on ergodic linear cellular automata over zm

On Ergodic Linear Cellular Automata over Zm

Gianpiero Cat taneo 1, Enrico Formenti 1, Giovanni Manzini 2,3, and Luciano Margara 4

1 Dipartimento di Scienze dell'Informazione, Universit~ di Milano, Milano, Italy. 2 Dipartimento di Scienze e Tecnologie Avanzate, Universit~ di Torino, Via

Cavour 84, 15100 Alessandria, Italy. 3 Istituto di Matematica Computazionale, Via S. Maria, 46, 56126 Pisa, Italy. 4 Dipartimento Scienze dell'Informazione, Universit~ di Bologna, Piazza Porta

S. Donato 5, 40127 Bologna, Italy.

Abs t rac t . We study the ergodic behavior of linear cellular automata over Z,~. The main contribution of this paper is an easy-to-check necessary and sufficient condition for a linear cellular automaton over Zm to be ergodic. We prove that, for general cellular automata, ergodicity is equivalent to topological chaos (transitivity and sensitivity to initial conditions). Finally we prove that linear CA over Zp with p prime have dense periodic orbits.

1 I n t r o d u c t i o n

Cellular Automata (CA) are dynamical systems consisting of a regular lattice of variables which can take a finite number of discrete values. The state of the CA, specified by the values of the variables at a given time, evolves in synchronous discrete time steps according to a given local ntle. CA can display a rich and complex temporal evolution whose exact determination is in general very hard, if not impossible. In particular, many properties of the temporal evolution of general CA are undecidable [4, 5, 12]. Despite their simplicity that allows a detailed algebraic analysis, linear CA over Zm exhibit many of the complex features of general CA. Several important properties of linear CA have been studied during the last few years (see for example [2, 7, 8, 11]) and in some cases exact results have been carried out. As an example, in [11] the authors present criteria for surjectivity and injectivity of the global transition map of linear CA. The qualitative behavior of CA is a main subject in CA theory. Quoting from [11]: "Criteria are desired for determining when the sequence o/ transations of a state- configuration o/ a cellular automata takes a certain type o/ dynamical behavior." In this paper we study the dynamical behavior of linear CA over Zm in the framework of ergodic theory. Ergodic theory has been recently applied to CA in a number of works. Some preliminary results can be found in [10, 13, 14, 16]. The main contribution of this paper is the solution to the two following open problems (Theorem 9).

(1) How to decide whether the global transition map of a given linear CA over Zm is ergodic.

Reischuk, Morvan (Eds.): STACS'97 Proceedings, LNCS 1200 (~) Springer-Verlag Berlin Heidelberg 1997

428 G. Cattaneo, E. Formenti, G. Manzini, L. Margara

(2) How to find ergodic linear CA over Zm.

The solution of problem (1) generalizes a result presented in [13] (Corollary 2, page 406), while the solution of problem (2) answers a question raised in [14] (Question 2, page 605). We also establish a connection between ergodic theory and topological chaos in the case of general CA.

Although a universally accepted definition of chaos does not exist, two properties are widely accepted as important features of chaotic behavior: topological transitivity and sensitivity to initial conditions. Sensitivity is recognized as a central notion in chaos theory because it captures the feature that in chaotic systems small errors in experimental readings lead to large scale divergence, i.e., the system is unpredictable. Transitivity guarantees that the system cannot be decomposed into two or more subsystems which do not interact under iterations of the map.

We prove (Theorem 6) that a CA is ergodic if and only if it is topologically transitive. Since in [3] one of the author proved that topologically transitive CA are sensitive to initial conditions, we conclude that , for CA, ergodicity is equivalent to topological chaos. In Theorem 7 we take advantage of the compactness of the space of the configurations on which CA are defined for proving that topologically transitive CA are surjective.

Another widely accepted feature of chaotic behavior is given by denseness of periodic orbits (see for example [6]). Here (Theorem 10) we prove that D- dimensional linear CA over Zp with p prime have dense periodic orbits, which is a generalization of a result in [7] (Theorem 4).

The rest of this paper is organized as follows. In Section 2 we give basic definitions and notations. In Section 3 we list our results. Section 4 contains the proofs of the theorems stated in Section 3. Section 5 contains some indications for further works.

2 B a s i c d e f i n i t i o n s

Let Z be the set of integers. Let m and D be positive integers. Let ,4 = {0, 1 , . . . , m - 1) be a finite alphabet of cardinality m. Let f , f : ,48 -+ A, s _ 1, be any map. We say that s is the size of the domain of f , or simply the size of f .

