Random Linear Cellular Automata: Fractals associated with random

Random Linear Cellular Automata: Fractals associated with random multiplication of polynomials R. Daniel Mauldin1 Department of Mathematics University of North Texas Denton, Texas 76203-305118 (U.S.A.) [email protected]

G. Skordev CeVis, Universitat Bremen Universitatsallee 29 D-28359 Bremen (Germany) [email protected]

Abstract Random multiplication of a given set of s polynomials with coecients in a nite

eld following a random sequence generated by Bernoulli trial with s possible outcomes is a (timedependent) linear cellular automaton (LCA). As in the case of LCA with states in a nite eld we associate with this sequence a compact set - the rescaled evolution set. The law of the iterated logarithm implies that this fractal set almost surely does not depend on the random sequence.

Keywords: random multiplication, polynomials, linear cellular automata, fractals

1 Introduction It has been observed that the evolution patterns of seeds (initial congurations) with respect to many cellular automata exhibit self-similarity properties, 19], 24]. This phenomenon is especially apparent for the evolution patterns of nite seeds with respect to the additive (linear) cellular automata with states in the residue classes of the integers. Moreover, the self-reproducing property, the main idea of the inventors of cellular automata, J. v. Neuman and S. Ulam, includes a kind of self-similarity property, 2]. For the mathematical understanding of the problem of deciphering the self-similarity structure of the pattern evolution (orbit) of a nite seed with respect to linear cellular automata (LCA) with states in the residue classes of the integers modulo a prime number (or a prime power), the idea of rescaling proposed by S. Willson is important. In a series of papers, S. Willson associated with such an automata a compact set - the so called rescaled evolution set, 20], 22]. This set is a fractal and its self-similarity structure codes the self-similarity properties of the evolution patterns of the LCA. The self-similarity structure of the evolution set of LCA was described as a special graph directed construction (in the sense of Mauldin-Williams, 15]), in 6], 5], 7], and 18]. The idea of a rescaled evolution set was used in 13] for the description of the self-similarity properties of some classical number sequences - Gaussian binomial coecients and Stirling numbers of the rst and second kind modulo a prime power. A generalization of the notion of a linear cellular automaton necessary for this purpose, is a periodic time dependent cellular automaton. Such a cellular automaton is generated by a periodic multiplication of a nite number of polynomials. The purpose of this note is the randomization of the notion of a periodic time dependent cellular automaton generated by s polynomials with coecients in a Galois eld and its geometrical 1

Partially supported by NSF Grant DMS-9801583

1

representation - the rescaled evolution set. For every point of the code or shift space on s symbols, by appropriately rescaling its geometrical representation, we obtain a sequence of compact sets. One main question is the convergence of this sequence with respect to the Hausdor distance. In general this sequence of compact sets does not converge, and even if it does converge, then its limit depends on the corresponding point of the shift space. The main result (Theorem 1) in Section 7 is that a natural rescaled evolution set exists for almost all points of the code space (with respect to the Bernoulli measure generated by a given probability vector). Moreover, this "expected" rescaled evolution set does not depend on the choice of the point in the code space. For the proof of this result we consider rst in Sections 3, a deterministic situation which includes the "expected " rescaled evolution set and some of its specializations (for one polynomial, in Section 5, and in Section 4, for several polynomials and a rational probability vector or parameters). In the last cases the rescaled evolution sets are anely equivalent with the rescaled evolution sets of appropriate linear cellular automata. In Section 8, we use this to calculate the Hausdor dimension of the expected evolution set in some cases. To make the notations more transparent we shall work with generating polynomials of the corresponding LCA or time dependent LCA. Acknowledgments The authors thank Heinz-Otto Peitgen for his interest, discussions and support, Anna Rodenhausen and Fritz von Haeseler for discussions and remarks.

2 Preliminaries and notations Let IFq be the Galois eld with q elements (q is a power of some prime number p) and let IFq x] be the ring of all polynomials with coecients in IFq . The most important and useful property of such polynomials r 2 IFq x] for this note is (r(x))q = r(xq ): This was called the q-Fermat property in 7] since for q prime number it is equivalent with the small Fermat theorem, 10],Ppp. 70, 77. Let S = (sn )n  sn (x) =  s( n)x 2 IFq x]. For the denition of the rescaled evolution set of the sequence S we need some notations. The set

X (S ) = fI ( n)js( n) 6= 0g

where I ( n) =   + 1]  n n + 1]  IR2 , is called evolution set of the sequence S . The set X (S ) codes some basic information (zero or non-zero) about the coecients of the polynomials sn 2 S , arranged with respect to n (the "time").

Examples 1 1. Linear cellular automaton L(r) corresponding to a nonzero polynomial r 2 IFq x].

