Characteristic quantities of multifractals-application to the Feigenbaum attractor

Physica D 54 (1991) 75-84 North-Holland

Characteristic quantities of multifractalsapplication to the Feigenbaum attractor Z. K a u f m a n n Institute for Solid State Physics, EiitclJs University, P.O. Box 327, H-1445 Budapest, Hungary Received 7 June 1991 Accepted 4 July 1991 Communicated by B.V. Chirikov

Quantities characterizing multifractals additionally to their asymptotic scaling properties are introduced, namely the generalization of the reduced R6nyi information and the decay rate of dimensions calculated by the ratio trick. They are applied to the Feigenbaum attractor, in which case they are universal functions. Asymptotic formulas are given for the dimensions and the multifractal spectrum of the 2~ cycle for large Iql by taking into consideration certain secondary scaling factors.

1. Introduction and summary

transition, it diverges in some cases even when

D(q) has only a break [8]. Calculating the generalized dimensions D(q) [1-5] has been proved to be a powerful method in describing multifractal properties in systems which have a characteristic measure. The generalized entropies [1, 6] are the appropriate quantities if time series of a quantity are studied. The often used static and dynamical multifractal spectra [4, 7] are closely related to the dimensions and entropies, so they offer an other possibility for describing the same properties. The dimensions can be conceived as the linear divergence rate of the R6nyi entropies as a function of the logarithm of the cell size in case of uniform partition. The constant part of this linear asymptotic law, called reduced R~nyi information (Iq. D), provides further details about multifractal properties [1, 8]. The nonanalyticities in the spectrum of dimensions, called phase transitions, are also reflected in Iq, D as function of the parameter q. In fact lq. D is more sensitive to the presence of a phase

Since the fractal investigated in this paper is the Feigenbaum attractor let us now turn to the dissipative systems that show an infinite sequence of period doublings [9, 10]. We know that for long period 2 n the map describing the return from the vicinity of a point of the cycle into itself is a 2 n times iterated map, therefore it possesses very strong dissipation and is very close to a onedimensional map. In the limit n ~ o0 the system is governed by a one-dimensional map, therefore the investigation of one-dimensional maps is of great importance. In the accumulation point of the period doublings these systems have a special attractor called 2 ~ cycle. In case of quadratic maximum a universal attractor, the so called Feigenbaum attractor, can be obtained by enlarging a small piece of the 2 ~ cycle around the maximum. The universal function g(x) of the 2 ~ cycle describes the corresponding return map [9, 10]. In this case the dimensions and the multifrac-

0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. All rights reserved

76

Z. Kaufmann /Characteristic quantities of multifractals

tal spectra of the 2 ® cycle are also universal [4]. Even the decay of the dimensions calculated on finite partition by the condition F = 1 follows a universal law and is related to the spectrum of dimensions [11, 12]. The aim of the present paper is to introduce quantities that describe such properties of the fractals that are not determined by the spectrum of dimensions or scaling indexes. The first such characteristic quantity is a generalization of Iq, o. Nonuniform partitions as compared to the uniform one are generally more natural and easier to handle not only in the case of the 2 = cycle but also in general [2, 5]. Therefore Iq, D, which applies uniform partition, is generalized here for the case of nonuniform partitions. To emphasize the difference we keep the notation Iq, o when the partition is uniform while the generalized quantity in other cases will be denoted by Jq, D" The generalized expression is the most natural one in the sense that in case of absolutely continuous measure R6nyi's integral formula [1] in terms of the density remains unaltered and correspondingly Jq, o agrees with Iq, 9. The quantities exp(Jq, D) change like D(q)-dimensional volumes for linear transformations. Especially, exp(J0, D) is suggested to be conceived as the D(0)-dimensional volume of the fractal. Jq, D of the Feigenbaum attractor is a universal function in the sense that it can be determined using any map that has a quadratic maximum and shows an infinity of period doublings. It is calculated numerically and in the limits q ~ ___~ also analytically. These results are presented in section 2. An other characteristic function is introduced in section 3 as the decay rate p(q) of the effect of finite partition on the dimensions when they are calculated by the ratio trick method (ref. [4], section Ill.B). It is analogous to the entropy decay rate [13, 14]. For the case of the Feigenbaum attractor, when it is also a universal function of q, it is determined analytically for q ~ _ oo and numerically in between. Attempts at obtaining the same decay rate from dimensions satisfying F = 1 are also discussed.

