cellular automata and multifractals: dimension spectra of linear cellular automata

$: Cellular automata and multifractals: Dimension spectra of linear cellular automata$
Physica D 45 (1990) 36-48 North-Holland

C E L L U L A R A U T O M A T A A N D M U L T I F R A C T A L S :

D I M E N S I O N S P E C T R A O F L I N E A R C E L L U L A R A U T O M A T A

Satoshi T A K A H A S H I

Department of Biophysics, Kyoto University, Kyoto 606, Japan

Received 9 January 1990 Revised manuscript received 26 April 1990

The dimension distribution of space patterns of linear cellular automata are studied. A dimension spectrum derived from the dimension distribution is shown to correspond to a singularity spectrum of a multifractal. Using this relation, the dimension spectrum is efficiently calculated.

1. Introduction

Cellular au toma ta are systems consist ing o f lat t ice

points. Each lat t ice poin t (cel l ) has a state with dis-

crete t ime steps by a local rule. A number o f relat ion-

ships between l inear cellular au toma ta and fractals

have been given by Willson [ 11-15 ]. He showed that

a series o f space - t ime pa t te rns o f a pr ime-s ta te l inear

cellular au tomaton converges to a fractal, which is

called the l imit set. He also devised an efficient com-

puta t ion method for the Hausdor f f d imens ion o f the

l imi t set. Previously, I ex tended Wil l son ' s result to

more general l inear cellular au tomata and showed the

re la t ionship between the t ransi t ion matr ix (def ined

by Wil lson [ 14 ] ) and the structure o f the l imi t set

[191.

In this paper , we concentra te on the d imens ions o f

space pa t te rns o f a l inear cellular au tomaton . In sec-

t ion 3, we define a d imens ion spectrum, which rep-

resents the d is t r ibut ion o f the d imens ions o f space

patterns. The not ion o f mult i f racta l is useful to com-

pute the d imens ion spectrum. The relat ion between

the d imens ion spect rum of a l inear cellular au toma-

ton and the singulari ty spec t rum of the correspond-

Present address: Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi, Osaka, 558, Japan.

ing mult i f ractal is shown in section 6. Using this re-

lation, we present an approx imat ion method for

calculating the d imens ion spectrum of l inear cellular

au toma ta (Markov weight t ransformat ion approxi-

ma t ion ) in section 8, and the efficacy o f this me thod

is shown in section 9.

2. Preliminaries

In this section, we int roduce some results o f my

previous work on l inear cellular au toma ta (LCA)

[9] .

Let 7/, ~, and d be the set o f integers, the set of nat-

ural numbers and a natural number , respectively.

Cons ider a d-d imensional lattice, each site o f which

is referred to as a cell. i~_ d indicates the locat ion o f a

cell and t e ~ denotes the t ime. Each cell is specified

as an integer in the range 0 through M - l, where M

is a natural number , represent ing the number o f the

states. We denote by a ~ e { 0, ..., M - 1 } the state o f the

cell located at the point i and t ime t. Cons ider a

ne ighbourhood o f cell i, {i+rl, ..., i+ r , , } , where

( rb . . . , rm ) e ( Z d ) " is called a neighbourhood index. I f

the states of the cells at t ime t - 1 de te rmine the states

of the cells at t ime t by the following transition rule,

0167-2789/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

S. Takahashi / Linear cellular automata and multifractals 37

a~ t - - I t - - I =ctat+,t +...+cmai+rm m o d M , (2.1)

where c~, ..., Cm are natural numbers, then we call this

system an M-state linear cellular automaton (abbre-

viated M-state LCA ). We consider the space-time pattern of an LCA in

a (d+ 1 )-dimensional Euclidean space and define the

limit set of this space-time pattern as follows. Let t, be a monotone series of natural numbers, and

S( t , ) be a subset of the ( d + 1 )-dimensional Euclid-

ean space defined by

S( t , ) ={( i , t) l t< t , , a ~ 0 } .

S (t , ) represents the space-time pattern of the LCA by successive steps up to time t . - 1. Multiplying each element of S ( t , ) by 1/tn, we obtain a contracted set S( t , ) / t , , i.e. (i / t , , t / t n ) eS ( t , ) / t , , when (i, t ) eS( t , ) .