The sequence of polynomials S (r) := (rn )n 0 is the orbit of the initial seed (conguration)  = ((0n) )ZZ with respect to L(r). Then X (S ) is a geometrical realization of this orbit in IR2, 6],

7], 8]. This is the set we see on the computer screen visualizing the evolution of the initial seed  under L(r), 24]. Usually we shall use the notation X (r) instead of X (S ). 2. Let r1  : : :  rs be nonzero polynomials with coecients in IFq and r = (r1  : : :  rs ). Let 1  : : :  s be a positive real numbers and  = (1 : : :  s ). The sequence of polynomials S (r ) := (Rn )n 0 , where 2

Rn =

Ys r n 

i=1

i

i]

is associated with the polynomials r1  : : :  rs and parameters 1  : : :  s . Here and later on we denote by a] the integer part of the real number a. We shall denote the evolution set of the sequence S (r ) by X (r ). 3. We shall consider also the sequence of polynomials S (r  !) := (Rn! )n 0 associated with the polynomials

Rn! =

Ys r! n

i=1

i

i(

)

and the point ! 2  = f1 : : :  sgIN , where !i (n) = cardfk j !(k) = i k  ng. The sequence S (r  !) for xed ! is generated by the multiplication of the polynomials r1 : : :  rs according to the point !: Rn! = Qsi=1 r!(i) . We shall denote the evolution set of this sequence by X (r  !). For m 2 IN m  2, the set

Xm (S ) = X (S ) \ (IR  0 m])

is compact and nonempty in case at least one of the polynomials sn of the sequence S is not zero. Then it is a point in the space K(IR2 ) of all nonempty compact subsets of the plane IR2 . In this space we consider the Hausdor metric H , generated by the l1-norm of IR2 , 11], pp. 214-215. It is dened as follows : for a given A 2 K(IR2 ) and a positive number , let

A = fx j jjx ; yjj1 <  for some point y 2 Ag where jjxjj1 = maxfjx1 j jx2 jg x = (x1  x2 ) 2 IR2 . Then the Hausdor distance H (A B ) between two sets A B 2 K(IR2 ) is dened by H (A B ) = inf f jA  B B  Ag: The metric space (K(IR2 ) H ) is complete, moreover, if K 2 K(IR2 ) the space (K(K ) H ) of all nonempty compact subsets of K is compact, 11], pp. 216, 407. The sets Xm (S ) are compact but their union is unbounded in IR2 . For this reason they are

rescaled by appropriate similitudes. The increasing sequences a = (a(n))n 0  b = (b(n))n 0 2 ININ such that maxf Ba((nn))  ab((nn)) g  C where dk = deg sk  B (n) = maxfdk j k  b(n) ; 1g and C is an appropriate constant are called scaling sequences if the sequence of compact sets (sa(n);1 (Xb(n) (S )))n 0 converges with respect to the Hausdor metric H . Here sc : IR2 ;! IR2 is the similitude sc(x y) = (cx cy) (x y) 2 IR2 . Observe that 3

The limit

sa(n);1 (Xb(n) (S ))  0 C ]  0 C ]: nlim !1 sa(n);1 (Xb(n) (S ))

is called rescaled evolution set of S (with respect to the scaling sequences a b 2 ININ ).

Examples 2 The sequences a = b = (qt )t 0 are scaling sequences for the sequence X (r) for every

nonzero polynomial r with coecients in IFq , 20]. The rescaled evolution set corresponding to these sequences is denoted by Aq (r). In general this is a fractal set - for example for r(x) = 1 + x 2 IF2 x] the set A2 (r) is the Sierpinski triangle. For more examples see 6], 5], 7], 8]. In this note we shall consider the scaling sequences a = (qt )t 0 and b = ( aqt ])t 0 for a positive number a. We shall call them standard scaling sequences. In the last section we consider also some other scaling sequences.

3 Existence of rescaled evolution set for several polynomials - general case Here we consider the nonzero polynomials r1  : : :  rs with coecients in the Galois eld IFq and positive real numbers (parameters) a 1  : : :  s . The evolution set of the sequence of polynomials S (r ) = (Rn )n 0 , where

Ys

Rn(x) = (ri (x))ni ] = 1

is the set

X R( n)x 



X (r ) = fI ( n)jR( n) 6= 0g where r = (r1  : : :  rs ) and  = (1  : : :  s ). The compact sets to be rescaled are

\

Xaqt](r ) = X (r ) (IR  0 aqt ]]):

The aim of this section is the following

Proposition 1 The sequence of compact sets (sq;t (Xaqt ](r ))t 0 (1) converges with respect to the Hausdor metric H . Proof The metric space (K(IR2 ) H ) is complete. Therefore the assertion is that the sequence (1) is a Cauchy sequence with respect to the Hausdor metric H . It suces to prove that there exists a constant C such that

H (sq;1 (Xaqt+1 ](r )) Xaqt ](r ))  C 4

(2)

for all t 2 IN. The estimation (2) follows from the inclusions

sq;1 (Xaqt+1 ] (r ))  (Xaqt] (r ))C  (3) Xaqt](r )  (sq;1 (Xaqt+1 ](r )))C (4) for all t 2 IN, where as dened earlier the subscript C indicates all points whose distance to the set is less than C . Proof of (3) Let

I (k n)  Xaqt+1 ](r ) (5) i.e., the coecient R(k n) of the polynomial Rn - the n-th element of the sequence S (r ) - is not zero and n < aqt+1 ]. We shall prove rst that the polynomial

Ys

R^ n(x) = (ri (x)) q ]i ]q = is a factor of the polynomial Rn , i. e.,

n

1

X R^( n)x 

(6)

Rn = R^ nR~ n

(7)

