Limit Measures for A ne Cellular Automata - Semantic Scholar

Limit Measures for Ane Cellular Automata Marcus Pivato and Reem Yassawiy October 2, 2000 Abstract

Let M be a monoid (e.g. N, Z, or ZD), and A a nite alphabet. A cellular automaton (CA) is a topological dynamical system F : AM ?! AM that commutes with all shift maps. If A is a group, then so is AM; a linear cellular automaton (LCA) is then a CA that is also a group endomorphism. We develop sucient conditions on  and F so that the sequence fFN g1N =1 weak*-converges to the Haar measure on AM, in Cesaro average. As an application, we show: if A = Z=p (p prime), F is any \nontrivial" LCA on A(ZD) , and  belongs to a broad class of measures (including all Bernoulli measures (for D  1) and \fast-mixing" Markov measures (when D = 1)), then FN  weak*-converges to Haar measure in Cesaro average.

 Department y Department

of Mathematics, University of Toronto, email: [email protected] of Mathematics, Trent University, email: [email protected]

1

1 Introduction Cellular automata (CA) exhibit a broad range of statistical behaviour. At one extreme are those automata manifesting strong self-organizing tendencies, evolving towards xed points, limit cycles, or other dynamical attractors. At the other extreme are automata which seem to obey a metaphorical \Second Law of Thermodynamics", evolving over time to a state of \maximum disorder". Computer simulations of such automata progress from initial order to seemingly random complexity. This capacity of some automata to \generate randomness" was investigated early on by Wolfram ([20], reprinted in [21]). However, a precise de nition of \randomness" is necessary to rigorously characterize the asymptotic statistical properties of these systems. One of the simplest classes of CA are linear cellular automata where we treat the alphabet, A, of possible states at each site as a group; the set of all possible combined states of a site and its immediate neighbours then has a product group structure, and the local transformation of the CA is then required to be a homomorphism from the latter group to the former. If the neighbourhood of a site is called U, the local transformation is a group homomorphism: f : AU ?! A

The appellation \linear" comes from the special case when the alphabet of local states is the cyclic group A = Z=p, for some prime p; Since Z=p is also a nite eld, this map is actually a linear map from the (Z=p)-vector space (Z=p)U into Z=p; it generally takes the form:





f [a] =

X

u2U

fu au

(1)

where a = au ju2U is an element of AU, and where ffu ; u 2 Ug is a set of coecients in Z=p. If the cellular automaton lives on a D-dimensional lattice ZD, then U  D Z can be thought of as some \neighbourhood around zero". The cellular automaton induced by the local transformation rule f is the map: D D F : A(Z ) ?! A(Z )

a j  in A(ZD), we de ne F(a) for any con guration a = ` `2ZD a0 jwhere,, where P f a . Concisely, 0 we can write this: a = ` `2ZD ` u2U u (u+`) X F [a]0 =

u2U

=

fu au

In general, the term \linear" is not really appropriate for these maps , since most groups are not elds. If the group A is abelian, then the terminology is still at least metaphorically accurate, since we can write the group operation of A as \addition", treat A as a Z-module, and the formula (1) is still valid, only 2

now with the coecients fu ranging over Z. However if A is nonabelian, the term \linear" is rather misleading; some authors have taken to calling these automata group automata for this reason. Intuitively, linear automata should do a good job of \generating randomness". For example, these automata are permutative [7] (suggesting that any local perturbation of a con guration is aorded the maximum opportunity of spreading outwards) and have been shown [6], [2] in many cases to be chaotic in the sense of Devaney [3]. Also, linear cellular automata are automorphisms of compact groups; when endowed with the invariant Haar measure, group automorphisms are one of the most \chaotic" classes of ergodic dynamical systems [14], [15]. It is desirable to validate this intuition, by precisely characterizing the asymptotic statistics of linear cellular automata. The rst investigation of this question was by Lind [10], who looked at the one-dimensional cellular automaton F : AZ ?! AZ, where F(a)0 = a(?1) + a1 , and A = Z=2  = f0; 1g. Using methods from harmonic analysis1 Lind showed that, starting from any nontrivial Bernoulli  (or \product") probability measure, , on AZ, the sequence of measures FN  ; N 2 N Cesaro -converge to the Haar (or \uniform") measure |that is: N 1X FN  = Haar : lim N !1 N n=1

(Here, FN  :=   F?N , and Haar is Haar measure.) also showed that it was vain to hope for the sequence of measures FNLind  ; N 2 N to itself converge to Haar measure; there is an extremely sparse but in nite subset J = f2n ; n 2 N g so that, for all j 2 J, the measure FJ  is quite far from Haar2. Ferrari et al. [13] proved weak* Cesaro convergence to Haar measure for any automaton of the form F(a)0 = k0 a0 + k1 a1 acting on the half-line AN, where A = Z=q, q = pn for some prime p, and k0 and k1 are relatively prime to p. Here, the measure  is the conditional probability distribution induced on AN by a probability measure  on AZ and a half-con guration a 2 A?(1:::0) ; the measure  must exhibit a sort of \rapid mixing" property such that a certain sequence of \correlation" coecients decays fast enough to be summable. For example, Bernoulli measures satisfy this condition. In [12], Ferrari et al. extend this result to the case when  is a Markov measure on AN . These results are summarized in [1], where the authors also prove that a Markov measure on AZ will Cesaro -converge to Haar, when A = (Z=2)  (Z=2), and f : A2 ?! A is de ned f [(x1 ; y1 ); (x2 ; y2 )] = (y1 ; x1 + y2 ). 1 Actually, in retrospect, Lind's proof can be understood as a special case of the method we develop in this paper. 2 This is closely related to \echo eect" of such automata: if you begin with a nite pattern on a background of zeros, then, after a long period of confusing noise, the original pattern and several disjoint, identical copies of it will reappear, on an otherwise zero background. This phenomenon was rst investigated by Edward Fredkin [4], who suggested such \parity automata" as the simplest cellular automata capable of exhibiting the \self-replicating patterns" originally sought by von Neumann [18] and Ulam [17]

