic sequece F (y m ( k, k; i particular p (k, m = 0. Fix k ad m: the sequece of sets (F i ([y( k, k] k R(k, m i 0 is periodic; sice µ([y( k, k] k R(k, m > 0 oe has, i the otatio of Lemma 3.2, W m (y( k, k = p(k,m 1 µ(r(k, m F i ([y( k, k] k p(k, m 1 p(k, m µ((r(k, m [y( k, k] k > 0. By Propositio 3.1 ad Lemma 3.2 the sequece W m (y( k, k is o decreasig ad teds to µ c ([y( k, k] k so that µ c ([y( k, k] k > 0 ad fially y S(µ c. 9

11 I particular whe F is oto ad S(µ = A Z oe has S(µ c = A Z. Let (X, F be a cellular automatohavig equicotiuity poits. For ay blockig word B, call E(F, B the set of all poits y X such that for ay atural iteger k, there is aother atural iteger m 0 such that m m 0 ad i p (k, m oe has [y( k, k] k F i (R(k, m. Propositio 3.4 The set E(F, B is a subshift; oe has F(E(F, B E(F, B, thus E(F, B is cotaied i the limit set W(F; if F is surjective, E(F, B = A Z. If X is trasitive (resp. strogly mixig for σ, the E(F, B does ot deped o the choice of the word B ad ca be deoted by E(F; it is also trasitive (resp. strogly mixig for σ. Proof: Sice the defiitio of E(F, B depeds oly o the blocks of coordiates of its poits, E(F is a subshift; the fact that F(E(F, B E(F, B derives from the same remark. Let X be trasitive, B ad B be two arbitrary blockig words; if [y( k, k] k F i (R(k, m, B, the [y( k, k] k F i (R(k, m, B F i (R(k, m, B provided m is big eough, which implies E(F, B E(F, B. Trasitivity or strog mixig result from the fact that two words i L(E(F ca occur i the image uder F of oe poit cotaiig oe or several blockig words betwee their respective occurreces. Propositio 3.5 If µ is equicotiuous for (X, F, S(µ c E(F. If moreover S(µ = X, the S(µ c = E(F. Proof: Fix B ad assume that y E(F, so there is a iteger k such that for ay iteger m ad for ay iteger i p (k, m oe has [y( k, k] k F i (R(k, m. Thus p(k,m 1 µ(r(k, m F (i+p (k,m ([y( k, k] k = 0. p(k, m Applyig Propositio 3.1 oe obtais µ c ([y( k, k] k = 0 ad y S(µ c. Hece S(µ c E(F. If S(µ = X it is sufficiet to prove that S(µ c E(F. If x E(F, for ay atural iteger k there is m N such that the uio of cylider sets G = F p (k,m ([x( k, k] k R(k, m is ot empty. Hece µ(g > 0. By Lemma 3.2 ad Propositio 3.1 the sequece idexed by m p(k,m 1 µ(r(k, m F (i+p (k,m ([y( k, k] k p(k, m is o decreasig ad sice µ(g > 0, Propositio 3.1 implies that µ c ([x( k, k] k > 0 ad fially x S(µ c. Examples I [10], [8], [2] oe ca fid examples of cellular automata that are surjective o 10

12 A Z ad have equicotiuity poits. Oe ca therefore apply Propositio 3.1, ad also Theorem 3.1 ad Propositio 3.3 if oe assumes that µ is for istace a Beroulli measure B(p 1, p 2, p 3 differet from the uiform measure. I [13] the automato called Gliders ad walls has equicotiuity poits without beig oto. Here is aother example: the cellular automatof : {0, 1, 2} Z {0, 1, 2} Z with radius 1 is defied by the local map f such that f(x 1, x 0, 2 = x 0, f(x 1, 2, x 1 = 2 ad whe x 1 {0, 1} ad x 0 2 the f(x 1, x 0, x 1 = x 0 +x 1 mod 2. F is oto; it has equicotiuity poits sice 2 is a blockig word. Let µ be a Beroulli measure with parameters {p, q, r} o A Z ; by cosiderig the cylider sets [2012] 0 ad [2112] 0 oe easily checks that the sequece µ F does ot coverge vaguely. By Theorem 3.1 it coverges i Cesàro mea but, still cosiderig the same two cylider sets, the limit caot be the Beroulli measure with parameters { p+q 2, p+q 2, r}. Questios Whe is µ c ergodic for F? Whe F is oto ad µ is the uiform measure, which is F ivariat i this case, µ c = µ is ever F ergodic (this would imply trasitivity of F, which i its tur implies sesitivity. Whe is it σ ergodic? Are there coditios for µ c to be the uiform measure, or at least Beroulli or Markov? Ackowledgemets We wat to thak A. Maass for may suggestios; we are also grateful to the referee, who sigalled various shortcomigs. Part of this work was doe by the secod author at Uiversidad de Chile i Satiago, thaks to the fiacial support of Fodap Modelamieto Estocastico ad Ecos Coicyt. Refereces [1] M. Boyle, B. Kitches, Periodic poits i cellular automata, prétirage (1999. À paraître, Idagatioes Math. [2] F. Blachard, A. Maass, Dyamical behaviour of Cove s aperiodic cellular automata, Theoret. Computer Sci. 163 (1996, [3] E. Cove, M. Paul, Edomorphisms of irreducible shifts of fiite type, Math. Sys. Th. 8 (1974, [4] M. Deker, C. Grilleberger, K. Sigmud, Ergodic theory o compact spaces. Lecture Notes i Math. 527, Spriger, Berli (1975. [5] P. Ferrari, A. Maass, S. Martíez, P. Ney, Cesàro mea distributio of group automata startig from measures with summable decay, preprit (1999. [6] R. H. Gilma, Classes of liear automata, Ergodic Th. & Dyam. Sys. 7 (1987,

13 [7] G. A. Hedlud, Edomorphisms ad automorphisms of the shift dyamical system, Math. Systems Th. 3 (1969, [8] P. Kůrka, Laguages, equicotiuity ad attractors i liear cellular automata, Ergod. Th. & Dyam. Sys. 217 (1997, [9] D. A. Lid, Applicatios of ergodic theory ad sofic systems to cellular automata, Physica 10D (1984, [10] D. A. Lid, Etropies of automorphisms of a topological Markov shift, Proc. Amer.Math.Soc. 99 (1987, [11] A. Maass, O sofic limit sets of cellular automata, Ergodic Th. Dyam. & Sys. 15 (1995, [12] A. Maass, S. Martíez, O Cesàro limit distributio of a classof permutative cellular automata, J. Statist. Physics 90 (1998, [13] J. Milor, O the etropy geometry of cellular automata, Complex Systems 2 (1988, [14] S. Wolfram, Theory ad applicatios of cellular automata, World Scietific,