The behavior of the dimensions D ( q ) of the 2 ® cycle at the accumulation point of period doublings is studied for q ~ + ~ in section 4. Asymptotic formulas are obtained for D ( q ) and for the static scaling function f ( a ) . While the leading term of D ( q ) is determined by the primary scaling factors apD , -1 apD-2 the correction is shown to be related to certain secondary scaling factors. The appendix contains details of the calculation for the 2 ~ cycle used in sections 3 and 4. The current approach describes finer details of the set than an approximation with a two-scale Cantor set.

2. Reduced R6nyi information As was mentioned in the introduction, lq, O is a useful quantity in the description of a multifractal. However, it is defined only for uniform partitions [1, 8]. In this section it is generalized to nonuniform ones. First Iq, o should be introduced. When the probabilities Pi belong to the cells of a uniform partition the Shannon entropy is N

S= - ~Pilnpi

.

(1)

i~l

Its generalizations are the R6nyi informations [1, 8]

1 Iq = ~----q l n E p q = S

i f q ~ 1, if q = 1.

(2)

These quantities depend strongly on the size of the cells. Characteristic quantities of the system that are independent of the partition can be extracted from them in the following way. The R6nyi informations Iq(1) for cells of equal size l are usually asymptotically linear in In I for small l, therefore order-q dimensions are defined in the

Z. Kaufmann /Characteristic quantities of multifractals

The straightforward generalization of the reduced R6nyi information to general partitions is

form [1, 2]

Io(l)

D(q)

77

= lim t--,0 - I n 1"

(3) 1

Jq, o= lni~m=i - - q lnF(q,~'(q),K,,),

(10)

In most cases the limit

Iq ' o

= lira [Iq(/) /~0

L

+D(q)lnl],

(4)

which is the constant part of the asyrnptotical linear form for Iq, also exists and is called reduced R6nyi information [1, 8]. At the same time 1o, o characterizes the convergence of (3):

where K. = {l~.)}/u=(~,)for n = 1, 2 . . . . are the partitions and lim. __.®max i 1}") = 0. Jq. n preserves two important properties of the quantity Iq, n. First, in case of an absolutely continuous measure, (10) leads to the following integral form as it is shown for Iq. o in refs. [1, 8]:

Ja, o=

Iq(l) lq.o -In/ =O(q) + -In/

1

fab[e(x)lqdx,

(11)

(5)

In many systems nonuniform partitions are more natural and more simple to handle. Therefore the reduced R6nyi information should be generalized for such partitions. Let l i, i = 1, 2 . . . . . N be the sizes of the cells. It is known that the generalized dimensions can be determined by the investigation of the partition function [4] N

r ( q , r, {/i}) = E p IU.

(6)

where P(x) is the density corresponding to the measure and [a, b] is the base interval. Accordingly, Jq, o = lq, o. In case of fractal measures this is not true in general. The other feature is that exp(Jq, o) (and, of course, exp(lq, o)) behaves like a D(q)-dimensional volume when the set is enlarged by a factor r: exp(Jq~,°D) =

r °(q) exp( Ja.o )"

(12)

i=1

Infinitely refining the partition this function becomes a step function:

r(q,z,[l,})=oo

lim

ifz>~-(q),

m a x l i ---, 0

=0

if ~"< z ( q )

(7)

and the position of the jump is related to the dimensions [4]: ~'(q) = ( q - 1 ) D ( q )

(8)

It is obvious that in case of uniform partition

Iq ' o

= lim n--,~

Especially for q = 0 it can be conceived as the D(0)-dimensional volume of the fractal, since exp(J0, o) = lira, _.® F(0, ~'(0), K~) is additive and it is especially the D-dimensional volume in case of a nonfractal set:

1 1-q

(9)

N

V = lim E lg = lim F ( 0 , z ( 0 ) , Kn). n~i=

1

(13)

n~

To calculate Jq, D for the case of the Feigenbaum attractor we can use any map with quadratic maximum that possess a 2 ® cycle as attractor. The interval of the nth partition that contains the position x 0 of the maximum should be taken and the part of the attractor lying in it should be determined. This can be done simply following the k × 2nth iterates (k = 1,2 . . . . ) of x 0. The

Z. Kaufmann / Characteristic quantities of multifractals

78 0.04

dq,D

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

0.05

0.02

2.502907875 is the universal scaling factor. With the normalization g(0) = 1 the size of the attractor is L = (apD + 1)/apD. Obviously for large q in the nth partition

r(q,r, K.)