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There are three dierent directions in which these results can be generalized: 1. Rather than a Bernoulli or Markov measure, we might let  come from a broader class of measures which still exhibit some kind of \mixing" property which can be exploited to generate randomness. 2. Rather than linear cellular automata, we might consider a natural generalization, ane cellular automata, which have local maps of the form: f [a]0 =

X

u2U

f u au + b

(2)

where b 2 A is some constant. 3. Rather then one-dimensional linear cellular automata (ie. on AZ or AN ), D we might look at higher dimensional cellular automata (eg. on A(Z ) ), or even cellular automata where the underlying geometry is \non-Euclidean"; for example we might consider automata acting on a con guration space AM , where M is some monoid (a class which includes N , Z, ZD, but also, for example, nonabelian groups and regular trees). In this paper, we explore all three generalizations simultaneously, by devel oping a sucient condition for the sequence of measures FN jN 2N to converge, in Cesaro average, to Haar measure, where F : AM ?! AM is a linear cellular automaton, A = Z=p for some prime p, and M is a nitely generated monoid. We require the measure  to have a kind of mixing property, called harmonic mixing |we demonstrate that, for example, Bernoulli measures (on AM , where M is any monoid) and \fast-mixing" Markov measures (on AT, where T is Z, N , or any regular tree) have this property. We also require the automata F to have a kind of \expansiveness" property, called diusion, which we show is fairly ubiquitous amongst linear automata. This paper is organized as follows: in x2, we develop some formalism: the compact abelian group structure of the con guration space AM and harmonic analysis on this group (x2.1); monoids (x2.2); and cellular automata (x2.3). In x3 we discuss harmonic mixing and exhibit some examples of it. In x4 we de ne diusion, and prove the main theorem. In x5, we show that, for any primeDp and D  1, if A = Z=p, then all \nontrivial" linear cellular automata on A(Z ) are D diusive; hence, such automata take harmonically mixing measures on A(Z ) into Haar measure.

2 Preliminaries

2.1 Harmonic Analysis on Con guration Space

Throughout this paper, A will denote a nite abelian group. The operation of A will be written as \+"; elements of A will be a; b; c; : : :. 4

Let T1 be the unit circle group. A character of A is a group homomorphism  : A ?! T1. The set of all characters of A forms a group, and will be denoted Ab.

Example 1: Let A 2 N be prime, and let A := Z=p be the cyclic group of order p. We identify the elements of A with the set [0::p) := f0; 1; : : :; p ? 1g. The group Ab is canonically isomorphic with A. First de ne : A ?! T1 by  2i  a :

(a) = exp p





2i Thus, for each k 2 A  = [0::p) and a 2 A, k (a) = exp p k
a :





Then Ab = k ; k 2 [0::p) , and the map A 3 k 7! k 2 Ab is an isomorphism. Let M be some in nite set. The product space AM has a natural structure as a compact abelian group, where the group operation is component-wise addition and the topology is the Tychono product topology induced by the discrete topology on each copy of A.

Notation: AM will be called con guration space; elements  of AM (\con gurations") will be denoted in boldface: a; b; : : :, where a = am jm2M . dM . The elements of AdM The group of characters of AM will be denoted by A are in bijective correspondence with the elements of the set



[m jm2M ] 2





 bM A

; m = 11 for all but nitely many m 2 M

If  m jm2M is such a sequence, then de ne  : AM ?! a = amjm2M is an element of AM , then (a) =

Y m2M

T1



so that, if

m (am ); (where all but nitely many terms in this product

are equal to 1.) In other words:  =

O

m2M

m :

 j  is called the coecient system of . m m2M Example 2: If A = Z=p, with p prime, then AdM is naturally isomorphic

to the group

n

[m jm2M ] 2 AM  = [0::p)M ; m = 0 for all but nitely many m 2 M 5

o





If  m jm2M is such a sequence, then de ne  : AM ?! T1 so that if a = amjm2M is an element of AM , then  Y m Y  2i (a) =

(am ) = exp p m
am : m2M

m2M

The rank of the character   is the number of nontrivial entries in the coef cient system m jm2M . If  is a measure on AM , then the Fourier coecients of  are de ned:

b[] = h; i =

Z

AM

 d

dM . The Fourier coecients of  completely identify . We will for all  2 A use the following basic result from harmonic analysis: Theorem 3: If 1; 2; : : : is a sequence of measures on AM , and  some

other measure, then

0 1 0 1 @ bn[] ???? ?!  in the weak*-topology A ?! b[] for all  2 AdM A () @ b ???? n!1 n!1

2.2 Monoids and Cayley Digraphs

Cellular automata are usually de ned so that the underlying space is a Ddimensional lattice. What this really means is that the \sites" which the automaton acts upon are the points in the Cayley digraph3 of the group ZD. Sometimes one-sided cellular automata are considered, acting on the \half-line", which is really the Cayley digraph of the monoid N . Cellular automata can be de ned in a similar fashion on the Cayley digraph of any monoid, and it is useful to do so. If lattices represent D-dimensional Euclidean space, then other Cayley digraphs can be imagined as various \nonEuclidean" geometries, still possessing enough homogeneity in their local structure for us to be able to well-de ne a cellular automaton. Let M be a monoid: a set is equipped with an associative \multiplication" operator , and an identity element, Id . Let U  M be some subset large enough to generate all of M ; intuitively, we imagine U to be a sort of \neighbourhood about Id ". We de ne a labelled digraph structure on M , so that there is an arc from element m to element n if and only if n = u  m for some u 2 U; this arc is then labelled by the element u (which in this context can be thought of as de ning the \direction" which n lies in, relative to m). This construction imposes a homogeneous geometric structure on M ; it is within this geometry that the automaton will act. For example: 3

For more on Cayley digraphs, see for example [19].