If there exists a t ime sequence t,, for which S ( t , ) / t, converges to a set, we call that set the limit set of

the LCA. The following assumption is made for the initial

states of cells:

a ° = l i = 0 ,

= 0 otherwise.

Main results of my previous work [ 9 ] are summa-

rized as follows:

Theorem I. Consider a pk-state LCA, where p is a prime and k is a natural number. I f t is divisible by pk- I, then a ~ =a~. I f t is divisible by pk and at least

one of the elements of i is indivisible by p, then a ~ = 0.

Theorem 2. For a pk-state LCA, if we take t, = p ' , then

the limit set lim,_o~ S ( t , ) / t , exists.

Theorem 3. The limit set of a pk-state LCA can be represented as a union of some members of a family of sets X i which have the following property: Each Xj is composed of n , 1/p-scaled Xq's. I f a transition matrix is defined by A= ( n , ) , then the Hausdorff dimension of the limit set is given by log#, where 2 is the maximum eigenvalue of the matrix A.

The following example clarifies the ideas of this

theorem.

Example 1. A one-dimensional 2-state LCA with a

transition rule,

a~=a~Zl +a~-l +a~+~ m o d 2 .

Figs. l a - l e show the sets S(2n) /2 n for n = 1-5, re-

spectively, which converge to a limit set (see fig. l f ) . As shown in fig. 2, the limit set is composed of two

triangle patterns designated X, if the symmetric figures on the right and left halves are regarded as equivalent. X is composed of two l /2-scaled X's and one 1/2-scaled square pattern designated Y. Y is composed of four l/2-scaled X's. Therefore, we have

n i i = 2, n 12 ~--- 4 , n 21 = 1 a n d n 2 2 = 0 , and the transition

matrix is given by

;) Since the maximum eigenvalue of the transition matrix is 1 + x/~, the Hausdorff dimension of the limit

set is given by log ( 1 + x /~) / log 2.

3. Definition of dimension spectrum

Consider a limit set of an LCA. From the definition of the limit set, the time coordinate of the limit set lies between 0 and 1. We denote the time of the limit set by z~ [0,1 ]. Let us cut the limit set perpen- dicular to the time axis at r and consider the dimension of the cutting surface (space pattern of the limit

set). A dimension spectrum H(~) of the limit set is de-

fined by the dimension of the set of time z's at which the space patterns have a dimension 6. The dimension spectrum is calculated by the following

procedure. We first count the number of non-zero cells of the

LCA at t ime t:

N( t ) = # {il a,'. q: 0}, (3.1)

where # denotes the number of the elements of a set.

38 S. Takahashi I Linear cellular automata and multifractals

1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(a) (b) (c)

1 , ' , , 1 1 1 , ~ : ~ ,

1 1 1 11 1 1 I , : : , , : , , : : ,

1 1 I 1 1 1 1 1 1 ,~, , ' , , ' ] ' 1 1 1 1 1 . . . . . . . . , , , , , , ,

1 1 1 . . . . . .

1 1 I ~ 1 1 1 1 , I '

1 1 1 1 1 1 1 1 1 , I

(d) (e) ( f )

Fig. 1. S(2")/2" for the LCA with the rule a[ =a~=~ + a [ - l +a[+ l rood 2. 1 l:epresents the points in set S(2")/2". They converge to the limit set. (a) n = i, (b) n=2, (c) n = 3 , (d) n=4, (e) n = 5 , ( f ) limit set.

(a)

(b) Fig. 2. The limit set of the LCA with the rule ai-aj_lt- t-, +air-l +a~+lt-i mod 2

x /x x /x

Is composed of two patterns X and Y as shown in (a) and (b) , respectively. X is composed of two 1/2-scaled X's and one 1/2-scaled Y, and Y is composed of four 1/2-scaled X's.


We then take the time series t, for which a limit set of the LCA exists, i.e. the series of the sets S ( t , ) / t , converges. Let us define the following quantity:

logN(t ) D ( t , t , ) - - - f o r t < t , . (3.2)

log t,

Thus the dimension spectrum H(5 ) is given by

H(5) = l im lim ,4(& t,, ¢ ) , (3.3)

where,

3(& t., ~)

= log # { t l & - ¢ < D ( t , t .) < 5 + E } (3 .4 ) log t .