0  ni ] ;

nq ]i ]q  q( + 1) i = 1 : : :  s

(8)

for some polynomial R~ n 2 IFq x]. This follows from the inequalities

where  = maxfi ji = 1 : : :  sg. In fact, let n = lq + j where l 2 IN and 0  j  q ; 1. Then

lqi ; q 

nq ]i ]q  ni] < lqi + q

which implies (8). The inequality (8) and (7) imply also that

deg R~ n  Dq( + 1) (9) P s where D = 1 di and di = deg ri  i = 1 : : :  s. The coecient R(k n) of the polynomial Rn is not zero. Then from (7) and (9) follows: there exists k1 2 IN such that the k1 -th coecient R^ (k1  n) of the polynomial R^ n satises

R^ (k1  n) 6= 0 k ; Dq( + 1)  k1  k: Consider the polynomial 5

(10) (11)

Ys

R nq ](x) = (ri (x)) q ]i ] = n

1

This is the nq ]-th element of the sequence S (r ). From (6) follows

R^n (x) = (R nq ](x))q =

q



X R( n ])xq :

Then (10) and (12) imply that

q



k1 = k2 q

and

(12) (13)

R(k2  nq ]) 6= 0

which gives (here we used nq ]  nq < aqt ). From (11), (13) and (14) follows and therefore

X R( n ])x:

I (k2  nq ])  Xaqt ](r )

(14)

sq;1 (I (k n))  (I (k2  nq ]))C1  (Xaqt] (r ))C1  for C1 = D( + 1) sq;1 (Xaqt+1 ] (r ))  (Xaqt] (r ))C1 :

(15)

Proof of (4) Let

I (k n)  Xaqt] (r ) i.e., the coecient R(k n) of the polynomial Rn - the n-th element of the sequence S (r ) - is not zero and n < aqt ]. Consider the polynomial

(Rn (x))q =

Ys (r (x)) n q = X R( n)xq  1

i



i]



(here we are using the q-Fermat property). Case n  1 + 1 The polynomial

Rn1 =

Ys r n 

 1 i]

1

6

i

(16)

is a factor of the polynomial (Rn )q for where

n1 = nq ; q ] ; 1

 = minfiji = 1 : : :  sg and n  1 + 1

i.e.,

(Rn )q = Rn1 R~ n1  (17) and R~ n1 is a polynomial with coecients in IFq . Moreover, Rn1 is the n1 -th polynomial of the sequence S (r ). In fact n1  0 and

nqi] ; ( q ] + 1) ; 1  n1 i ]  q ni ]  nqi ]

since and Moreover,

(18)

n1 i ]  n1 i  nqi ; q i  q(ni ; 1)  q ni ]  nqi ]

n1 i]  n1 i ; 1 = nqi ; i ( q ] + 1) ; 1  nqi ] ; ( q ] + 1) ; 1:

(19) 0  n ; nq1  1q ( q ] + 1)  1 + 1: The coecient R(k n) of the polynomial Rn is not zero, then from (16) and (17) follows that there exists k1 2 IN such that the k1 -th coecient R(k1  n1 ) of the polynomial Rn1 is not zero and Then (18) implies Therefore

kq ; deg R^ n1  k1  kq:

(20)

deg R^ n1  Df( q ] + 1) + 1g:

(21)

I (k1  n1 )  Xaqt+1 ](r ):

(22)

From (19) , (20), (21), and (22) follows

I (k n)  (sq;1 (I (k1  n1 )))C2  (sq;1 (Xaqt+1] (r )))C2 for n  1 + 1 and C2 = maxf 1 + 1 Dq f( q ] + 1) + 1g. Case n < 1 + 1 7

(23)

The inclusion

I (k n)  (sq;1 (Xaqt+1] (r ))) D ( 1 +1) 

follows since

0k and

(24)

Xs d n ]  Xs nd i  D ( 1 + 1) 1

i

deg Rnq 

i

i

1

 

Xs d nq ]  q D ( 1 + 1) i

i

  and there exists l with 0  l  q D ( 1 + 1) with I (l nq)  Xaqt+1 ](r ). Then jk ; ql j  D ( 1 + 1). 1

Combining (23) and (24) we obtain

Xaqt ](r )  (sq;1 (Xaqt+1 ] (r )))C3 

where C3 = maxfC2  D ( 1 + 1)g. From (25) and (15) follows (2) with C = maxfC1  C3 g.

(25)

2 We denote by Aq (r  a) the limit of the sequence (1) with respect to the Hausdor metric H :

Aq (r  a) = tlim !1 sq;t (Xaqt ](r )):

Remark 1 For s = 1 = a = 1 r = r1 the set Aq (r 1 1) is the rescaled evolution set Aq (r)

associated with polynomial r (or with the linear cellular automaton generated by the polynomial r), 20], 7], 8]. For dierent values of the parameter a the sets Aq (r  a) are dierent and in some cases nonhomeomorphic. For example let q = 2 s = 1 r(x) = r1 (x) = 1 + x 2 IF2 x] 1 = 1. The sets A2 (r 1 3) and A2 (r) are not homeomorphic. The second set is the Sierpinski triangle and the rst is given by

A2 (r 1 3) = s4 (A2 (r)) \ ( 0 3]  0 3]): (26) The sets A2 (r 1 3) and A2 (r) are not homeomorphic, since the Sierpinski triangle has only 3 points - (0 0) (1 1) (1 0) - with branching index 2 and from (26) follows that the set A2 (r 1 3) has 5 points with branching index 2 - (0 0) (0 3) (1 3) (2 3) and (3 3), 16], pp. 125-126.