Z D A D-dimensional CA based on a local rule f of size s is a pair (A , F) , where

A z~ = z " A)

is the space of configurations and F, F : A zD -+ .4 zD, is the global transition map defined as follows. For every c E -4 zD and for every v E Z o

[F(c)](v) = f (c(v + r i b ( l ) ) , . . . , c(v + rib(s))), (1)

where nb : {1 , . . . , s} --+ Z D, is the neighborhood structure map.

On Ergodic Linear Cellular Automata over Zm 429

Example 1. We describe a 2-dimensional binary CA whose evolution is governed by a local rule f which computes the sum modulo 2 of its 4 input values selected by the neighborhood map rib. In this example nb selects the cells to the north, west, east, and south of the cell we are considering. Let D = 2, A = (0,1}, and s = 4. Let nb be defined by rib(l) = (0,1), rib(2) = <-1,0>, nb(3) = (1, 0), rib(4) = ( 0 , -1 ) . The local rule f is defined by f ( x l , x2, x3, x4) = (Xl + x2 + x3 + x4) mod 2. The global transition map F is defined by

[F(c)](i,j) = (c( i , j + 1) +c( i - 1, j ) +c( i + 1, j ) + c ( i , j - 1)) rood 2.

In the case of 1-dimensional CA, we use the following simplified notation. Let f , f : A 2~+1 -> .4, be any map. A 1-dimensional CA based on the local rule f is a pair (A z, F) , where A z is the space of configurations and F, F : A z -> A z, is defined by

[F(c)](i) = f (c ( i - r ) , . . . , c(i + r)), c E A z, i E Z. (2)

We say that r is the radius of f . Note that , even if f must depend on at least one between x - r and xr, in general f does not depend on all the 2r + 1 variables X - - r , � 9 x r -

Throughout the paper, F(c) will denote the result of the application of the map F to the configuration c, c(v) will denote the value of the entry with coordinates v of the configuration c, and v~ will denote the i-th component of the vector v. We recursively define Fn(c) by Fn(c) = F(Fn- I (c ) ) , where F ~ = c.

We now give the definition of linear local rule over Zm-

D e f i n i t i o n 1. Let A = {0, 1 , . . . , m - 1}. A map f , f : A s -+ A, is linear over Zm if and only if it can be written as

where Ai E Z.

From now on, we say that a CA defined over a finite alphabet Jt -- { 0 , . . . , m - 1} is linear over Zm if the local rule on which it is based is linear over Zm. Note that for a linear D-dimensional CA, equation (1) becomes

8

[F(c)](v) = Z A~c(v q- rib(i)) rood m. (3)

For 1-dimensional linear CA of radius r, the notation can be further simplified by writing

f ( X - r , . . . , X r ) = ~ aixi m o d m , i -~ - - r


so that equation (2) becomes

[F(c)](i) = ~ ajc(i+j) mod m, c E A z, i E Z. j=-r

It is often useful to introduce a distance over the space of the configurations. Let A: .4 • .4 -+ (0, 1} be such that

A(i, j)= {O' i f / = j , 1, i f i ~ j .

Given a, b E A zD the Tychonoff distance d(a, b) is defined by

b(v)) d(a,b) : Z 2max(v) ' (4 )

v ~ Z D

where max(v) is the maximum of the absolute value of the components of v. It is easy to verify that d is a metric on .4 zD and that the metric topology induced by d coincides with the product topology induced by the discrete topology of ,4. With this topology, .4 z~ is a compact and totally disconnected space and F is a (uniformly) continuous map.

We now recall the definition of three properties, namely topological transitivity, sensitivity to initial conditions, and denseness of periodic orbits which are widely accepted as important features of chaotic behavior for general discrete time dynamical systems. Here, we assume that the space of configurations X is equipped with a distance d and that the map F is continuous on X according to the topology induced by d. For CA the Tychonoff distance satisfies this property.

Defini t ion 2. A dynamical system (X, F) is topologically transitive if and only if for all nonempty open subsets U and V of X there exists a natural number n such that Fn(U) N V ~ 0.

Intuitively, a topologically transitive map has points which eventually move under iteration from one arbitrarily small neighborhood to any other. As a consequence, the dynamical system cannot be decomposed into two disjoint open sets which are invariant under the map.