Above procedure gives H(~) from the following reason. Consider a space pattern of the limit set at an arbitrary time z ( e [0,1 ] ). For a given n, take a time t (n ) such that

t (n ) t ( n ) + l - - < - C < - - t, t,

Then we can cover the space pattern at t ime z by N ( t ( n ) ) boxes

{ ( z ' , i ' ) l t ( n ) / t , < z ' < [ t ( n ) + l ] / t , ,

( i , - r ) / t , <i" < ( i s + r ) / t , , i = (il , ..., id) ,

i ' = ( i i , ..., i D , a~ ~") # 0 } ,

where r is the range of neighbourhood and is given by max{ I r~ I ..... I r,,I }. As n tends to infinity, the dimension of the space pattern at t ime z is evaluated by

lim l o g N ( t ( n ) ) = lira D ( t ( n ) , t . ) = ~ . .~oo log tn n ~

We next consider an interval [t i t , , ( t + l ) / t , ] (t=O . . . . , t . - 1 ) and assign D(t, tn) to that interval. The argument of the logarithm in the numerator of the r.h.s, of (3.4) indicates the number of the intervals in which D(t , t , ) lie between ~ - ¢ and ~+¢. As n tends to infinity, D(t , t , ) approaches the dimension of the space pattern and ,4(& t . , ~) converges to the dimension of the set ofz ' s at which the dimension of the space pattern lies between t~-e and ~+¢. At the limit of ~ to zero, H(O) gives the dimension of

the set of z's at which the dimension of the space pattern is &

Note. To prove eq. (3.3) rigorously, we need the law of large numbers and theorem 14.1 in ref. [2].

We omit the proof due to limitations of space and

time.

4 . C o m p u t a t i o n o f H ( 6 )

Example 2. Consider a 2-state LCA satisfying the fol-

lowing rule:

a~=a~:]+a~+] m o d 2 .

Taking the time series t, = 2", we have the limit set of this LCA as shown in fig. 3. This limit set is called Sierpifiski gasket, a typical fractal figure.

The dimension spectrum of this limit set is given

by

- ~ l o g ~ - ( 1 - ~ ) log( 1 - ~ ) H(t~) = (4.1)

log 2

The derivation of (4.1) is given as follows. We can show from theorem 1 that

N(2t ) = N ( t ) (4.2)

and

N ( 2 t + 1 ) = 2 N ( t ) . (4.3)

Let t= Y.~-~ tj2 j, where t j=0 or 1 (i.e. t = t , _ l t , _ 2 ...to

1"=1/7 z=1/5 "r=1/3

~=1/2

.•.•.r.. & = 1/3 & =1/2

Z& Z&':' A A A A

Fig. 3. The limit set of the LCA with the rule a, fa,_; '-' +a j+,'-1 rood 2. J indicates the dimension o f the space pattern at t ime z.

40 S. Takahashi / Linear cellular automata and multifractals

in diadic expansion). Then by iterating (4.2) and (4.3) we have N( t )=2 xtj. Let m= Y~tj and ~=m/n, then from (3.2), we obtain D(t, 2") = 6. Substituting this relation into (3.4), we have

log.C. A(~, 2", e ) - - -

n log 2

for sufficiently small e. Applying Stirling's formula leads to

A(a, 2", c ) = -c~ log 6 - ( 1 - ~ ) log( 1 -c~)

log 2

for sufficiently large n and for sufficiently small ~. Thus we obtain (4.1).

5. Multifractals and singularity spectra

Singularity spectrum f ( a ) represents the distribution of measure on a multifractal [ 5-7 ]. To correlate the dimension spectrum of an LCA with the singularity spectrum of a multifractal, here we consider a set with weight in stead of a measure. In a set with weight, total weight varies depending on the way of division of the set.

Consider a set which is divided into N pieces. Let l~ and mi be the size and the weight of the ith piece, respectively. If the set has a uniform structure to which a uniform measure is assigned, then m~ is pro- portional to l~, where D is the dimension of the set. In analogy, we define the singularity of the ith piece,

oti, by

m,~l~' (l~--,0). (5.1)

ot~ indicates the dimension of the ith piece in the limit of It to zero.