6

 L1 lattice: Let M = ZD, and let U = f0g [ fed ; d 2 [1::D]g, where ed = (0| ; :{z: : ; 0}; 1; 0; : : :; 0): d?1

. Then U corresponds to the unit ball on ZD in the L1 norm.  L1 lattice: If M = ZD, and U

=

(X D d=1

ad ed ; ?1  ad  1; 8d

)

then U corresponds to the unit ball on ZD in the L1 norm.  D-ary tree: Let M is the free monoid on D generators, say m1 ; m2 ; : : : ; mD . Let U = fm1; m2 ; : : : ; mD g. Then the Cayley digraph of M is a D-ary tree, with its root at the identity element.  Free group:1Let M1is the free1group on D generators, say m1 ; m2 ; : : : ; mD . Let U = fm1 ; m2 ; : : : ; mD g. Then the Cayley digraph of M is a collection of 2D distinct (2D ? 1)-ary trees, all joined together at the identity element.  If M be the group generated by D, the set U = fm1 ; m2 ; : : : ; mD g, with relations m21 = m22 = : : : m2D = Id , then the Cayley digraph of M is D distinct (D ? 1)-ary trees, joined at the identity. M

The action of M on itself by translation induces a natural shift action of on con guration space: for all e 2 M , and a 2 AM , de ne  e [a] = [bm jm2M ]

where, 8m; bm = ae:m .

2.3 Cellular Automata

Fix a nite subset U  M (a \neighbourhood of the identity") and a map f : AU ?! A. The cellular automaton induced by f is function F : AM ?! AM   de ned in the following way: for any am jm2M in AM , F(a) = [bmjm2M ] ;

U

where for all m 2 M , bm = f [au:mju2U] :

is called the radius of F, and f is called the local transformation rule.

Notation: We will use capitalized Gothic letters (eg. F, G) to denote cellular automata. We will use the corresponding lower-case Gothic letters (eg. f, g) to denote local transformation rules. 7

Example 4: Let M = Z, and U = f? 1; 0; +1 g. Let f : Af?1;0;+1g ?! A.   Z Then for any sequence a 2 A , F(a) = bmjm2Z , where for all m 2 M , bm = f [am?1 ; am ; am+1 ] : If A is a group, and AM is endowed with the natural product group structure, then a linear cellular automata is a cellular automata which is also a group endomorphism from AM to itself. This is equivalent to requiring the local transformation rule f to be a group homomorphism from AU into A. Suppose A is an abelian group. Then the set End [A] of group endomorphisms from A to itself is a ring, with operations of composition and pointwise addition. In other words, if f; g : A ?! A are group endomorphisms, then so are f  g and f + g.

Example 5: If A = Z=p, with p prime, then there is a bijective correspondence between the elements of End [A] and the ring Z=p. Denote the elements of Z=p by f[0]; [1]; : : :; [p ? 1]g. If [k] 2 Z=p, then de ne f[k] : A ?! A by: f[k] (a) = (k:a (mod p)). Notice that, for any [k]; [j ] 2 Z=p f[k]  f[j] = f[k
j] ; and f[k] + f[j] = f[k+j]

The following can be proved using simple algebra:

Lemma 6: Suppose A is abelian, and let Hom AU; A be the set of group homomorphisms from AU into A. There is a natural bijection between (End [A])U and Hom AU; A , as follows: For each u 2 U, suppose that fu 2 End [A]. De ne f : AU ?! A by: f [au ju2U] =

X

u2U

fu (au ):

Then f : AU ?! A is a group homomorphism. Every group homomorphism 2 from AU into A arises in this manner.

Example 7: If A = Z=p,?with
p prime, and U = f?1; 0; +1g, then End [A] = U Z=p, and thus, End [A]  = Z=p f?1;0;+1g : We will therefore identify elements of End [A] with elements
? of Z=p when describing coecients of a map. For example, if f = f(?1); f0 ; f1 2 ?Z
f?1;0;1g, then de ne f : Af?1;0;+1g ?! A so that, for any triple (a ; a ; a ) 2 =p (?1) 0 1 Af?1;0;1g ,
?
? f a(?1) ; a0 ; a1 = f(?1)
a(?1) + (f0
a0 ) + (f1
a1 ) (mod p): 8

M As a consequence of this, any linear cellular automaton F : AM ?! A  U with radius U is uniquely de ned by an element of (End [A]) . If fm ju2U 2 (End [A])M , then the corresponding linear cellular automaton F is de ned:

F(a) = [bu ju2U] where, for each m 2 M , bm =

X

u2U

fu [au:m ]:

Notation: The collection fu ju2U is called the coecient system of the cellular automaton F, and will be denoted by the boldface letter \f ". If A = Z=p, then we will abuse notation and identify elements of End [A] with elements in (indicated  by italic letters); in this case the coecient system will take the form: f = fu ju2U , where fu 2 Z=p. We can use these coecients to express the map F as a \polynomial" of shift maps:

Z=p

X

F =

u2U

fu   u

In the case when A = Z=p, and the coecients fu are elements of Z=p, and this polynomial formalism becomes quite appropriate:

Example 8: Suppose M = Z, and f3 = 1; then, for any a 2 AZ, F(a) =

U

= f0; 1; 2; 3g, and f0 = 1; f1 = 3; f2 = 0,

a + 3 (a) +

 3 (a)

What this formalism really represents is an identi cation of cellular automata with elements of the group ring with ring coecients in End [A] and group coecients in M (if M is a monoid, then this is a monoid ring). In fact, this identi cation is an isomorphism: composition of cellular automata corresponds to multiplication of these polynomials, which in turn corresponds to convolution of the coecient systems. To be precise, for any two coecient systems f = fnjn2M  and g = gnjn2M ,  de ne the f  g to be the coecient system hn jn2M , where, for all k 2 M , hk =

X

m;n2M n:m=k

fm  gn

(recall that End [A] is a ring, and that all but nitely many terms in this summation are zero). It is then a straightforward exercise to show:

Proposition 9: If F; G : AM ?! AM are linear cellular automata with coecient systems f and g, respectively, then F  G is a linear cellular automaton 2 with coecient system f  g. 9

Z=p, with p prime. Let M = Z. Let f = f jExample , and g10:= Suppose g j  Abe=sequences in Z=p, where all but nitely many n n2Z n n2Z entries are zero. Then f induces a linear cellular automata F : AZ ?! AZ so   that F(a) = bn jn2Z , where X

bn =

fma(n+m) (mod p):

m2Z





and likewise for G. The convolution of f and g is the sequence h = hn jn2Z , where

hk =

X

fm g(k?m) (mod p): m2Z P P If we write F = n2Zfn n and G = m2Zgmm , then FG =

X

fn 

n2Z

! X n


!

gmm =

m2Z

X

!

hk k :

k2Z

2.4 Characters and Cellular Automata

The convolution formula given for composition of linear cellular automata has an analog for composition of linear cellular automatawith characters. a linear , and  isIf aF ischaracter cellular automata with coecient system f = f j n n2M     with coecient system  = m jm2M , then de ne   f = m jm2M , where, for all k 2 M , k is the element of Ab de ned

k =

Y

m;n2M n:m=k

(n  fm )

To make sense of this, recall that the composition of a character with an endomorphism is also a character, and that Ab is a group, so the product of characters is also a character. Note also that almost all terms in this product are equal to 1: if fm is the zero endomorphism, then n  fm (a) = n (0) = 1, and if n = 11 , then of course n  fm (a) = 1. The proof of the following proposition is along the same lines as that of Proposition 9.

Proposition 11: If  2 AdM and F : AM ?! AM are determined by coecient systems  and f respectively, then   F is also a character, and is 2 determined by coecient system   f .

10

3 Harmonic Mixing

A measure  on AM is called harmonically mixing if, for all  > 0, there is dM , some R > 0 so that, for all  2 A

0 1 0 1 @ rank [] > R A =) @ jb[]j <  A

Intuitively, if a 2 AM is a con guration, then (a) is a way of measuring some sort of \relationship" between the values a at dierent sites in M . Thus, if  is a probability measure on this con guration space, then the Fourier coecients of  describe \correlations" between dierent sites. Harmonic mixing is therefore a form of \mixing" because it says, \the amount of correlation goes to zero as the number of sites involved becomes large".

Example 12: Haar Measure The Haar Measure on AM (also called the equidistributed Bernoulli measure)

is de ned:

Haar = where Haar

O `2M

HaarA

A is the probability measure on A assigning each element probability 1=Card [A]. For all  except the trivial character 11 , we have: h; Haar i = 0. Thus, the Haar measure is harmonically mixing. Not all measures on AM are harmonically mixing. For example, if  is a measure on AM , and m 2 M , we ? say 
is m-quasiperiodic if there is an orthonormal basis fn jn2Ng of L2 AM ;  , consisting entirely of eigenfunctions of the shift map m . It is not dicult to show:

Proposition 13: If  is m-quasiperiodic for any m 6= Id, then  is not harmonically mixing. 2 Also, if  : AM ?! T1 is a nontrivial character, then the Markov subgroup ([15], [11], [9]) induced by  is de ned:





AM := a 2 AM ;   m (a) = 1; 8m 2 M : If AM is nontrivial, it is a subshift of nite type4 . If  is a stationary measure on AM , then  cannot be harmonically mixing: if m1 ; : : : ; mJ 2 M are spaced widely enough apart, and  := 4

K Y

k=1

   mk

In fact, if A = Z=p, with prime, then AM  must be periodic. p

11

then h; i = 1, no matter how large K becomes. dM; However,  may still be harmonically mixing relative to the elements of A see Corollary 20.

3.1 Harmonic Mixing of Bernoulli Measures

Proposition 14: Let A = Z=p, where p is prime. Let be any measure on A which is not entirely concentrated on one point. Let M =

O

m2M

be the corresponding Bernoulli measure on AM .

Then M is harmonically mixing.