2-nq(

--,-"

C+L)

2n,

~PD"

(15)

0.01

Then 0.00

r(q) -0'01-

1'5 . . . . . '-' ~'6 . . . . . . 2 5 . . . . . . . . (~. . . . . . . . .

g ........

iO . . . . . . . q

enlargement of t h e set obtained this way is almost identical to the Feigenbaum attractor even for moderate values of n. Therefore Jq, o calculated for this set is very close to Jq, o of the Feigenbaum attractor. This procedure was done numerically using the logistic map, the result is shown in fig. 1. Rapid convergence was found when increasing n. The set was rescaled here to unit length. Since the obtained function is independent of the map it is a universal function in addition to the dimensions in the universality class of maps with quadratic maximum. In contrast to D(q), Jo, o is found not to be monotonous. The limits for q --* +_oo can also be obtained analytically using the universal function g(x) for mapping. The nth approximation of the attractor can be given by the 2 ~ interval between the first 2 "+l iterates of the maximum point x 0 = 0. The measures of these intervals are equal: pi = 2 - " . The partition function (tr) is dominated by the contribution of the smallest (largest) interval for q --* o, ( - oo). The lengths of these intervals scale as [9] -- X l [ = c + L a ~ 2n

1(2) ~ [ X2n+' -- X2n ] = C _tOlpd,

In2

(16)

q21napo

i5

Fig. 1. Numerical results for the generalized reduced information of the F e i g e n b a u m attractor w h e n it is rescaled to length l = 1.

l(ni)n -- [ X z . + l

=

(14)

follows, and after rescaling to L = 1,

Jq, o(K) = q

1J®o,

In2 J=, o = - - 2 I napD In c + = 0.03308359559.

(17)

Similarly for large negative q:

Jq, D = J-=,D = O,

(18)

while for q ~ oo the convergence is like 1/q for q ~ -oo exponential decay is found. It should be mentioned, while lq, D characterizes the convergence of dimensions on uniform partitions, Jq, o is related to convergence on the nonuniform partitions. The dimensions calculated by the condition [4]

F(q,%(q),K.)

= 1,

(19)

%(q) = ( q - 1 ) D n ( q )

(20)

follow the asymptotics [11, 12]

D,,(q)-D(q)

Aq I n

= -,,

L

- L(q)

(21)

if the size L of the set is close to a certain L(q) (in the case when Pi = m - ~ with some integer m this restriction in L is not needed). L(q) is expressed in form [12]

1

where c _ = 1, c + = 5lg"(O)lapD/(apD + 1) = 1.09152875t, Xk is the k t h iterate and a p t ) =

q

L(q) =

( l i m F ( q , ' c ( q ) , r . , L = l ))"" , n--ooo

(22)

Z. Kaufmann / Characteristic quantities of multifractals

where it is indicated that F should be calculated in case L = 1. From (22) and (10) it follows that

Using (24)

ln( p~n)/p~'-I)) q ( q - 1)Dn*(q ) = ln(lln)/lln-l))

L

Ja, D = D(q) In

79

(23)

L(q) '

[ In PI

= ~~ from which it can be seen that Jq. O contains the L-dependence of the amplitude of (21).

3. Dimension decay rates An other characteristic function can be extracted from the convergence of the dimensions that are determined from the condition [4]

F ( q , r , Kn) = r ( q , z , K . _ , ) .

(24)

If r (or the dimension) is expressed from (24) in function of q it is denoted by r*(q) (D*(q)) and the inverse function is q*(r). It is enlightening to show the main features of the methods using (19) and (24) on a general case when the most concentrated (or the most rarefied) interval scale as

c3P~ ( 1 +

(25)

c4p" ) .

(27)

Here c 5 and c 6 are constants depending on P~, Lt, c I and c3. In (27) an exponentially fast convergence is obtained instead of the 1/n-like in (26). Similar behavior can be expected for moderate q values. More can be proven for multifractals similar to the Feigenbaum attractor in which increasing the level n the intervals of the partition are subdivided into the same number of nonempty cells, say m, and their measures are equal. Then N = m ~ and Pi = m-n. The partition function separates into a product: mn

/ ' ( q , r , Kn) =m-nq E (l~")) -~ i=1

=--m-"qo'n(r).