Assume that all the pieces have the same size/, and the distribution of ot~'s has the following form:

# {i la < ai < o t + d a } ~ d a l -ft~) . (5.2)

In the limit of I tending to zero, f (o t ) represents the dimension of the set of points with a singularity a. Thus we c a l l f ( a ) singularity spectrum.

Direct numerical calculation off(or) by using (5.2) is a formidable task. However, the calculation can be

considerably simplified by introducing a partition function [ 5 ]:

r(q , z, l )= ~ mq/P. (5.3)

Assume that in the limit of l to zero, the following relation holds:

mq ~ l r(q) , (5.4)

where z(q) is given by

r(q) =sup{rl lim F(q, z, 1)=0} 140

=inf{rl lim F(q, z, l )=oo} . (5.5) /~0

On the other hand, from (5.1) and (5.2), we have

mq ~ J da l-Y~'~)l~ (l--.0) . (5.6)

Because l is very small, only a(q) that minimizes q a - f ( a ) contributes to this integral. Therefore we have

df(ot) =q (5.7) d a -=-cq) "

Then, comparing (5.4) and (5.6) yields

z( q) =qa( q) - f ( a( q) ) . (5.8)

Combining (5.7) and (5.8), we have

dr(q) dot d f ( a ) dot dq =a+q-~q d a dq = a " (5.9)

Therefore, if we obtain z(q) from ( 5.5 ), f ( a ) is calculated by using (5.8) and (5.9).

Notice that an approximate value of z(q) is usually calculated from the equation F(q, z(q), 1) = 1 for a finite l ( < 1 ) with Ym~= 1. If the measure is distributed in a self-similar manner, this procedure gives an exact z(q), because F(q, z, l") is given by F(q, z, l)" (see ref. [ 5 ] ).

Example 3. The multifractal corresponding to the LCA of example 2 is generated by the following procedure (see fig. 4). Take a unit interval and assign weight l to it. Divide it into two same-sized intervals and assign weight 1 to the left interval and l /2 to the right interval. Continue infinitely the same process


I

1

1 1

[ z ( q ) = l o g ( l - a ) log2 " (5.13)

Substituting (5.12) and (5.13) into (5.8), we have the singularity spectrum as

1 L I

1 1 • t 2 i I

1 1 1

1 1 1 1 1 1

21412141418 I

Fig. 4. The multifractal of example 3. Number assigned to each interval indicates the weight of that interval.

of dividing intervals and assigning the weight of the previous interval to the left half and 1/2 of that value to the right half. Since this multifractal has a self-similar structure, the singularity spectrum is calculated by letting (5.3) be equal to unity at first division, thus we have

( 1 / 2 ) q + 1 = ( 1 / 2 ) ~.

Taking logarithms of both sides, we have

r(q) = - log[ ( 1 /2)q+ 1 ] log2 (5.10)

From (5.9), a is given by

(1 /2 ) q a = (172)q+ 1 , (5.11)

or

log a - - l og ( 1 - a ) q= - log2 (5.12)

Using (5.11 ), (5.10) is rewritten as

- a log a - ( 1 - a ) log( 1 - a ) I f(ot) = log 2

The same result as above can also be obtained by the method used in section 4.

6. Dimension spectra and singularity spectra

We calculated the dimension spectrum H(~) for the case of example 2. However, it is not possible to calculate analytically dimension spectra for general LCA. In this section, we show how to construct a multifrac-

tal whose singularity spectrum f ( a ) is the same as the dimension spectrum H(d ) of a given LCA.

Consider a LCA in which a set S ( t , ) / t , (defined by (2.2) ) converges to a limit set for a time series t,. To generate a multifractal corresponding to this limit set, we take a unit interval [0, 1 ], and divide this interval into t, equivalent intervals. Assign weight mt = l / N ( t ) to the interval [t / t , , ( t+ 1 ) / t , ] , where N ( t ) is the number of non-zero cells at time t in the LCA as defined in (3.1). In the limit of n tending to infinity, we have a multifractal.