4 Rescaled evolution set for several polynomials and rational parameters i

Here we shall specialize the situation from the previous section. We consider the nonzero polynomials r1  : : :  rs 2 IFq x], a positive real number a and positive rational numbers 1  : : :  s . Assume that i = kli , where l and ki  i = 1 : : :  s are positive integers and a = l. By Fl : IR2 ;! IR2 we denote the ane map dened with Fl (x y) = (x yl ) for (x y) 2 IR2 . 8

From Proposition 1 we know that the sequence (sq;t (Xlqt (r )))t 0 converges to the set Aq (r  l) with respect to the Hausdor metric H . Here we shall describe this set. For this description we shall use the representation of the rational numbers i = kli and shall write Xlqt (r ( kl1  : : :  kls )) instead of Xlqt (r ). What we show is that the limit set associated with this vector of polynomials and rational parameter values is the ane image of the limit set associated with a single polynomial.

Proposition 2 The sequence of compact sets (sq; (Xlq (r ( kl1  : : :  kl ))))t 0 converges to the set t

Fl;1(Aq (r1k1    rsks )).

s

t

Proof The assertion follows from the inequality

H (Fl (Xlqt (r ( kl1  : : :  kls ))) Xqt (r1k1    rsks  1))   P for all t 2 IN, where  = s1 di ki and di = deg ri .

(27)

The inequality (27) follows from the following two inclusions

Fl (Xlqt (r ( kl1  : : :  kls )))  (Xqt (r1k1    rsks  1)) Xqt (r1k1    rsks  1)  (Fl (Xlqt (r ( kl1  : : :  kls ))))

(28) (29)

Proof of (28) Let

I (a n)  Xlqt (r ( kl1  : : :  kls ))

i.e., the coecient R(a n) of the polynomial Rn - the n-th element of the sequence S (r ( kl1  : : :  kls )) is not zero and n < lqt . Let n = lm + j 0  j  l ; 1 m 2 IN and let jki = ui l + vi  0  vi  l ; 1 ui 2 IN i = 1 : : :  s. Then and

ni ] = n kli ] = mki + ui  i = 1 : : :  s

Rn =

Ys r n 

1

i

i]

= (r1ki    rsks )m r1u1    rsus :

P

(30)

Therefore the polynomial Rml (x) = (r1 (x)ki    rs (x)ks )m =  R( ml)x is a factor of the polynomial Rn . Since the coecient R(a n) of the polynomial Rn is not zero, then there is an integer a1 such that

R(a1  ml) 6= 0

a ;   a ; deg r1u1    rsus  a1  a: 9

(31) (32)

Here Rml is the m-th polynomial of the sequence S (rk1    rks  1) From (31) and (32) follow

I (a1  m)  Xqt (r1ki    rsks  1) Fl (I (a n))  (I (a1  m)) :

The last two inclusions imply (28). Proof of (29) Assume that

I (c m)  Xqt (r1k1    rsks  1) i.e., the coecient R(c ml) of the polynomial Rml = (r1ki    rsks )m - the m-th element of the sequence S (rk1    rks  1) is not zero and m < qt . This polynomial Rml is also the ml-th element of the sequence S (r ( kl1  : : :  kls )). Therefore I (c lm)  Xlqt (r ( kl1  : : :  kls )) and

I (c m)  (Fl (I (c lm)))1  (Xlqt (r ( kl1  : : :  kls )))1 

which implies (29) since 1  .

2

Remark 2 The limit of the sequence (sq; (Xlq (r ( kl1  : : :  kl ))))n 0 depends on the representation of the rational numbers i as fractions:

t

s

t

uk uk Ful;1 (Aq (rk1u    rks u )) = tlim s ;t (Xulqt (r ( 1  : : :  s )) for u 2 IN: q !1 lu lu

and these sets are in general dierent.

5 Rescaled evolution set for one polynomial Here we shall consider another specialization of the situation considered in the Section 3: s = 1 r = r1  a = 1 and  a positive real number. Let F : IR2 ;! IR2 be the ane map dened with F (x y) = (x y) for (x y) 2 IR2 . From Proposition 1 we know that the sequence (sq;t (X qt ] (r )))t 0 converges to the set  Aq (r  1 ) with respect to the Hausdor metric H . Here we shall describe this set.

Proposition 3 The sequence of compact sets (sq; (X ](r )))t t

F ;1(Aq (r)), i.e., Aq (r  1 ) = F ;1 (Aq (r)).

10

qt 

0

converges to the set

Proof The assertion follows from the estimation

H (F (X qt ] (r )) Xqt (r))  C  where C = maxfd( + 1)  + 3g and d = deg r, which is equivalent with the inclusions F (X qt ] (r ))  (Xqt (r 1))C   Xqt (r 1)  (F (X qt ] (r )))C : 

(33) (34)

Proof of (33) Let

I (k n)  X qt ](r ) i.e., the coecient R(k n) of the polynomial (r(x))n] =



X R( n)x 

is not zero. The polynomial rn] is the n-th element of the sequence S (r ). It is also n]-th element of the sequence S (r 1). Therefore It follows that The last inclusion implies (33).