Defini t ion 3. A dynamical system (X, F) is sensitive to initial conditions if and only if there exists a 5 > 0 such that for any x E X and for any neighborhood N(x) of x, there is a point y E N(x) and a natural number n, such that d(Fn(x), F~(y) ) > 5. 5 is called the sensitivity constant.

If a map possesses sensitive dependence on initial conditions, then, for all prac- tical purposes, the dynamics of the map defies numerical approximation. Small errors in computation which are introduced by round-off may become magnified upon iteration. The results of numerical computation of an orbit, no matter how accurate, may be completely different from the real orbit.

In [3] it has been proved that, for CA, topological transitivity implies sensitivity. Thus, for CA, the notion of transitivity becomes central to chaos theory.


D e f i n i t i o n 4 . Let P(F) = {x E X[ 3n E N : Fn(x) = x} be the set of the periodic points of F . A dynamical system (X, F ) has dense periodic orbits if and only if P(F) is a dense subset of X , i.e., for any x E X and e > 0, there exists y E P(F) such that d(x,y) < e.

Denseness of periodic orbits is often referred to as the element of regularity a chaotic dynamical system must exhibit. The popular book by Devaney [6] isolates three components as being the essential features of chaos: transitivity, sensitivity to initial conditions and denseness of periodic orbits. Finally, we recall the definition of ergodic map.

D e f i n i t i o n 5. Let (X, ~ , #) be a probabilistic space. Let F, F : X ~ X, be a measurable map which preserves #, i.e., for every subset E E 5 r we have # (E) = # ( F -1 (E)). Then F is ergodic with respect to # if and only if for every E E ~ r

(E = F-I(E)) :=~ (#(E) = 0 or # (E) -- 1).

In the following we will adopt the Haar probability measure over A zD which is defined as the product measure induced by the uniform probability distribution over .4.

3 S t a t e m e n t o f n e w r e s u l t s

Surjective ~- Topologically r Ergodic ~ gcd(m, ),2,. . . , ),8) = 1 Transitive

Sensitive to initial F surjective and conditions Vn E N I - F n surjective

Fig. 1. Implications among properties of global transition maps associated with D-dimensional linear CA over Z,~. Solid arrows represent results proved in this paper

In this section we state the main results of this paper. The same results are summarized in the diagram of Fig. 1.

Our first result shows that transitive CA are ergodic with respect to the normalized Haar measure. Since the converse of this fact is already known (see Theorem 13), we establish that for CA the concepts of transit ivity and ergodicity are indeed equivalent.


T h e o r e m 6. Topologically transitive cellular automata with respect to the met- tic topology induced by the Tychonoff distance are ergodic with respect to the normalized Haar measure. []

Our next result establishes that transitive CA (hence also ergodic CA in view of Theorem 13) are surjective.

T h e o r e m 7. Topologically transitive CA with respect to the metric topology induced by the Tychonoff distance are surjective. []

The following two theorems show that for linear CA there exists a simple characterization of ergodic maps based on the coefficients of the local rule. Note that Theorem 8 is indeed a special case of Theorem 9. We decided to state ex- plicitly both theorems since the former has a much simpler proof and gives a clearer picture of the role of the coefficients.

T h e o r e m 8. Let F denote the global transition map of a linear 1-dimensional CA over Zm with local rule f ( X - r , . . . , X r ) = ~']~=--r aixi mod m. The global transition map F is ergodic if and only if

gcd(m, a - r , . . . , a - l , a l , . �9 at) = 1,

where gcd denotes the greatest common divisor. []

T h e o r e m 9. Let F denote the global transition map of a linear D-dimensional CA over Zm defined by

8

[F(c)](v) = ~ Aie(v + nb(i)) . i = 1

Assume nb(1) = O, that is, ~1 is the coefficient associated to the null displace- ment. The global transition map F is ergodic if and only if

gcd(m, ,k2,/k3,..., As) = 1.

[]

Theorem 9 generalizes a result given in [13] where the authors proved that 1- dimensional linear CA over Z2 based on a local rule different from the identity map are ergodic. In addition, Theorem 9 allows us to answer the following question raised in [14]. Are all surjective 1-dimensional CA defined over {0, 1} z (with the exception of the identity and the inversion map) ergodic? Consider the sub- class of linear CA over Zm. If m is prime, the answer is yes, otherwise one can easily construct a 1-dimensional non trivial linear CA over Zm which is not ergodic. In the case of general CA, the answer is No. In fact, let F be the global transition map of the (global) injective binary 1-dimensional CA defined in [1]. Since F = F -1 we have that F 2 is the identity map and then F is not ergodic.