Relation (5.1) for this multifractal is written as

1 / N ( t ) ~ ( 1 / t ~ ) ~ . (6.1)

Comparing (3.2) and (6.1), we can see that D(t, t ,) in (3.2) corresponds to a in (6.1). It thus follows from definition (3.4) and (5.2) that the singularity spectrum of this multifractal is given by the same function as the dimension spectrum of the LCA.

The relationship between the dimension spectrum of an LCA and the singularity spectrum of the corresponding multifractal is summarized in table 1. Fig. 5 shows the relation between the limit set of the LCA given in example 2 and the corresponding multifractal shown in example 3. Using this relationship, we


Table 1 The relationship between the parameters of the dimension spectrum and the singularity spectrum.

Dimension spectrum 1/t. 1 / N ( t ) D ( t, t . ) ~ H ( J )

Singularity spectrum I m log m/log I a f ( a )

I I 1 1 /2

_1_ 1/2 /2 _1_1/4

A A I 1/4

-1.1/2 ~,~ J,. 1 /4 ,,8' ,4 -1-1/2 ~ 1/4

1/4 J~ 1/4 ~,,

1/8 -1-1/4 .L 1/8 1/1"

Fig. 5. Correspondence between example 2 and example 3.

will find a good approximation of H(~) for general LCA as will be shown in section 8.

7. Partially self-similar limit sets

E x a m p l e 4. In general, the limit set of an LCA con- sists of some elementary patterns, which in turn, comprise contracted patterns of those elementary patterns. This type of fractal structures is called to be partially self-similar [4], while fractals composed of single elementary patterns are called to be self-similar (see fig. 3 for an example of a self-similar set).

As an example of a partially self-similar set, we consider the LCA in example 1, whose limit set is composed of two elementary patterns X and Y (see fig. 2). To obtain the dimension spectrum of this LCA, we divide the elementary patterns, X and Y, into upper and lower halves by a horizontal center line. The upper part of X contains one 1/2-scaled X, and the upper part of Y contains two 1/2-scaled X's (fig. 6a). The lower part of X contains one 1/2-scaled

X and one 1/2-scaled Y, and the lower part of Y contains two 1/2-scaled X's. Let Ao (A l ) denote a matrix whose ijth (i, j = 1, 2) element (Ao)~j ((A,)u) is given by the number of the 1/2-scaled ith pattern contained in the upper (lower) part of the jth pattern, where the ith pattern represents X for i= 1 and Y for i= 2. Thus we have

Ao=(10 20), A ,= ( l l 20). (7.1)

Using the matrices Ao and A ~, the number of non- zero cells at time t, N(t) , as defined by (3.1) is given by

N ( t ) = ½cAt~Atz. . .Atsil , (7.2)

where u = (2), e= ( 1 2) and t=tsts_t.. .tl in diadic expansion. A brief sketch of the derivation of (7.2) is given below. This derivation is similar to that given by Willson [ 14 ].

Suppose that a neighbouring pair ofceUs (say i and i + 1 cells ) at time t are at state (a ~ a ~+ ~ ) = ( 1 0 ). We can see from theorem 1 that t 2ta2t 2t Ka2i ~2i+1 a2 i+2)

S. Takahashi I Linear cellular automata and multifractals 43

(a)

Xx l Yx l

(b)

Xx2

Fig. 6. Numbers of the 1/2-scaled X's and 1/2-scaled Y's contained (a) in the upper part of X and Y, (b) in the lower part of X and Y.

= ( l 0 0) . At the next t ime 2t + 1, 2t+ l 2 t+l ~2t+l'~ is ca lcula ted as (1 1 0) (fig. 7a) . a 2 i a 2 i + l t~2i+2 /

I f we call a neighbouring pa i r o f cells a block, the

above result can be res ta ted that a block ( 1 0) at t ime

t yields one ( l 0) at t ime 2t and one ( l 0) and one

( 1 1 ) at t ime 2 t + 1, when we count blocks al lowing

overlaps between them. Similarly, a block (1 1 ) at

t ime t yields two (1 0) at t ime 2t and two ( l 0) at

t ime 2 t + 1, i f we regard block (0 1 ) as equivalent to

block (1 0) , because o f the symmet ry o f the transi-

t ion rule (fig. 7b) . Therefore we see that the ijth ele-

ment o f the matr ix Ao (A ~ ) in (7.1) equals the num-

ber o f block i at t ime 2t ( 2 t + l ) generated f rom block

j at t ime t, where block 1 represents block ( 1 0) and

block 2 block ( l l ). Let us now focus on the config-

ura t ion at t ime t and count the numbers o f blocks

2kl 2i 2i+1 21"+2 2i+3

i /'+1 2t 0 ~ 0

1 0 2t+1 (10)x l (10) x l (1 1)xl

(a)