(28)

Now it is easier to determine q in function of ~'. From (19),

Zln) - ClL'~(1 + c2pn), p~n) =

)

+ c6P" q.

q'(~')=

In o',,(r) nlnm '

(29)

and from (24), Such a law is found when some transformation generates the fractal and its eigenvalues determine the asymptotic behavior as for the case of the 2 ® cycle does the renormalization transformation [9, 10]. The contribution of the interval (25) dominates (6) for q >> 1 (or q << - 1). Then the result on the basis of (19) is

(q - 1)Dn(q)

In p~n) In l~n) q

q.(~.) = ln(tr,,(~')/tr,,_,(r)) In m

(30)

Instead of (25) exponential correction is assumed for F at the exact r(q) value

F(q, r(q), x.) -- F=(q) (1 + cp")

(31)

for large n. Then In F®(q(7)) + cp" q n ( r ) = q(~') +

--- [ In L 1 +

q"

(26)

nlnm

(32)

Z. Kaufmann /Characteristic quantities of multifractals

80

and

0.2

q~(f) =q(f)

- - C ( I __p)pn--I + ~ , ( # g )

(33)

0.1

f)q

+ +

+ ÷ + +

+

+

0.0

follow, where @ denotes the order of the correction and Ip21 < Ip[, Ip31 < Ipl. In (33) the 1/n convergence of (32) is improved in agreement with (27). It is worth noting that the convergence (33) can also be obtained by fitting a linear function of 1/n to q,(r) and extrapolating to 1/n = O. For precise values of ql('r),q2(r),...,qn('t) the best result is found by fitting to the last two of them:

(l('t)=nqn(r)-(n-1)qn_,(r

)=q*(z).

(34)

t~(~-) is at the same time the slope of the line fitted to nq, in function of n. The form of (34) is similar to the construction of truncated entropies [13, 14] but the role of q and r is exchanged. It should be emphasized that q*(r) can be expressed in form (34) only for cases in question, when Pi = m-". In the other method used in ref. [11], starting from eq. (19), a linear fit is applied to the D,(q) values. But the result obtained this way typically has a term decaying like 1/n z. To show this fact the simple case can be taken when the lengths are equal and I~ ") = c , L ~ ( 1

+ c2p" ) .

Then

D,(q)

=

In m - I n L I - ( l n c I + c2P")/n ln m c7 @ [ 1 ~ -lnL-------~+ n + [ )~-7

(35)

follows. The linear fit removes the term proportional to 1/n but the next one, ~ ( 1 / n 2) remains. Special results are proven in the appendix for the 2 = cycle of one-dimensional maps with quadratic maximum that have period-doubling sequence. With a detailed investigation the follow-

-0.1

+ +

-0.2 +

-0.3 +

-0.4

+

4"

+

+

+

-0.5

q Fig. 2. The numerically calculated decay rate of D*n(q) for the 2~ cycle of a map with quadratic maximum.

ing decay is proven:

D*(q)-D(q)~(-apD)-"

ifq<<-l,

O*(q) - D ( q ) ~ ap~"

if q >> 1.

(36)

Similar convergence,

D*(q)-D(q)

~ [p(q)]",

(37)

is found numerically also for smaller ]ql values. For a map given by small perturbation of g(x) the fitted values of p(q) are shown in fig. 2. They are in correspondence to (36): p ( - o o ) = - 1 / a e D , p(oo) = 1/Ot2D. AS p ( + o0) is shown to be universal in the appendix, p(q) is also expected to be universal for maps of quadratic maximum following period-doubling sequence. This can be supported by the following arguments. Evidently the deviation of (30) from q ( r ) is also proportional to [p(q)]" for the given map. Since the scaling properties of the attractor are preserved when changing the map the interval lengths change by a factor that smoothly depends on the position. This way terms of cr are multiplied by approximately the same factors as the corresponding terms of ~r,_ 1. The small deviations between the factors give corrections to the ratio in (30), which sum up according to the asymptotical structure of

Z. Kaufmann /Characteristic quantities o f multifractals

the attractor and lead to a certain asymptotic form for n ~ oo in case of typical changes of the map. If the caused change Aq* of q* is asymptotically larger than q*(r)-q(z) then Aq* determines the new value of p(q). In this case the original map was a special one possessing an untypical value of p(q). For typical maps Aq* has faster or equal decay than q*(~')-q(r), therefore p(q) does not change. The invariance property of p(q) was checked numerically for conjugations in case of a two-scale Cantor set.

The results of the appendix give also approximate expressions for the exact dimensions of the 2 ® cycle:

q1

q -

D(+oo)

Iri--------2- / 3 -

for q ~ + 0%

(38)

where

D(-~)

In 2 = In a p D ' 1

/3-=l[g-

'

0.8 i ......... i ........ ,IL, ...... ,i ........ ,i,

. . . . . . .