I (k n])  Xqt (r 1): F (I (k n))  (I (k n]))1 :

Proof of (34) Let

I (k n)  Xqt (r)

i.e., the coecient r(k n) of the polynomial

(r(x))n =

rn

X r( n)x 

is not zero ( is the nn -th polynomial of the sequence S (r)). The polynomial r  ]] is a factor of the polynomial rn since

n ;  ; 1 

n ]]  n: Since the coecient r(k n) of the polynomial rn is not zero, then there is a natural number k1 n  n ] ]  such that the coecient R(k1   ]) of the polynomial r satises 11

R(k1  n ]) 6= 0 k ; d( + 1)  k1  k:

(35) (36)

The polynomial r n ]] is the n ]-th element of the sequence S (r ). Then from (35) follow From (36) and the inequality follows which implies (34).

I (k1  n ])  X qt ](r ): 

0  n ;

n ]]   + 2]

I (k n)  (F (I (k1  n ])))C 

2

6 Rescaled evolution set for a perturbation of the sequence S (r

)

Here we shall consider some perturbations of the sequence of polynomials S (r ) under which the rescaled evolution set is stable. Let r1  : : :  rs be nonzero polynomials with coecients in IFq and let a i  : : :  s be positive real numbers. We consider also the functions h gi : IN ;! IN satisfying The function h is nondecreasing,

limn!1 h(nn) = 0, jgi (n) ; ni ]j  h(n) for all n  N , where N is xed natural number.

The function h is o(n) for n ! +1 and the functions gi are small (h-small) perturbation of the functions i : IN ;! IN, given by i (n) = ni ], for n 2 IN. The condition that the function h is nondecreasing is not important but technically convenient. We shall choose the natural number N so big that h(n)    and h(n)  1 for n  N (37) n 2( + 2) where  = minfi ji = 1 : : :  sg and  = maxfi ji = 1 : : :  sg. Consider the polynomials

Ys

Rng (x) = (ri (x))gi (n) = 1

X R~( n)x 

and the sequence S (r g) = (Rng )n 0 , where r = (r1  : : :  rs ) and g = (g1  : : :  gs ). The sequence S (r g) is a small perturbation of the sequence S (r ). The goal of this section is the following 12

Proposition 4

lim s ;t (Xaqt ] (r g)) = Aq (r  a) t!1 q

i.e., the rescaled evolution set of the sequence S (r g) does not depend on the small perturbation g. Proof The assertion follows from the estimation

H (Xaqt ](r g) Xaqt ] (r ))  Ch( aqt ]) t for some constant C and for all suciently large t 2 IN, since limt!1 h(qaqt ]) = 0.

The last estimation is equivalent with the inclusions

Xaqt](r g)  (Xaqt] (r ))Ch(aqt ])  Xaqt ](r )  (Xaqt ](r g))Ch(aqt ]) :

(38) (39)

Proof of (38), case n  N Let N  n < aqt ]. Dene

m = n ; h(n) ]:

Then from (37) follows that m is a nonnegative integer. Moreover, n ; h(n) ; 1  m  n ; h(n)  which implies 0  n ; m  2 h( aqt ]):



The polynomial Rm is a factor of the polynomial Rng , since ni ; h(n)  ;   mi]  gi (n)  ni + h(n): Let

Rng = Rm R^ m

for some polynomial R^ m with coecients in IFq . From (41) follows deg R^  D(  +  + 1)h( aqt ]): m

Assume that



I (k n)  Xaqt ](r g) 13

(40) (41) (42) (43)

i. e., the coecient R~ (k n) of the polynomial Rng is not zero. Then (42) implies that there exists a natural number k1 such that the k1 -th coecient R(k1  n)of the polynomial Rm is not zero, which means that I (k1  m)  Xaqt ] (r ). Moreover, (43) implies (44) k ; D(  +  + 1)h( aqt ])  k  k Then from (40) and (44) follows



1

I (k n)  (I (k1  m))C1 h(aqt]) 

(45)

where C1 = maxf 2  D(  +  + 1)g. Proof of (38), case n < N Assume

I (k n)  Xaqt ](r g) and n < N

then

0  k  DM

P where D = s di  di = deg ri , and M = maxfgi (n)jn < N g.

(46)

1

Let I (l n)  Xaqt ](r ). Since the degree of the n-th element Rn of the sequence S (r ) is bounded from above by the number DN , then From (46) and (47) follows

0  l  DN :

I (k n)  (I (l n))C2  where C2 = D maxfN  M g, since jk ; lj  C2 . From (45) and (48) follow (38) for C  maxfC1  C2 g. Proof of (39), case n  2N Let 0  2N  n < aqt ]. Dene From (37) follows that m  N . Therefore and

m = n ; 2 +  h(n)]:

gi(n)  mi] + h(m)  ni ; h(n)( + 1) + i  ni ; 1  ni]  ni gi(m)  ni ; f(2 + )  + 2gh( aqt ]): 14

(47) (48)

Then 0  ni ] ; gi (m)  f(2 + )  + 2gh( aqt ]):

(49)

Rn = Rmg Rm 

(50)

deg Rm  Df(2 + )  + 2gh( aqt ]):

(51)

Therefore the polynomial Rmg - the m-th polynomial of the sequence S (r g) - is a factor of the polynomial Rn - the n-th polynomial of the sequence S (r ): and from (49)

Suppose I (k n)  Xaqt ](r ) and 2N  n  aqt ]. Then (50) imply that there exists a natural number k1 such that k1 coecient R~ (k1  m) of the polynomial Rmg is not zero and moreover, from (51) follows

k ; C3h( aqt ])  k1  k

for C3 = Df(2 + )  + 2g. From the last inequality and 0  n ; m  2 +  h( aqt ])



follows where C4 = maxfC3  2+

 g.