21"-1 2i 2/+1 2i+2 2i+3 i /+1

2t 0 ~ 0 (10)x2 1 1 2t+1 ~ (10)x2

(b) Fig. 7. Blocks ( 1 0) and ( I 1 ) at time t determine the blocks at time 2t and 2t+ 1. (0 1 ) is regarded as the same block as ( 1 0). (a) block (1 0), (b) block (1 1).


( 1 0) , (0 1 ) and ( 1 1 ), allowing overlaps between

different blocks. Take a vector v whose first element

represents the sum of the numbers of the blocks ( 1 0)

and (0 1 ), and the second element represents the

numbers of the block ( 1 1 ) at t ime t. Then Aov and

A lV give the numbers of blocks at t ime 2t and at t ime

2 t + 1, respectively. Let a counting vector c = (1 2) ,

which represents the numbers of l ' s contained in each

block, and let a unit vector u = (2), which indicates

the number of blocks at t ime 0. Then for a t ime

t=tj~_l...t~ in diadic expansion, N(t) is given by

½cAt At2...Atsu, where the coefficient ~ is introduced

to cancel the effect due to overlaps of blocks.

Finally we show why the matrices Ao and A t origi-

nally defined for the transit ion of the patterns X and

Y (fig. 6 ) also specify the transit ion of blocks. Recall

the procedure of constructing the limit set o f the LCA:

we take the space- t ime pat tern up to t ime 2" and

contract it by a rate 1/2 ". Choose an arbi trary block

( 1 0) in a contracted space- t ime pat tern of t ime 2 ".

We focus on the square region under this block (see

fig. 8a). This square region is occupied by a pattern

1 0 0

1 l 0

at t ime 2 "+ 1 . Iterating the same procedure for each

block in the square region, the pattern of 1 in the

square region converges to the 1/2"-scaled X. Simi-

larly, a square region under a block ( 1 l ) becomes

the pattern Y (fig. 8b). Therefore the blocks ( 1 0 )

and (11 ) in the space- t ime pat tern up to 2 ~ t ime

steps precisely correspond to 1/2~-scaled X's and Y's

in the limit set. Thus the number of 1/2~-scaled X's

and Y's contained in the region between two lines

z = ( t + l ) / 2 " and z = ( t + l ) / 2 ~ is given by

AtlAtu...At~u.

0 1 0 0 0 o 1 0 o o 0 o .

o . . . . . . . . . . . . . . . . . . . . . . . . . . .

l : . . . . . . . . . . . . . t . . . . . . . . k . . . . . . . . . . . . i :

i , i ~i ~ii ili i!~ii iiii iili i!

(a)

0 1 0 1 0 o , 0 0 o ! o . . . . . . . . . . . . . .

I 0 1 o

.......... t ) : t

)

(bl

Fig. 8. Block ( 1 0) generates the pattern X, and block ( 1 1 ) generates the pattern Y. Figures correspond to S(t)/t, S(2t)/2t, S(4t)/4t, S(8t)/St from left to right.


It has been shown in the theory of random matrices that IogN( t (n ) ) / l og t , defined by (3.1) converges to a certain value 50 for almost all time sequence [ 3 ]. Therefore almost all space patterns take a dimension 8o. For the case of example 2, space patterns almost certainly take a dimension 1/2, since

H(tf) < 1 ((f# 1/2). In general, however, it is diffi- cult to obtain rio analytically.

8. Markov weight transformation approximation

We have shown in secion 5 that if the limit set of an LCA is self-similar, then the weight of the corresponding multifractal is distributed in a self-similar manner, and hence the dimension spectrum of the LCA is calculated exactly (see example 3 ). This does not apply to partially self-similar limit sets like example 4, because the weight is not distributed self- similarly in the corresponding multifractals.