Our final result shows that linear CA over an alphabet of prime cardinality have dense periodic orbits.


T h e o r e m 10. Let F denote the global transition map of a linear D-dimensional CA over Zp with p prime. Then the set P ( F ) of periodic orbits of F is dense in ,4 z ~ . []

This result is a generalization of Theorem 4 of [7] where it has been proved that 1-dimensional linear CA over Z2 have dense periodic points.

4 P r o o f of the main theorems

We now prove the results stated in Section 3. In our proofs we make use of the following known facts about CA.

W h e o r e m l l . [11] Let .4 = { 0 , 1 , . . . , m - 1 } . Let f , f : .4s _+ .4, be a linear map defined by

The CA based on the local rule f is surjective if and only if gcd(m, ,~1, . . . , ,~n) = 1. []

T h e o r e m 12. [13] Let G be a compact abelian group with normalized Hear measure tt, and let 0 be a continuous surjective endomorphism of G. Then, the following two statements are equivalent.

(i) 0 is ergodic. (ii) For every n > 1, I - O n is surjective.

[]

T h e o r e m 13. [16] Ergodic CA with respect to the normalized Hear measure are topologically transitive with respect to the metric topology induced by the Tychonoff distance. []

P r o o f o f T h e o r e m 6. Let Vk, k > 1, be the set of all the D-dimensional vectors vi such that Ilvilloo < k. One can easily verify that the cardinality of Vk is n k D -= (2k + 1) D. Let e l , . . . ,an~D be n k D arbitrarily chosen elements of .4. We define a D-dimensional cylinder C y l ( a l , . . . , ankD) C_ .4 zD as follows:

C y l ( a l , . . . ,ankD) : { a : e ( v i ) : ai, vi E Vk, 1 < i < nkD } .

Assume that there exists a D-dimensional CA (A zD , F ) which is topologically transitive but not ergodic. Then, there exists E C A zD such that

0 < # ( E ) < l a n d F - I ( E ) = E .

One can easily verify that since #(E) > 0, for h > 0 there exists a cylinder Cyl (a l , . . . , anhD) entirely contained in E. In an analogous way, since #(E) <


1, there exists another cylinder Cyl(bl,.. . ,bnjD), j >_ 0 entirely contained in

A zD \ E. Since Cyl(al , . . . ,anhD) C_ E and F - I ( E ) = E we have tha t for every n E Z

Fn(Cyl(a l , . . . ,anh, ) ) C_ Fn(E) C_ E.

Since Cyl(bl , . . . , bnjD) C A zD \ E we have tha t

F n ( C y l ( a l , . . . , anhD ) ) [ ~ C y l ( b l , . . . , bnjD ) : ~,

i.e., F is not transit ive which is a contradiction. []

P r o o f o f T h e o r e m 7 ( s k e t c h ) . Assume by contradiction tha t F is not surjective, tha t is, there exists x E X such that , for all y E X , F(y) ~ x. Since F is transitive, we can find a sequence Yn such tha t d(F(yn), x) < 1/n. Since (X, d) is a compact metric space, we can find a subsequence y,~ which converges to z C X. We have

d(F(z), x) < d(F(z), F(yn~)) + d(F(y~,), x).

Since the right-hand te rm goes to zero as n~ -+ c~ we have F(z) = x as claimed. []

Let F denote the global transit ion map of a linear CA over Zm. In order to prove Theorems 8 and 9, we need to study the s tructure of F n for n _> 1. It is well known tha t F n is a linear CA; in the following we derive a simple relationship between the coefficients of the local maps f and f(n) associated with F and F ~ respectively. We first consider the simpler case of 1-dimensional CA. Let f denote a linear local map of radius r

f ( x - - r , . . . , X-- l , XO, X l , . . . , Xr) : ~ a i x i mod m . i -~-r

By substitution we get

f (2) ( X _ 2 r , . . . , Z2r) : ai a j x i + j mod m, i-~--r j

More in general, we can prove by induction tha t f (") is a local map of radius nr such that

f(~) (x -~ r , . . . , Xnr) = nT

bix~ rood m, bi = i-=--nr - r < i l ..... in_<r

iI -~,,,-~in =i

aQ ai2 . . . ai~ .


The fundamental observation is tha t the coefficient bi is equal to the sum of all n- term products ai~ ai2 "" �9 a i , such that i l + . . . + in = i.

