The infinite convolution product of the µn converges in the weak* sense to a probability measure µ on [0, 1] which is known as a coin tossing measure [9], µ =n=1∗∞ µn

VOL. LXIX 1995 FASC. 1

A MULTIFRACTAL ANALYSIS

OF AN INTERESTING CLASS OF MEASURES

BY

ANTONIS B I S B A S (IRAKLION)

1. Introduction. Let µn = pnδ(0) + (1 − pn)δ(1/2ⁿ), n = 1, 2, . . . , where pn ∈ [0, 1] and δ(x) denotes the probability atom at x. The infinite convolution product of the µn converges in the weak* sense to a probability measure µ on [0, 1] which is known as a coin tossing measure [9],

µ =_n=1∗^{∞ µn.}

Let x = P∞

n=1εn(x)/2ⁿ, where εn(x) ∈ {0, 1}, be the 2-adic expansion of x ∈ [0, 1]. It is not difficult to see that if

dνa,N =

N

Y

n=1

(1 + anrn(x))dλ, N = 1, 2, . . . ,

where a = (an)n≥1, λ denotes the Lebesgue measure, rn(x) = 1 − 2εn(x) is the nth Rademacher function and pn= (1 + an)/2, then

N →∞lim νa,N = µa

in the weak* sense and µ = µa (see also [12]). So we have two ways to describe the same measure. In this work we shall use the second way. The characterizations of the sequences (an)n≥1which give continuous or singular measures are given in [5], [6], [9], [11]. In a previous work [4] we have proved that

lim inf

n→∞

log µa(En,k(x))

−n log 2 = δa µa-a.e., where

δa = 1 − lim sup

N →∞

1 N log 4

N

X

n=1

log[(1 + an)^1+an(1 − an)^1−an]

1991 Mathematics Subject Classification: Primary 28A78.

Key words and phrases: Hausdorff dimension, multifractal, Rademacher Riesz pro- ducts.

[37]

and En,k(x) is the segment [k/2ⁿ, (k + 1)/2ⁿ) containing x, for some k = 0, 1, 2, . . . , 2ⁿ− 1. From this relation we deduce that µ_a is δa-dimensional [8] and dim µa = δa, where dim µa = inf{dim E : µa(E) = 1} and dim E denotes the Hausdorff dimension (HD) of the Borel set E (see [1]). If there are infinitely many c ∈ R such that dim Ec> 0, where

Ec=



x : lim inf

n→∞

log µa(En,k(x))

−n log 2 = c

 ,

then we say that µa is multifractal [8], [10]. We have seen [2], [3] that some special cases of Markov measures are multifractal. In Section 2 we shall give a necessary and sufficient condition for µa to be multifractal under the condition sup_n|a_n| < 1. In Section 3 we give an application which permits us to give a lower bound for the HD of a set Mβ(b), where

(1) Mβ(b) =



x : lim inf

N →∞

1 N

N

X

n=1

βnεn(x) ≤ b

 ,

b, βn∈ R, β = (βn)n≥1, |βn| ≤ M , M > 0. In some special cases our method gives equality.

2. A multifractal analysis. We need the following lemma, which can be deduced from [4]:

Lemma 1. Let γ = (γⁿ)n≥1 and µγ be the corresponding coin tossing measure. If sup_n|a_n| < 1, then

lim sup

N →∞

1 N

N

X

n=1

log (1 + anrn(x))

= lim sup

N →∞

1 2N

N

X

n=1

log [(1 + an)^1+γn(1 − an)^1−γn] µγ-a.e.

Theorem 1. If the sequence a = (aⁿ)n≥1 is such that sup_n|an| < 1, then µa is multifractal if and only if

lim sup

N →∞

1 N

N

X

n=1

|an| > 0.

P r o o f. (i) Suppose that lim sup_{N →∞}(1/N )PN

n=1|a_n| = 0. Then by the Cauchy–Schwarz inequality we have equivalently limN →∞(1/N )PN

n=1|a_n|²

= 0. Since µa(EN,k(x)) = νa,N(EN,k(x)) = 2^−NQN

n=1(1 + anrn(x)) and Ec =



x : 1 − lim sup

N →∞

1 N log 4

N

X

n=1



log(1 − a²_n) + rn(x) log 1 + a_n 1 − an



= c

 ,

using the uniform convergence of the Taylor series for the function log(1+x),

|x| ≤ sup_n|a_n|, we see that E_c = ∅ if c 6= 1 and Ec= [0, 1] if c = 1 and so µa is not multifractal.

(ii) Suppose that

lim sup

N →∞

1 N

N

X

n=1

|a_n| > 0.