When we divide an interval into two intervals for constructing the corresponding multifractal, the transformation of weight is achieved by multiplying the matrices Ao o r a l given by (7.2).

In this section, we devise an approximation method to calculate the dimension spectra of partially self- similar limit sets of LCA. We explain the approximation method by applying it to example 4.

Consider N(t ) defined by (7. l ). Let j =j~j~_ t...Jl in diadic expansion. The N(j) is given by

NO') = ½cAj, Aj2...Ajsu. ( 8.1 )

(8.1) is approximated by

(l) N(j)~-b½cAj, Aj~...Aj, 1 , (8.2)

where b is given by

b = ( 1 1 )A j, ...Aj~u (1 1) .G. . .AjAI) "

This approximation is valid for large s, since direc- tions of Aj, ...Ainu and A j, ...Ai~( ] ) approach each other as s increases (see fig. 9 ):

1) for large s . (8.3) Aj~...Aj, u_~Aj,...Aj, l

Let i= isis_ l...il in diadic expansion. Then N(t) at t= i+2s j is given by

N ( i T 2•) = ½CiilJi2 ...Ai~A),Aj2 ...Ajsu . (8.4)

By using relation (8.3) again, we approximate (8.4)

as

1 1 N(i+2V)~-B,jb~cAimA;v..Ai~(l), (8.5)

where B o is given by

Bij = ( 1 1 )Ai, ...Ai, Aj, ...Ai,(l ) (1 1 )A,...Aa(1)

Similarly N( k + 2 si + 2 z~j) is approximated by

N(k+2q+22"j)~-Bk, Bob½CAklAkv..Ak,(1), (8.6)

where k=k~ks_ 1...kl in diadic expansion and Bk~ is given by

Bki = ( 1 1 )Ak, ...Ak~Aa ...A,( I ) ( 8.7 ) ( l 1 )&, . . .A~( I )

Since b½cAaA~ :...AiAI ) is bounded for any i, only the product by B o contributes to the dimension of a space pattern, 6, and hence H(6) . Therefore, it is reasona- ble to assume that the ratio of N 0 + 2 s t ) and N(i+2~j+2zst) for any t is given by Bij.

Recalling the relationship between the number of the elementary patterns of an LCA and the weight of the corresponding multifractal as shown in table 1, we define a 2~× 2 ~ weight transformation matrix M whose element m~j is given by the inverse of B~j. We construct a multifractal using this weight transformation matrix by the following procedure.

Consider a unit interval, which we label as 0 (singularity spectrum f ( a ) of the interval does not de- pend on the first choice of the label). Divide the interval into 2 s intervals with length l= (1 /2) ~ and assign to the jth interval [/'/2 s, j /2~+l] =- [lj, lj+l] ( j --0 .... ,2 s - 1 ) weight mjo and label that interval as

j. The jth interval is further divided into 2 ~ equiva-


Y

X

- A 1 A o

_ A 1 A 1

AoAo AoA1

X

Y AI AoAo A1AoA1

A1A1A1

A1A1Ao

oAoAo oA1Ao

X AoAoA1 AoA1A1

Y

AIAoAoAo A1AoAoA~ A~AoAIAo A1AoA1A1

A~AIA~Ao

• : A1A1A~AI

~ AoAoAI Ao AoA1A1Ao X AoAoAIA1 AoAIA1A1

Fig. 9. Operation of multiple products of the matrices A0 and AI confines the first quadrant to lines, if the sequence of A~ contains at least one Ao, and the dotted region converges to a line as multiplicity increases.

lent intervals. We assign to the ith interval [Ij+ (i/2s)l, l j+[ ( i+ l ) /2S] l ] ( i = 0 .. . . . 2 ~ - 1 ) weight

mijmjo and label that interval as i. Repeating this procedure, we obtain the desired multifractal in the limit (see fig. 10). The weight transformation in this multifractal is not self-similar, and depends on both label i and label j like a Markov process. Therefore we call this weight transformation a Markov weight transformation.