,i

. . . . . . . .

Dq O.

7 l

i

t

0.6

~ ~j

0.5 0.4 0.3

-~5

-~o

-s

o

s

.... ii$ ....... is q

Fig. 3. Comparison of the precise numerical results for the dimensions of the Feigenbaum attractor (solid lines) and the asymptotic formulas (38) (dashed lines).

4. A s y m p t o t i c s for q --* +

D(q)=oq_~lD(+oo) _ _

81

In 2 D ( ~ ) = ap-----~' 21n

It is conjectured that scaling factors Of higher orders and corresponding corrections to D(q) and f(a) can also be extracted. Comparing to numerical results expression (38) proved to be surprisingly precise, up to 2 digit s for Iql > 3 and 4 digits for q < - 6 . 5 and q _> 8 (see fig. 3). In the two-scale approximation of the attractor (with scales P1 = P2 = 3, 1 L I - l/aeD, L2 = 1/a2D) apD±I appears in place of/3 ± in (38). Therefore (38) also reflects finer details of the structure. From (38) asymptotic forms for the f(a) spectrum [4] follow:

1

( x ) ] Ix=0 = 2.244914699

a n d / 3 + = Ig"(0)l = 3.055265994. For q --+ +oo one of the terms of F dominate, therefore D(+oo) are determined by the extreme scalings [4] (14). When decreasing Iql the intervals whose size is close to that of the dominant one (secondary intervals) play important role. /3± is .just their typical ratio to the dominant interval in the limit n ~ ~. It is more specific to say that 13± is the asymptotic ratio of the secondary interval farthest from the dominant one when the universal map Xk+l=g(Xk) is used (see the appendix). In this respect apD and a2D are primary scaling factors determining the first term of D(q) and /3 ± are secondary ones responsible for the second term.

f(a)

= -In(

ln2

e D 2 ( + w ) In/3+ a - D ( ± ~) × D( -t- ~) In/3 +

[a-D(-i-=)]) (39)

if a---D(+oo). It can be seen that f(a) at its endpoints is a function like - x In x, similarly to the two-scale Cantor set, but the coefficients differ from that of the two-scale approximation.

Acknowledgements

I would like to thank P. Sz6pfalusy for valuable discussions. This work was partially supported by

82

Z. Kaufmann /Characteristic quantities of multifractals

the Hungarian Academy of Sciences under Grants No. AKA 28-3-161 and OTKA 819.

where n

(41)

= 1+ E

j=O

Appendix Approximating expressions will be derived here for the D * ( q ) dimensions of the 2 ® cycle for large positive and negative q. First the q << - 1 case is investigated. Normalization of the map can be fixed with the place and value of the maximum fmax = f(0) = 1. It is well known that the attractor can be approximated with the (Xk, Xk+2,) intervals for k = 1,2 . . . . . 2 ", which lie between the first 2 n+~ iterates of the maximum point x 0 = 0. (The endpoints of the intervals are not necessarily written in increasing order.) The nth partition K, is formed from these intervals. Since these intervals are visited periodically they have the same measure Pi = 2-n" Then (30) can be used. For large negative r the relatively large intervals dominate F and also or, therefore they should be sorted according to their approximating sizes. For this purpose it is sufficient to consider the approximation with the two-scale Cantor setwhose scales are - 1/aVD and 1 / a 2 D . The largest interval is situated around the origin, A (') = (X2n , X2n+l) ,

whose size scales as A (n) ~ ap-i~. The next intervals in order of magnitude are

Using the connection between the endpoints of B) ~) and A ('), their ratio can be expressed as

B)"'/A'"'---14,;(O) + a:l, rej(xr)

A~ ) =

-

X2n -- X2n+l

(43)

Here ~by(x) = f(-2J)(x), the inverse of the 2 j times iterated map, and the branch of the inverse is taken on that f(2J)(0) lies. T is a transformation subtracting the linear part of the function: T F ( x ) = F ( x ) - F(O) - F'(O) x.

The ratio (43) can be estimated using the properties of the renormalization transformation. Its first few eigenvalues, largest in absolute value, are [9, 15] 6,--apD, 1 , - 1/apD. The first eigenmode does not play a role in the stage when the 200 cycle exists. The second would appear only if the place of the maximum would be shifted. The third one is absent if the normalization is properly done with an extra factor c, then after j renormalization steps the fourth eigenvalue determines the order of the correction: [(--OlpD)J/¢]f'2')(C(--OlpD)-Jy) --~g(y) + (--O/pD)-JU(y),

B) n) _~ (x2n_2j, X2n+l--2J ) ,

(42)

(44)

j = 0 , 1 . . . . . n - 1, and inverting it,

B)') ~ ap-~-'.