 Proof: For any k 2 [0:::p), let ck := k ; , where : A ?! T1 is the character described in Example 1. Claim 1: jck j < 1, unless k = 0, in which case c0 = 1. Proof: p is prime; therefore, for any k 6= 0, and any a; b 2 A, if a 6= b, then k (a) 6= k (b). Thus,



k ; 

=

X k (a) fag < X k (a)
fag a2A a2A

= 1:



The (triangle) inequality here is an equality i the elements k (a) ; a 2 A all have the same phase angle, which they cannot, since they are distinct elements on the unit circle. ................... 2 [Claim 1]

 log() 

Let c := 0max jc j: Thus, c < 1. Given  > 0, de ne R := log(c) : 0 so that J \ [N::1)  J.

dM , (for details, see [14], Remark 6.3, Chapter 2, or [8]). Thus, for all  2 A jlim !1 h(  F j2J

n ) ; i = 0:

2

[Theorem 19]

The same reasoning applies stationary measures supported on shift-invariant subgroups of AM :

Corollary 20: Let F : AM ?! AM be as in Theorem 19. Suppose that G  AM is a closed subgroup, and  is a measure supported on G . Suppose  is harmonically mixing relative to the elements of Gb, and F(G )  G . If F is diusive (in density), then jlim Fj  = Haar G ; in the weak* topology. !1 where convergence is either absolute or in Cesaro mean, as appropriate, and where Haar 2 G is the Haar measure on the compact group G . To extend these results to ane cellular automata, use the following:

Proposition 21: Let A be any nite group. Let F : AM ?! AM be a linear cellular automata with local transformation f : AU ?! A. Let c 2 A be some constant, and de ne the local transformation g : AU ?! A by g(a) = f(a) + c. Let G be the corresponding ane cellular automaton. Let  be a measure on AM , and let J  N so that j aar in the weak* topology. jlim !1 F  = H j2J

j aar in the weak* topology. Then we also have jlim !1 G  = H j2J

Proof: Let c 2 AM be the constant con guration whose entries are all equal to c. Thus,Pabusing notation, we can P write: G = F + c. Now, suppose that F = u2U fu  u . Let f = u2U fu , an endomorphism of A. Then F(c) = c1 , where c1 is the constant con guration whose entries are equal to c1 = f (c). More generally, for any N , FN (c) = cN , where cN is the constant con guration whose entries are equal to cN = fN (c). Thus, GN = FN + (c1 + c2 + : : : + cN ). 18

For all N , de ne hN = (c1 + c2 + : : : + cN ), and then de ne HN : AM ?! AM to be translation by hN |that is: HN (a) = a + hN . Thus, we can write: GN = HN  FN : dM is a character, then for any a 2 AM , Now, if  2 A   HN (a) =  (a + hN ) =

KN
(a)

where KN =  (hN ) is some element of T1 . Concisely:   HN = KN
. Now, FN  converges in density to the Haar measure, in the weak* topology, which is equivalent to saying: for every nontrivial character , there is a subset J  N of density one such that



j jlim !1   F ; j2J

But for any j ,

  Gj ; 







 = 0









=   Hj  Fj ;  = Kj
  Fj ;  = Kj
  Fj ; 

But jKj j = 1, and thus,

  Gj ; 

=

  Fj ;  ???? j2?! J j!1

0

Since this is true for each character, we conclude that GN  also converges in density to the Haar measure.

2

[Proposition 21]

5 Diusion on Lattices Diusion on trees was relatively straightforward, because distinct terms in the polynomial expansion of FN never \cancel". In a lattice, however, there are many distinct \paths" between any two sites, which means that many terms of the polynomial expansion of FN may end up overlapping, and possibly canceling. The proof of diusion on lattices is thus much more involved, since we must track these cancellations and ensure that \enough" uncancelled terms remain for the rank to become large. Say that a linear cellular automata F on AM is nontrivial if F is not merely a shift map or the identity map. If we write F as a \polynomial" of shift maps (as in Example 8 in x2.3), then F is nontrivial if this polynomial contains two or more nonzero coecients.

19

Theorem 22: Let p be a prime number, and A = Z=p. Let D  1. Then D any nontrivial linear cellular automaton on A(Z ) is diusive in density. The proof of this theorem will occupy the rest of this section. We will eventually accomplish a reduction to the case when D = 1; hence, the reader may initially nd it helpful to assume D = 1, and to treat all elements of ZD (indicated as vectors, eg. \m ~ ") as elements of Z instead (indicated as scalars, eg. \m"). It will also be helpful to rst work through the details of the proof in the special case when p = 2; we will make reference to this special case in footnotes. We will represent automata using the \shift polynomial" notation introduced Example 8 in x2.3. It will be convenient to write these polynomials in a special recursive fashion. For example, suppose D = 1, and suppose that g0 ; g1 ; g2 2 [1::p), and `0 ; `1 ; `2 2 Z. Let G be the linear CA on AZ de ned: G = g0 `0 + g1 `1 + g2 `2 . Then we can rewrite G as:
? G = g0
F  `0 ; where F = Id + f1m1 (1 + f2 m2 ) ; (3) with m1 = `1 ? `0 , m2 = `2 ? `1 , f1 = g0?1 g1 and f2 = g1?1 g2 (with inversion in the eld Z=p)6 . More generally, we have the following:

Lemma 23: Let g0; g1; : : : ; gJ 2D [1::p), and ~`0; ~`1; : : : ; ~`J 2 ZD, and suppose that G is the linear CA on A(Z ) de ned: G = g0 ~`0 + g1 ~`1 + : : : + gJ ~`J :   Then G = g0
F  ~`0 ; where: F =

Id +

0 f1 m~ 1 @Id

+

? ? f m~ 2 : : : Id + f 2

J ?1

~ J ?1 ?Id m

(4) (5)

+

1


fJ m~ J : : : A

and, for all j 2 [1::J ], m ~ j = ~`j ? ~`j?1 , and fj = gj??11
gj .

2

Composing with the shift ~`0 and multiplying by the scalar g0 does not aect convergence of the measure; hence, it is sucient to prove Theorem for polynomials like (5). On rst reading, it may be helpful to assume that J = 2, as in (3). By Proposition 9, the powers FN of the linear cellular automaton F correspond to powers of the corresponding polynomial. To prove Theorem 22, we will therefore need to develop some machinery concerning multiplication of polynomials with coecients in Z=p. We will be making use of a theorem of Lucas describing the binomial coecients, mod p. 6

If = 2, we can assume that p

f1

=

f2

= 1.