The partition function of this multifractal is obtained by using a partition matrix, F(q, z), whose ijth element is given by F#=m~/ l ~, where l= 1/2 ~. The partition function with 2 '~ equivalent intervals is given by

( 1 1 ...1 ) (F (q , r) )"

From the definition of z (q) given by (5.5), we have z(q) such that the maximum eigenvalue ofF(q, z(q) ) is unity. Substituting z(q) for (5.8) and (5.9), we obtain f (o t ) , which approximates the dimension

spectrum H ( 6).

9. Simulation results

Here, we demonstrate efficacy of the Markov weight transformation approximation. We first calculate the dimension spectrum H(6) of the LCA given in example 4. Fig. 1 1 shows the result of the calculation of A(6, 2", (). These graphs approach a real H(6) as we take more time points t,. However, they converge very slowly and still remain far from H(6) , even if we take millions of time points.

I f we use the Markov weight transformation approximation, even the second step (i.e. s = 2 in sec-


1 (o)

F

moo/~o

moo m~ o (o) (1)

I I moo~m,o mo l~ rn , ,

moomoo rn~ornoo mo, m,o m.m~o (0) (I) (0) (1)

I I I I I

Fig. 10. A multifractal generated by the Markov weight transformation.

t ion 8) gives a good approx imat ion of H ( ~ ) and it

seems to converge almost complete ly at tile th i rd or

fourth step (fig. 12a). The Markov weight t ransfor-

ma t ion approx ima t ion gives quite good results for

o ther pr ime-s ta te LCA (figs. 12b, 12c).

We also calculate the d imens ion spectrum of a pk.

state LCA, where p is a pr ime and k is an integer

greater than I. In this case the Markov weight trans-

fo rmat ion approx imat ion does not converge rapidly,

as shown in fig. 12d. However , we can show that the

dimension spectrum of the correspondingp-sta te LCA

that evolves according to the t rans i t ion rule (2. l )

with m o d p instead o f m o d p k gives the same d imen-

sion spect rum as that o f the original pk-state LCA.

Singulari ty spectra o f mult i fractals have often been

calculated on the assumpt ion tha t the measure is dis-

t r ibuted in a self-similar manner as discussed in sec-

t ion 5. We compare this convent ional me thod (here

HI5] HI6] 1 l

(a) {s c5

(b)

/' 'x L . . . . . L\

(a)

H[8] 1

/ 1 i 5

(b)

Fig. 11. Direct calculations ofzt(t~, 2% c) of the LCA in example 1. n= 15, 17, 19, 21. As n increases, the peak of the bell-shaped curves tend to shift upwards. (a) ~=0.1, (b) ~=0.05.

He8] H[8] 1 l

(c) I c5

(d)

Fig. 12. Calculation of H(t~) by the Markov weight transformation approximation, s=2-5 in (a), (b), (c) and s=2, 5, 8 in (d). (a) a~=a~-l+a~-t+a~¥~ rood 2; (b) a~=a~El+a~ -~ +2a~7-I mod 3; (c) a~=a~-l+a~-~+a~¥~ mod 2; (d) a~= a ~- ~ + a ~+ ~ mod 4. In ( d ), the graph shifts to the left as s increases.


HI:6]

6

Fig. 13. Calculation of H(c~) by the self-similar weight transformation approximation for the LCA in example 1. s=4 , 8, 12, 16. The graph shifts to the left as s increases. Calculation by the Mar- kov weight transformation approximation at s = 5 is also shown with a dashed line, which takes the correct value 0 at c~= 1.

called the self-similar weight transformation approximation) with the Markov weight transformation approximation. To calculate the dimension spectrum of an LCA by using the self-similar weight transformation approximation, the weight of the ith interval,of the sth step is defined by the inverse of the number of non-zero cells at time i, i.e. m~=N(i) -I. z(q) is given by Y~mq/lT= 1, where l= 1/tn= 1/2 s. In fig. 13, the results from the self-similar weight transformation approximation at s=7 -16 and the Markov weight transformation approximation at s=5 are shown. For small c~, the error of the self-similar weight transformation approximation is not large, while the error is notable for large c~. Convergence of H(c~) with respect to s is also not rapid in the self-similar weight transformation approximation. Therefore the Mar- kov weight transformation approximation is more suitable to calculate dimension spectra of LCA.

Acknowledgement

I am very grateful to Dr. Nanako Shigesada for her careful reading of the manuscript and many helpful suggestions.

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