I (k n)  (I (k1  m))C4 h(aqt]) 

(52)

Proof of (39), case n < 2N Since and

deg Rn < 2ND

deg Rng  M1 D where M1 = maxfgi (n)jn < 2N g then for I (k n)  Xaqt ](r ) and I (l n)  Xaqt ] (r g) follows

I (l n)  (I (k n))C5 

where C5 = D maxfM1  2N g. Then (39) follows from (51) and (52) for C  maxfC4  C5 g. Therefore the proposition is proved for C = maxfC1  C2  C4  C5 g.

(53)

2

15

7 Random multiplication of polynomials - the main theorem The goal of this section is the proof of the theorem:

Theorem 1 Let r = (r1  : : :  rs) r1P : : :  rs 2 IFq x] (nonzero polynomials) and let a  =

(1  : : :  s ) a > 0 i > 0 i = 1 : : :  s s1 i = 1. Then for almost all points ! 2  = f1 : : :  sgIN with respect to the Bernoulli measure  , induced by the probability vector  follows: a) The sequence of compact sets (sq;t (Xaqt ] (r  !)))t 0 converges with respect to the Hausdor metric and

Aq (r  a) = tlim !1 sq;t (Xaqt ](r  !)): b) For s = 2 r1 = r r2 = 1 1 =  2 = 1 ;  the sequence of compact sets (sq;t (X qt ] (r  !)))t 

converges with respect to he Hausdor metric and

0

Aq (r) = tlim !1 sq;t (X qt ] (r  !)): 

= ( kl1  : : :  kls ) l k1  : : :  ks 2 IN the sequence of compact sets

c) For a rational probability vector  (sq;t (Xaqt ] (r ( kl1  : : :  kls ) !)))t 0 converges with respect to the Hausdor metric and

ks k1 Aq (r1k1    rsks ) = tlim s ;t (Xaqt ] (r (  : : :  ) ! )): q !1 l l

Proof For ! = (!(n))n 2 IN let

!i (n) = cardfk j !(n) = i k  ng:

Here we shall consider the polynomials

Ys

Rn! (x) = (ri (x))!i (n) = 1

X s^( n)x 

and the sequence S (r !) = (Rn! )n 0 . On the probability space (  ) we consider the random variables Ynj :  ;! f0 1g, dened by

(1

!(n) = j Ynj (!) = 0 otherwise : The expected value of Ynj with respect to the measure  is E (Ynj ) = j and the variance var(Ynj ) = j (1 ; j ) = j2 . Then (Ynj )n 0 is i.i.d. sequence of random variables. The law of iterated logarithm, 12], p. 42, implies that for almost all points ! 2  with respect to the Bernoulli measure  ,

Pnk lim sup q n

Ykj ; nj = 1: 2j n log log n Therefore for almost all ! 2  there exist a natural number N = N (!) such that =1 2

p

p

nj ] ; 2 2n log log n]  !j (n)  nj ] + 2 2n log log n] 16

P for all n  N and  = maxfj2 j j = 1 : : :  sg. Here we used that !j (n) = nk=1 Ykj (!). Dene the functions gj! : IN ;! IN and h : IN ;! IN with gj! (n) = !j (n) j = 1 : : :  n

and

p

h(n) = 2 2n log log n]: Then the functions h and gj!  j = 1 : : :  s satisfy the conditions of the Proposition 4. Therefore for almost all ! 2  the sequence S (r !) is a small perturbation of the sequence S (r ) and the theorem follows from the Proposition 4.

2

8 Hausdor dimension of rescaled evolution sets The Hausdor dimension of the rescaled evolution set Ap (r) = Ap (r 1 1) was calculated by S. Willson in 22], using the Perron-Frobenius eigenvalue  of the transition matrix (associated to the polynomial r), introduced by him: dimH Ap (r) = lnln p . Later on F. v. Hassler at all., 7] described the rescaled evolution set Aq (r) using the attractor of a special graph directed construction - matrix substitution system. The transition matrix of S. Willson coincides with the transition matrix of this graph directed construction. Since the graph directed construction of v. Haeseler et all. satises the open set condition, then the formula of S. Willson follows from the formula of Mauldin-Williams,

15], Theorem 3, for the Hausdor dimension of the attractors of graph directed construction, 15]. In addition, from Mauldin-Williams, 15], Theorem 3, follows that the Hausdor measure in the dimension is positive and nite, since the transition graph of the graph directed construction of Haeseler et all. is strongly connected, 9]. About the Hausdor dimension see 3]. The goal of this section is the following proposition

Proposition 5 Let r = (r1  : : :  rs) r1  : : :  rs 2 IFq x] i  a 2 IR a > 0 i > 0 i = 1 : : :  s. Then a) The Hausdor dimension of the rescaled evolution set Aq (r  a) does not depend on a, i.e., dimH Aq (r  a) = dimH Aq (r  1):

b) For s = 1 r1 = r 1 =  follows

dimH Aq (r  a) = dimH Aq (r) c) For i = ( kl1  : : :  kls ) l k1  : : :  ks 2 IN follows dimH Aq (r ( kl1  : : :  kls ) a) = dimH Aq (r1k1    rsks ): P d) For i > 0 i = 1 : : :  s and s1 i = 1 let  be the Bernoulli measure on the space  = f1 : : :  sgIN . Then for almost all ! 2  with respect to the Bernoulli measure  follows dimH Aq (r  ! a) = dimH Aq (r  1):