20

De nition 24: p-ary expansion, Index set. N If n 2 N , then n has a unique p-ary expansion, P(n) = fn[i] g1 i=0 2 [0::p) , such that

n=

1 X i=0

n[i] pi :

The index set S (n) of n is de ned to be:





S (n) = i 2 N ; n[i] 6= 0 :

If m is a number, then the congruence class of m, mod p, will be denoted [m]p .

Theorem 25: Lucas, [5] Let N; n 2 N , with p-ary expansions as before. Then

N 

n p

where we de ne

0 0

= 1;

=

1  N [k]  Y

k=0

and

n[k] p

0 n

= 0; for any n > 0:

As a consequence, note that

N 

n p 6= 0

!

2

1 0 () @ n[k]  N [k] , for all k 2 N A

Thus, we will partially order p-ary expansions as follows: we write n  N whenever n[i]  N [i] for each i 2 N .

De nition 26: Lucas Set If n 2 N , then the Lucas set of n, denoted L(n), is the set L(n) := fk 2 N ; k  ng : The following elementary arithmetic observation will be used later.

21

Lemma 27: Let n1, n2; : : : ; nL 2 N. If M > 0, and for all ` 2 [1::L] and

i  M , n[li] = 0, then

L ![i] X

for all i  M + dlogp Le;

l=1

nl

= 0

(here, dne 2 N is the smallest integer such that dne  n.) Proof: The \logp " term comes from the fact that, in summing L distinct p-ary numbers, there is the possibility of up to logp [L] digits of carried value spilling forward.

2

[Lemma 27]

With Lucas' theorem, one can obtain expressions for powers of linear automata. For example, if f 2 Z=p, and F is the linear automata on AZ de ned by F(x) = x + f
 (x), then Lucas' Theorem tell us that FN (x) =

X N  k k k f
 x:

k 2L(N )

p

Next, if m1 ; m2 2 Z, and f1; f2 2 Z=p, and F is as in (3), then FN =

=

X  N  k1 m1k1 f1
 (1 + m2 )k1 k 1 p k1 2L(N ) 1 0   X N k1 m1k1 @ X  k1  k2 m2 k2 A f2
 f1
 k1 2L(N ) k1 p k2 2L(k1 ) k2 p ! X X  N   k1 

f1k1 f2k2
m1 k1 +m2 k2 : k k 2 1 p p k1 2L(N ) k2 2L(k1 ) X X f(k1 ;k2 )
m1 k1 +m2 k2 ; = =

k1 2L(N ) k2 2L(k1 )

where we de ne f(k1 ;k2 ) :=

 N   k1  k1 k2 k1 k2 f1 f2 : p

p

A similar argument works in ZD, and for an arbitrary number of \nested" polynomial terms of this type. This leads to the following

Lemma 28: If m~ 1; m~ 2; : : : ; m~ J 2 ZD, and f1; : : : ; fJ 2 Z=p, and F is as in

(5), then

FN

where:

=

X k2LJ (N )

f(k)hk;mi;

(6)

m := [m~ 1; m~ 2; : : : ; m~ J ] ; [k1 ; k2 ; : : : ; kJ ] 2 N J ; kJ  kJ ?1  : : : k2  k1  N ;

LJ (N ) :=

22

and, for any such k = [k1 ; k2 ; : : : ; kJ ], we de ne

hk; mi := k1 m~ 1 + k2 m~ 2 + :: : + kJm~ J and f := N k1 : : : kJ ?1 f k1 f k2 : : : f kJ : (k)

k1 p k2 p

kJ p

1

2

J

2

(the dependence on N is suppressed in the notation \f(k)".)

Proof of Theorem 22: It suces to prove the theorem for polynomials

F like (5). So, suppose F is not diusive in density. Thus, there exists some (ZD) , some R 2 N , and a subset B  N (of \bad" nontrivial character  2 A\ numbers), of upper density  > 0, so that, for all n 2 B , rank [  Fn ]  R.

Now, for each ~n 2 ZD, let pr~n : A(ZD) ?! A be projection onto the ~nth coordinate: pr~n (a) = a~n . Let : A ?! T1 be the character

(a) = exp

 2i  p
a

(where we identify A = [0::p) in the obvious way). Thus, there is a nite subset Q  ZD, and a collection of coecients f~q 2 [1::p) ; ~q 2 Qg so that  is de ned7 : (a) =

Y ?


~q
pr~q (a)

(7)

q~2Q

Thus, if FN is as in (6) of Lemma 28, and  is as in (7), then, by Proposition 11, the character   FN has the following expansion8:   FN =

Y Y

~q2Q k2LJ (N )





~q
f(k)
pr(hk;mi+~q) :



(8)

Note that, for every ~q 2 Q and k 2 LJ (N ), the factor ~q
f(k)
pr(hk;mi+~q) is nontrivial: f(k) is never  a multiple of p, and thus, if ~q is nontrivial, then

~q
f(k)
pr(hk;mi+~q) is also nontrivial. Thus, the only way the coecients of the character de ned by (8) can be trivial is if two terms of the form 7

In the case when = 2, we can write this: (a) = (?1)a~q

Y

p

~q2Q

8

.