~j(X) -- C( --O/pD) -J[ g - ' ( y ) From (30) it follows that

2q*(~)= [ A ~n) 1 -~ ton(~") ~ A (n-1) J

~n_l(q')'

+ ( - - a p D ) - i q J ( y ) ] y=t(_~pDV/ClX, (45) (40)

where u(y) and qJ(y) are smooth and not strongly varying functions, and the right branch of g-1 is

Z. Kaufmann /Characteristic quantities of multifractals taken. This implies

4,;.(o) = -/3_+

(-a~o)-%'(0),

(46)

13_=[[g-'(x)]'lx= o.

Since f(2') is assumed to have a unimodal form in the neighborhood of the origin and its images, even for j = 0

~;(o) < 0.

(47)

From (44), for y = 0,

Substituting into (40), (41), the first term of (51) appears both in oJ, and to,_ l for every j but only in ton for j = n - 1. The sum in (51) contributing to t% also appears in the (B)" T l)/A("-o)-, term of to,_ i. The exception is the case j = 0, but then the sum of (51) is of order ~ ' ( ( - a e o ) - " ) . Finally, 2q*")=a~,O(1 +/3_-')[1 + ~ ' ( ( - - a e o ) - " ) ] is obtained. From this formula,

(48)

(53)

follows. @ indicates the asymptotic behavior of the correction. Then from (43), (45) and (48),

and inverting it,

A~.n)=

O~*(q) =

apD

((

--apD)n-J

X[Tg-l(yn)- Tg-X(y.+l) ] + ( --apD)"-2"/[ T 0(Y.)

- T

q -- (In 2) - l f l - o t - = ) q q-1 O(-~)

+ C_(q) (--apD)-",

0(Yn+,)] )

X [1 + @ ( ( - - a p o ) - " ) ] ,

In2 D ( - o o ) = In O/pD'

(54)

(49)

with the notation y,=[(--apD)J/C]X2,. Since TO(y) starts at y = 0 as a quadratic function the terms containing TO give a contribution of order (--apD)-n to (49): A~n )

(52)

q,(~.) = r In apD + ~-~" --. ln2 +C(r)(-apt)) ,

X2..=C(--apD)-n[I+~'((--apD)-")]

apD + 1

83

where C(¢) and C_(q) are suitable factors. It can be seen from (42), (46) and (50) that B)")/A ¢") fl_, when simultaneously n ~ 0%j ~ oo, n - j ~ oo. Moreover, if the map is the universal one Xk+ l = g ( X k ) , then 0 = 0, therefore limn_. ® B¢om/ACm=

fl_.

a PD

A similar calculation can be done for large ~-. Then the very small intervals dominate in F and or. The smallest one is

apD + 1 ( --apo)'-J[Tg-I( y")

-Tg-'(y.+,)] + @((--apD)-" ) = A(on-J) + t ~ ( ( - - a p D ) - n ) ,

(50)

z(n) = ( X2n + l, Xl) ~ap-D2n, and the order of A~:) is (--apD):-". Using (42), (47), (50) and (46),

and the next ones are

[,:.,/A,°,]

~(")= (x2.+l +2-x, +2,) ~ a ~ "+', j=0,1 ..... n-1.

=

i - a
+ : ( ( - .pD)-°).

(51)

A modified renormalization transformation must be used that applies a magnification with scale a2D around the x = 1 point. Its eigenvalues are ~ , a ~ D , l , 1 / a E D . . . . . Its fixed point function is

84

Z. Kaufmann /Characteristic quantities of multifractals

d e n o t e d by

G(x). T h e result is

q - ( I n 2 ) - - 1 / 3 ~ O(°°)q

D*( q) =

q- 1

D(oo)

+ C+(q) ap~ ~, where /3+ = IG'(1)I = Ig"(0)l a n d l n 2 / 2 1 n apo. In this c a s e Yjtn)/Ztn) -~

(55) D(oo)=

fl+ in t h e

limit n ~ oo, j --. oo, n - j ~ oo and for the m a p Xk+ I

=

G(Xk ) , lim,, ._,~ Yo~)/Z ~) =/3+.

References [1] A. R6nyi, Probability Theory (North-Holland, Amsterdam, 1970), appendix. [2] H.G. Hentschel and I. Procaccia, Physica D 8 (1983) 435.

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