17

Proof a) Recall that

Aq (r  a) = tlim s (X t (r )): !1 q;t aq ]

Set  = lnln aq ] + 1. Then for t big enough t1 = t +  is a nonnegative integer and

qt+;2  aqt  qt+:

Therefore

Xqt+;2 ](r )  Xaqt ] (r )  Xqt+ ](r ):

The last inclusions imply

sq;2 (Aq (r  1))  Aq (r  a)  sq Aq (r  1)):

The assertion follows, since the ane maps do not change the Hausdor dimension. b) The assertion follows from the Proposition 3 and a). c) The assertion follows from the Proposition 2 and a). d) Follows from the Theorem 1 and a).

2

Remark 3 In general, the set Aq (r  1) and its Hausdor dimension depend on , e.g. in the case  = ( k1l u  : : :  k lu ) u l k1  : : :  ks 2 IN the Hausdor dimension of the set Aq (r ( k1l u  : : :  k l u )) i

s

depends in general on u. The formula of S. Willson gives the dimension of the rescaled evolution set for  vector with rational coordinates. For all other cases we do not have a formula for calculating it.

9 Some remarks 1. The eld of coe cients We considered polynomials with coecients in the Galois eld IFq . All assertions proved in this note hold in a more general situation - for m-Fermat polynomials with coecients in a commutative ring with 1. Let R be a nite commutative ring with 1 and r a polynomial with coecients in R. We say that the polynomial r has the m-Fermat property (or is an m-Fermat polynomial) if

r(x)m = r(xm) for a given natural number m  2.

Examples 3 1. For R = IFq , the Galois eld with q = ps elements (where p is a prime number),

every polynomial p(x) 2 IFq x] is a q-Fermat polynomial, 10], pp. 62, 65. 2. For Rs;=1 ZZ=ps ZZ, (p - prime number), the integers modulo ps  s  2, every polynomial p(x) = q(x)p 2 R x] is a p-Fermat polynomial, 17], 22]. 3. For more examples and comments see 1], Lemma 1, 2 , pp. 11-12. 18

2. Scaling sequences Here we shall consider the simplest case - the sequence of polynomials S (r) for r 2 IFq x]. The set of scaling sequences for a given polynomial is quite big: for arbitrary sequences a = (a(n))n 0 and b = (b(n))n 0 satisfying the conditions dn ab((nn))  C , where d = deg r and C is an appropriate constant, the sets sa(n);1 (Xb(n) (r)) are subset of 0 C ]  0 C ] = 0 C ]2 . The space (K( 0 C ]2 ) H ) is compact. Therefore there exist subsequences a0 = (a(nk ))k 0 and b0 = (b(nk ))k 0 such that the sequence (sa(nk );1 )(Xb(nk ) (r)))n 0 converges, i.e., the sequences a0 and b0 are scaling sequences for X (S (r)) (or for the polynomial r). In general these scaling sequences depend on the polynomial r. We want to have scaling sequences independent on r. Such are the standard sequences a = (qt )t 0 and a = ( aqt ])t 0 for every positive real number a. For a = 1 these are not only "universal" scaling sequences but the corresponding rescaled evolution set Aq (r) have a self-similar structure generated by some special graf-directed system, 7]. Moreover, the rescaled evolution set Aq (r 1 a) is determined by and

Aq (r 1 a) = Aq (r) \ (IR  0 a]) for a < 1 Aq (r 1 a) = sqt1+1 (Aq (r)) \ (IR  0 a]) for qt1  a < qt1 +1 :

We shall call Aq (r) the standard rescaled evolution set. For some specic polynomials the self-similarity structure of the rescaled evolution set with respect to some other scaling sequences is simpler in comparison with the self-similarity structure of the standard rescaled evolution set. In 6] are given such examples. One of them is the following. 3n ;1 2 Let r(x) = 1 + x + x 2 IF3 x]. The sequences a = b = ( 2 )n 0 are scaling sequences for the polynomial r. Denote by B the rescaled evolution set with respect to these sequences: n B = nlim !1 s 3n2;1 (X 3 2;1 (r)):

The set B is not anely equivalent with the standard rescaled evolution set A3 (s) s 2 IF3 x] ( the last set is the rescaled version of the geometrical representation (zero - non-zero) of the binomial coecients modulo 3, 4]. The sets B and A3 (r) are not homeomorphic and the self-similarity structure of B is simpler. Similar observations imply that the sequences a = b = (n)n 0 are not scaling sequences for all polynomials with coecients in IF2 . Let r = 1 + x 2 IF2 x]. The standard rescaling evolution set A2 (r) of this polynomial is the Sierpinski triangle. Assume that the sequences a = b = (n)n 0 are scaling sequences for the polynomial r. Then