When = 2 = , and = 1 the expansion is:   Fn (x) = J

p

D

Y Y

Y

q2Q k1 2L(n) k2 2L(k1 )

23

:

(?1)(k1 m1 +k2 m2 +q)













~q
f(k)
pr(hk;mi+~q ) and ~q
f(k)
pr(hk;mi+~q) cancel out, which

can only occur when

hk ; mi + ~q = hk; mi + ~q:

(9) This is an equation of D-tuples of integers, and hence, is only true if, for all d 2 [1:::D],

hk ; mi(d) + q(d) = hk; mi(d) + q(d)

where the subscript \(d)" refers to the dth component of the D-tuple.

(10)

The idea of the proof is thus as follows: In order for the rank of the character   FN (for N 2 B ) to be less than R, most of the terms in the expression (8) must cancel out; this requires a speci c kind of \destructive interference" between the the index sets S (N ) and various translations of S (N ) so that virtually all elements (k; ~q) 2 LJ (N )  Q must be paired up as in equation (9), so as to cancel with each other. Our goal, then, is to show that the equation (9) is hard to achieve, so that, after the dust settles, more than R nontrivial coecients remain. We will show that, the set B (indeed, any set of nonzero density) must contain numbers for which sucient cancellation fails to occur. Reduction to Case D = 1: In order for cancellation of terms (ie. equation (9)) to occur in ZD, equation (10) must be true for every d 2 [1::D] simultaneously. Hence, it is enough to disrupt the equation in one dimension. Hence, at this point, we can reduce the argument to the case when D = 1. We will treat m1 ; : : : ; mJ as elements of Z, and m = [m1 ; : : : ; mJ ] as a J -tuple of integers; thus, for any other J -tuple k = [k1 ; : : : ; kJ ], we have hk; mi = k1 m1 + : : : kJ mJ . Likewise, Q will be some nite subset of Z. Gaps in the Index set: We will use an ergodic argument to show that any subset of N of nonzero density must contain numbers N possessing large \gaps" in their index sets: i.e. P[N ] has long blocks of 0's terminated by 1's. We can then nd elements k1 2 L(N ) also exhibiting these long gaps. The gap in such a k1 is long enough that it is impossible to nd some other element (k; q) 2 LJ (N )  Q so that the terms in the expression hk; mi + ~q sum together to \cancel" the terminating 1 in the gap of S (k1 ). Since there are many of these gaps, there are many such elements k1 , and thus, there will be at least (R +1) distinct 1's that remain uncancelled, and thus at least (R + 1) nontrivial terms in expression (8), contradicting the hypothesis  that rank   FN  R for all N 2 B . We can assume that, when we transformed expression (4) into expression (5), we had `0 < `1 < : : : < `J ; hence, we can assume that m1 ; : : : ; mJ > 0. Thus, they have well-de ned Lucas sets, S (m1 ); : : : ; S (mJ ): So, to begin, de ne: 24

? :=

2J 3 [ max 4 S (mj )5 j =1

+

2 0J 1 3 X 66logp @ Card [S (mj )]A + logp(J )77 + 2 6 7 j =1

? stands for \gap", and is the size of the gaps we will require. Let q1 be the smallest element of Q, and de ne

Q1 :=

8 9 < = q ? q 1; q2Q ; : ;

and U :=

[ q2Q1

S (q ) :

Next, let w be the element of [0::p)?+1 de ned: w := (0| ; :{z: : ; 0}; 1) ?

We will be concerned with the frequency of occurrence of w in the p-ary expansions of integers.

Notation: If s = s0s1 : : : sn is a string in [0::p)n, then we de ne the frequency of the word w in s, denoted by fr[w; s], as   C ard fi 2 [0; n ? 1] : si s(i+1) ::::s(i+?) = wg fr[w; s] := n

If si s(i+1) : : : s(i+?) = w, we'll say that w occurs at si . Claim 1: For any   >0, there exists M  such that, for any M > M , there is a set GwM ()  1:::pM so that:





? ) : Card GwM () > (1 ? )pM ; and; 8g 2 GwM (); fr[w; g] > (1p(?+1)





Proof: Consider the ergodic dynamical system [0::p)N; Haar ;  , where Haarnis the Haar measure ando : [0::p)N ! [0::p)N is the shift action. The set a 2 [0::p)Z ; a[0; ?] = w has measure p???1 . The result now follows from Birkho's Ergodic Theorem.

......................

2

[Claim 1]

 aar 1 ?  ?1?? : In particular, let  := 2p : Also, let  := 1 ? 2 H (w) = 2 p Claim 2: There exist M and N such that the following conditions are satis ed: 1. M > R + 2; 2. U  [0; M], 25

3. N 2 B \ [0; pM ), 4. fr[w; N~ ] > (1 ? ) p?1?? . Proof: B has upper density , so there is some sequence fnk g1k=0 such that,

Card [B \ [0; nk ]] ???? ?! : k!1 n k

Find K so that, for k > K; Card [B \n [0::nk ]] > 2 : k Then choose M large enough to satisfy [1] and [2], and such that pM ?1  nk  pM . Thus,





Card B \ 0; pM

  Card [B \ [0; n ]]  nk  pM ?1 k 2 2 M  p2p :

(11)

Also, by Claim 1, let M be large enough so that there is a subset GwM ()  [0::p)M so that









Card GwM () > (1 ? )pM = 1 ? 2p pM ; (12) and fr[w; a] > (1 ? ) p?1?? ; for all a 2 GwM ().      Now, if G := n 2 1::pM ; P(n) 2 GwM () , then Card [G ] = Card GwM () .  Thus, combining (11) and (12), we see that Card B \ 0; pM + Card [G ] > pM ; hence, the two sets must intersect nontrivially. Let N 2 B \ 0; pM \

then N satis es [3] and [4]. ......................... 2 [Claim 2] Claim 3: Let Q = dMe. Then w occurs more than R times in the string: (N [Q+1] N [Q+2] N [Q+3] : : : N [M ] ). Proof: N satis es condition [4] of Claim 2, and of course w occurs at most Q times in the string (N 0 N [1] : : : N [Q] ): Thus, beyond position Q, w must occurs at least G;

?(1 ? )p?1??M
? Q  ?(1 ? )p?1??M
? M ? 1  1?