This formula can be easily generalized to dimensions greater than one. As- suming F is given by (3), we have

8(n) [ F n ( c ) ] ( v ) = E b i c ( v + nb ' ( i ) ) ,

i = 0

where

bi = E )~i1~i2 . . . ~i~. l ~ i l , . . . , in ~ s

nb(i 1 )+ ' " -+r ib( in )=nbt(~)

In other words, b~ is equal to the sum of all n- term products )% ~i2 "'" 3~i. such that the sum of the displacements rib(i l ) + . . - + rib(in) is equal to rib'(i). There is no simple way to determine the size s ( n ) of the local rule f(n). However, it is clear tha t s ( n ) < c~, that is, there is only a finite number of displacements rib' with a nonzero coefficient.

P r o o f o f T h e o r e m 8. Let q = g c d ( m , a _ r , . . . , a - l , a l , . . . ,a t ) . We first prove that q = 1 implies F is ergodic. By Theorem 11 we know that F is surjective. We prove that F is ergodic by showing that I - F n is surjective for n > 1 and using Theorem 12. For a fixed n let b -nr , . �9 b - l , b0, b l , - . . , bar denote the coefficients of F n. By Theorem 11 we know that I - F n is surjective if and only if

g c d ( m , - b - n r , . . . , - b - l , 1 - b0 , -b l , �9 �9 �9 - b a r ) = 1. (5)

For i = 1 , . . . , r , let

t i = gcd(m, a - r , �9 �9 �9 a - i , a i , . . . , at) , (6)

If tr = 1 the result is proven since

n ~b gcd(m, a - r , at) = 1 ~ gcd(m, a_r , a r ) = gcd(m, b - n r , bar) = 1

which implies (5). Assume now tr > 1, and let P denote the set of prime factors of tr. To prove (5) we show that for all p E P there exists j ~ 0 such that p)(bj . Given p, let i denote the greatest integer such that pJ(ti but p l t i + l . Such an integer must exist since by hypothesis tl = 1. We have that p divides all coefficients a - r , . . . , a - i - l , ai+l , . �9 �9 at , but p does not divide at least one between a - i and a~. Assume for simplicity that pXai . We prove that , as a consequence, pXbni. We know that bni is equal to the sum of the n- term products ajl aj2 . . . ajn with j l + "'" + j n = n i . Among the terms of the sum there is a~ which cannot be divided by p. We prove that p~bni by showing that every other term in the sum is a multiple of p. In fact, if Jl + "'" + j n = n i and 3k : jk ~ i at least one index j h must be greater than i and the product a j la j2 . . . aj~ can be divided by p. This proves that (5) holds which implies tha t F is ergodic.


Now we prove that gcd(m, a - r , . . . , a - l , a l , . . . , at) = 1 is a necessary condition for ergodicity. Assume by contradiction that F is ergodic and

q = gcd(m, a - r , . . . , a - l , a l , . . . , aT) > 1.

Since ergodicity implies surjectivity (Theorems 13 and 7), by Theorem 11 we know that gcd(m, a - r , . . . , a - l , no, a l , . . . , at) = 1. Hence there exists a prime p such that Plq and pXao. We establish our result by proving that I - F p-1 is not surjective which is impossible by Theorem 12.

Let n = p - 1, and let b - n T , . . . , but denote the coefficients of F n = F p-1. We want to show that

Pl gcd(m, - b - n r , . . . , b - l , 1 - bo, bl,. �9 �9 -bnr ) . (7)

Since Plq, P divides the coefficients b - n r , . . . , b - l , b l , . . . , br. Consider now the coefficient b0. We know tha t bo is equal to the sum of all n- te rm products a j l a j 2 . . . a j ~ with j l + "'" + Jn -=- 0. The fundamental observation is tha t if some of the j~'s is ~ 0 the product ajl . - - a j . is a multiple of p. Hence

( 1 - b 0 ) m o d p = ( 1 - a ~ ) m o d p = ( 1 - a p - 1 ) m o d p - 0 .

where the last equality follows from Fermat Theorem and the fact tha t by hypothesis pXao. Since pl(1 - b0), p satisfies (7) as claimed.