A2 (r) = tlim !1 s2;t (X2t (r)) = tlim !1 s3:2;t (X3:2t (r)) = s3 (tlim !1 s2;t (X3:2t (r))) = s3 (A2 (r 1 3)) i.e., the sets A2 (r) and A2 (r 1 3) are anely equivalent, which is not possible since they are not homeomorphic. At the end we shall prove that the sequences a = b = (3n )n 0 are not scaling sequences for the polynomial r(x) = 1 + x 2 IF2 x]. The evolution set X (r) of this polynomial is the geometrical representation (zero - non-zero) of the sequence of binomial coecients modulo 2 and its rescaled evolution set 19

n A2 (r) = nlim !1 s2;n (X2 (r))

is the Sierpinski triangle. 2 The number  = log log 3 is irrational and its simple continued fraction  = 0 a1  a2  : : :] is innite. Let

pk = 0 a  : : :  a ] gcd(p  q ) = 1 1 k k k qk be the k-th convergent of the continued fraction 0 a1  a2  : : :], 10], Ch. X. Then

jqk ; pk j  1  for k = 1 2 : : : 

qk

10], p. 140. The last inequality implies

qk j1 ; 2pk j  2  3 qk k p 3 1+ 2  qk 2qk

(54) (55)

for k = 1 2 : : :. In 6] is proved

A2 (r) = klim s ;q (X q (r)) = klim s ;p (X p (r)): !1 2 k 2 k !1 3 k 3 k

(56)

Since lim qk = 1 this assertion follows from the following inequalities

H (s2;qk (X2qk (r)) s3;pk (X3pk (r)))  1 2qk

H (X2qk (r) X3pk (r)) + 2q1k H (X3pk (r) s2qk 3;pk (X3pk (r)))  d j2qk ; 3pk j + 1q j1 ; 2pqk jdiamX3pk (r)  (1 + 2 ) 4d : 2 k 3 k qk qk

2qk

Assume that the sequence (s3;n (X3n (r)))n 0 converges. Then from (56) follows that its limit is the Sierpinski triangle A2 (r). We know that the sequence (s2;n (X3:2n (r)))n 0 converges and its limit is the set

A2 (r 1 3) = s4(A2 (r)) \ ( 0 3]  0 3]):

The last set is not homeomorphic to the Sierpinski triangle. From (54) follows

pk +1

H (X3:2qk (r) (X3pk +1 (r))  3dj2qk ; 3pk j  2d3q

Therefore, the sequence (s3;pk (X3:2qk (r)))k

0

converges and

k

:

lim s3;pk (X3:2qk (r)) = klim s ;p (X p +1 (r)) = s3 (A2 (r)) !1 3 k 3 k

k!1

(since the sequence (s3;pk ;1 (X3pk +1 (r)))k

0

is a subsequence of (s3;n (X3n (r)))n ). 20

From (54) and (55) follows

H (s2;qk (X3:2qk (r)) s3;pk (X3:2qk (r))) = 2q1k H (X3:2qk (r) s2qk 3;pk (X3:2qk (r)))  1 2qk

therefore

j1 ; 32pqkk jdiamX3pk (r)  6qkd 

A2 (r 1 3) = klim s ;q (X q (r)) = klim s ;p (X p +1 (r)) = s3 (A2 (r)) !1 2 k 3:2 k !1 3 k 3 k i.e., the sets A2 (r 1 3) and A2 (r) are anely equivalent, which is impossible since they are not

homeomorphic. With similar arguments it follows that the sequences a = b = (mn )n 0  m  2 are not scaling sequences for the polynomial r(x) = 1 + x 2 IF2 x] in the case the natural number m is not a power of 2. This is a geometrical counterpart of the result of Allouche et all, 1], that the sequence of ;  n binomial coecients ( k mod 2)nk is m-automatic if and only if m is a power of 2. 3. Polynomials in k-variables All results of this note also hold for polynomials r 2 IFq x1  : : :  xk ]. Then the rescaled evolution sets are fractal subsets in the k + 1-dimensional Euclidean space IRk+1 . For the standard scaling seqiences a = a = (qt )t 0 and the parameters a = 1 =    = s = 1 this is proved in 6]. For example the standard rescaled evolution set A2 (r) of the polynomial r(x1  x2 ) = 1 + x1 + x2 2 IF2 x1  x2 ] is the Sierpinski pyramid, 14]. 4. Hausdor dimension of rescaled evolution sets does not depend on scaling sequences The squeezing trick of S. Willson, 23], applied in section 8 implies: if a = b = (a(n))n scaling sequences and

then

0

are

Aa (r ) = nlim !1 sa(n);1 (Xa(n) (r ))

dimH Aa (r ) = dimH Aq (r  1): S. Willson proved this for s = 1 1 = 1 with the following argument. For every n choose the natural number kn such that qkn  a(n)  qkn +1 , then and

Xqkn (r )  Xa(n) (r )  Xqkn +1 (r )

sq;kn;1 (Xqkn (r ))  sa(n)q;kn ;1 (sa(n);1 (Xa(n) (r )))  sq;kn ;1 (Xqkn +1 (r )): (57) By choosing a suitable subsequence if necessary we may assume that limn!1 q;ak(nn;) 1 = . By taking

limit in (57) we obtain

which implies the assertion.

sq;1 (Aq (r ))  s(Aa (r ))  Aq (r ) 21

10 Open questions Let r1  : : :  rs be polynomials with the coecients in the Galois eld IFq and let 1  : : :  s be positive real numbers. Is the Hausdor dimension (1  : : :  s ) = dimH Aq (r  1) a continuous function on  = (1  : : :  s ) for a xed r = (r1  : : :  rs )? Is there a simple formula for the Hausdor dimension of dimH Aq (r  1)?

22

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