= (1 ? )p?1?? M ? M 2 p?1?? ? 1 = M 1 ?2  p?1?? ? 1 = M ? 1 26

times and so, by condition [1] of Claim 1, at least R+1 times. 2 [Claim 3] Say w occurs at some positions N [j1 ] ; N [j2 ]; : : : ; N [jR+1 ] beyond Q. Thus, for each r 2 [1:::R + 1], we have: N [jr +k] = 0 for 0  k < ? and N [jr +?] = 1. In particular,

8r 2 [1:::R + 1]; pjr +? 2 L(N ): (13)   Now, N 2 B , so rank   FN  R. This means that in the expression

(8), all but at most R of the terms are cancelled by a like term. In other words, for all but R of the elements: (k ; q ) 2 LJ (N )  Q, there exists some (k; q) 2 LJ (N )  Q so that

hk ; mi + q = hk; mi + q: (14)   |we say that (k ; q ) is annihilated by (k; q). +1 , and thus, there are However, there are R + 1 elements in the set fjr gRr=1 ?
  ( j +?) r R + 1 pairs of the form (kr ; q1 ), where kr = p ; 0; : : : ; 0 . Hence there exists some r such that the pair (kr ; q1 ) is annihilated by some other pair (k; q). De ne n := jr + ?; then hkr ; mi = m1 pn , so we can rewrite (14) as: m1 pn = hk; mi + (q ? q1 ); (15) where k = [k1 ; : : : ; kJ ] is some other element in LJ (N ). Claim 4: For all j 2 [1::J ], and all i  n ? ?, we have: kj [i] = 0.

Proof: First we'll show k1[i] = 0 for i  n. The RHS and LHS of (15) must come from dierent terms of the expansion (8), which means that either q= 6 q1 or kj 6= 0 for some j > 1 ; either way, one of the other terms on

the RHS is positive besides \m1 k1 ", and therefore, m1 k1 < m1 pn . Thus, k1 < pn , and thus, k1[i] = 0 for all i  n. Next we'll show k1[i] for n ? ?  i < n. Recall that k1 2 L(N ), and by hypothesis, N [i] = 0 for all i 2 [jr :::(jr + ?)), where n = jr + ? and n ? ? = jr . Thus, k1 [i] = 0 for n ? ?  i < n. Since kJ  kJ ?1  : : :  k2  k1 , the same holds for k2 ; : : : ; kJ .

2

[Claim 4]

Claim 5:

(mj kj

)[i]

For all j 2 [1::J ], and all i  n ? 2 ? logp (J ), we have:

= 0. Proof: Fix j 2 [1::J ]. For any s 2 S (mj ), it follows from Claim 4 that



mj[s] ps kj

[i]

= 0; for all i  n ? ? + s.

27

Hence, by Lemma 27, (mj kj )[i] =

1[i] 0 X @ mj[s] ps kj A s2S (mj )

= 0; 8i  n ? ? + max [S (mj )] + logp (Card [S (mj )]) : The claim now follows from the de nition of ?. ......... 2 [Claim 5] Claim 6: For all i  n ? 3, (q ? q1)[i] = 0. Proof: By de nition, n ? 3 > n ? ? = jr > Q  M. Recall that condition [2] de ning M was: U  [0; M]. Thus, (q ? q1 )[i] = 0 for i  M . ............................................. 2 [Claim 6] Claim 7: For all i  n ? 1, (hk; mi + (q ? q1))[i] = 0. Proof: Recall that hk; mi = (k1 m1) + : : : + (kJ mJ ); thus, it follows from Claim 5 and Lemma 27 that hk; mi[i] = 0; 8i  n ? 2: Thus, the claim follows from Claim 6 and Lemma 27. ... 2 [Claim 7]

Now, by hypothesis, hk; mi + (q?q1 ) = m1 pn . Hence, (hk; mi + (q ? q1 ))[i] = (m1 pn )[i] for all i 2 N . In particular, if I := min [S (m1 )]  0, then (hk; mi + (q ? q1 ))[I +n] = (m1 pn )[I +n] = m[1I ] 6= 0: But I + n  n, so this is a contradiction of Claim 7.

Conclusion

2

[Theorem 22]

We have shown that harmonically mixing measures on AM , when acted upon by diusive linear cellular automata (possibly with an ane part), will weak*converge, in Cesaro mean, to the Haar measure. This sucient condition is broadly applicable: in particular, if A = Z=p, then any \nontrivial" linear cellular automata acting upon a \fast-mixing" Markov measure (in Z, a regular tree, or a free group) or a nontrivial Bernoulli measure (in ZD) will converge to Haar in Cesaro mean. Many questions remain unanswered, however. What other classes of measures on Z or ZD are harmonically mixing? Measures on AM exhibiting quasiperiodicity cannot be harmonically mixing; what is the Cesaro limit of such a measure, if anything? Also, what linear automata are diusive, when M is neither a lattice nor a free group, or when A is some group other than Z=p? 28

Acknowledgments: We would like to thank David Poole of Trent University for introducing us to Lucas' Theorem, and Dan Rudolph of the University of Maryland for reminding us that, for nonnegative measure, Cesaro convergence is equivalent to convergence in density.

References

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