This completes the proof. []

P r o o f o f T h e o r e m 9 ( ske t ch ) . The proof is a generalization of the previous one. Let bo, b l , . . . , bs(~) denote the coefficients of F n. The value b~ is associated to the displacementnb~(i) , and we assume tha t rib'(0) = 0. It suffices to show that

gcd(m, A2, . . . , )%) = 1 ~ gcd(m, 1 - bo, - b l , . . . , -bs(~)) = 1, (8)

since this implies that I - F ~ is surjective and we can apply Theorem 12. We reorder the coefficients A2, . . . , A~ according to the euclidean norm of the displacements rib(i), ( that is, i < j ~ Ilnb(i)ll < Ilnb(j)ll), and for i = 2 , . . . , s we define

ti = gcd(m, A~, Ai+ l , . . . , As).

As in the proof of Theorem 8, we have that t~ > i implies (8). In addition, for each prime p such that plt~ we can find a coefficient bj such that pXbj. Hence (8) holds and the map F is ergodic.

The implication gcd(m, A2, . . . , As) = 1 ==~ F ergodic is obtained by repeating verbatim the proof given for the 1-dimensional case. []

P r o o f o f T h e o r e m 10 ( ske t ch ) . In order to prove that F has dense periodic points it is sufficient to prove that every cylindric subset of .A zD contains at least one periodic point. Let f and nb be the local rule and the neighborhood structure on which F is based.


We first prove this thearem in the 1-dimensional case. Let a l , . . . , a2k+l be 2k+ 1 arbitrarily chosen elements of .4. Since F is topologically transitive, there exists a positive integer n such that

F n ( C y l ( a l , . . . , a2k+l)) N C y l ( a l , . . . , a2k+l) • ~.

Let b e F '~ (Cy l (a l , . . . , a2k+l))A C y l ( a l , . . . , a2k+l). Let ei ~ A z be the configuration defined by

1 if j = i and e~(j) -- 0 otherwise,

Let

M = m a x { i : [Fn(b+ ei)](k) # [F'~(b)](k)},

m = rain { i : [Fn(b + ei)]( -k) # [F~'(b)](-k)).

When i is out of the interval [ -k , k], b(i) may be different from [F"(5)](i). Let bt E A z be any configuration such that bl(i) = b(i), i = m , m + 1 , . . . ,M . It takes a little effort to verify that , since the cardinality of`4 is prime, it is possible to find suitable values for b l ( m - 1) and bl (M + 1) such that bl(i) --- [F'(bt)](i) inside the interval [ - k - 1 , k+ 1]. By repeating the same process we may construct a Cauchy sequence of configurations bh, h E N, such that bh(i) = [Fn(bh)](i) for - k - h < i < k + h. Since A z is complete bh converges to a configuration c which is also a fixed point for F n. Since a l , . . . , a2k+l can be arbitrarily chosen, we conclude that F has dense periodic orbits. For simplicity of notation we prove this theorem in the 2-dimensional case by using a general decomposition technique which can be easily applied to the D- dimensional case.

A 2-dimensional additive CA can be seen as a 1-dimensional CA defined over the infinite alphabet B = `4z. The local rule g of the new CA is the sum of a finite number of surjective mappings (1-dimensional CA) gi, gi : B --+ B, suitably defined according to f and rib. The local rule f of Example 1 can be written as . f(xl , x2, x3) = gt ( x l )+ g2 (x2)+ g3(x3), where gi, g~ : {0, 1} z ~ {0, 1} z, 1 < i < 3, are defined as follows. For any a E {0,1} z, gl(a) = a, g2(a) = b, and g3(a) = a, where b(i) = a ( i - 1) + a(i + l), i E Z .

Since each gi is surjective (being the local rule of an additive 1-dimensional CA defined on an alphabet of prime cardinality) we may prove the thesis by applying the proof technique used in the 1-dimensional case (with some little adjustments) to the new CA based on the local rule g.

A D-dimensional additive CA can be seen as a 1-dimensional CA definite over the alphabet • = A zD-1 . The local rule g of the new D-dimensional CA is the sum of a finite number of surjective mappings from/3 to B. The thesis follows from the surjectivity of (D - 1)-dimensional additive CA defined on alphabets of prime cardinality. We wish to emphasize that in the D-dimensional case, the surjectivity of the ( D - 1)-dimensional mappings which form the local rule is of fundamental importance and allows us to construct the Cauchy sequence of D-dimensional configurations whose limit is the fixed point for F n. []


5 Further works

We are currently investigating the topological behavior of linear cellular au- t o m a t a over Zm with the aim of providing easy-to-check characterizations of the following dynamical properties: expansivity, strongly transitivity, topological mixing, sensitivity to initial conditions, and denseness of periodic orbits (for composite m).

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on ergodic linear cellular automata over zm

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