It is clear that c must be such that

(2) 1 − lim sup

N →∞

λN ≤ c ≤ 1 − lim sup

N →∞

κN, where

λN = 1 N log 2

N

X

n=1

log(1 + |an|) and κN = 1 N log 2

N

X

n=1

log(1 − |an|), otherwise the set Ec is empty. We define the function

f (y) = lim sup

N →∞

[κN + y(λN − κ_N)], y ∈ [0, 1].

If 0 ≤ y0< y ≤ 1, then using the properties of lim sup we obtain 0 ≤ f (y) − f (y0) ≤ (y − y0) lim sup

N →∞

(λN − κ_N).

This implies that f (y) is continuous on [0, 1]. Since f (0) = lim sup_{N →∞}κN

and f (1) = lim sup_{N →∞}λN, from the intermediate value theorem there is γ0∈ (−1, 1) such that f ((1 + γ₀)/2) = 1 − c ∈ (f (0), f (1)), f (0) ≤ 0 < f (1).

We consider the measure µγ, where γ = (γn)n≥1 with γn = γ0sgn log 1 + a_n

1 − an



(sgn is the sign function, sgn 0 = 0). From Lemma 1 we have lim sup

N →∞

1 N log 4

N

X

n=1



log(1 − a²_n) + rn(x) log 1 + an

1 − an



= lim sup

N →∞

1 N log 4

N

X

n=1



log (1 − a²_n) + γ0log 1 + |an| 1 − |an|



= lim sup

N →∞



κN +1 + γ0

2 (λN − κN)



= 1 − c µγ-a.e.

From this we get µγ(Ec) = 1 and so dim Ec≥ dim µ_γ = δγ > 0, for infinitely many c ∈ R (f (1) > 0). This means that µa is multifractal.

3. Application. We consider the set of (1). It is clear that we can find a sequence a = (an)n≥1 such that

βn = log 1 + a_n 1 − an

 , with sup_n|a_n| < 1. We also have

N →∞lim 1 N

N

X

n=1

|βn| = 0 ⇔ lim

N →∞

1 N

N

X

n=1

|an| = 0.

If limN →∞(1/N )PN

n=1|β_n| = 0, then dim M_β(b) is 0 if b < 0 and is 1 if b ≥ 0.

Suppose that lim sup_{N →∞}(1/N )PN

n=1|a_n| > 0. From (1) we see that b must be such that

lim inf

N →∞

1 N

N

X

n=1 βn<0

βn ≤ b ≤ lim inf

N →∞

1 N

N

X

n=1 βn>0

βn,

or equivalently, (3) lim inf

N →∞

1 N

N

X

n=1

log 1 + an

1 + |an|



≤ b ≤ lim inf

N →∞

1 N

N

X

n=1

log 1 + an

1 − |an|

 , Otherwise dim Mβ(b) = 0 or 1.

Let b > 0 and

c = 1 + b

log 2 − lim sup

N →∞

1 N log 2

N

X

n=1

log(1 + an).

Using elementary properties of lim sup, lim inf and (3) we easily see that c satisfies (2). From the proof of Theorem 1 we have

dim Ec ≥ dim µγ, where

Ec=



x : 1 − lim sup

N →∞

1 N log 4

N

X

n=1

[log (1 − a²_n) + (1 − 2εn(x))βn] = c



and γ = (γn)n≥1, γn= γ0sgn βn with lim sup

N →∞



κN +1 + γ0

2 (λN − κ_N)



= 1 − c.

If x ∈ Ec then

lim inf

N →∞

1 N

N

X

n=1

βnεn(x) ≤ b

and so Ec⊂ M_β(b), which means that dim Mβ(b) ≥ dim µγ.

R e m a r k. If βn > β06= 0, n = 1, 2, . . . , b > 0, using the above method we get

β0= log 1 + a0

1 − a0

 , c = 1 + b

log 2− 1

log 2log (1 + a0), γn= γ0sgn βn

and

1 − c = log (1 − a0) log 2 + β0

log 4(1 + γ0).

This gives

b β0

= 1 − γ0

2 . Since

Mβ(b) =



x : lim inf

N →∞

1 N

N

X

n=1

εn(x) ≤ b β0



and

dim µγ = 1 − 1 log 2

 b β0

log 2b β0

 +

 1 − b

β0

 log

 2

 1 − b

β0



, using Eggleston’s Theorem [7] we get dim Mβ(b) = dim µγ for b ≤ β0/2.

REFERENCES

[1] P. B i l l i n g s l e y, Ergodic Theory and Information, Wiley, New York, 1965.

[2] A. B i s b a s, A note on the distribution of digits in dyadic expansions, C. R. Acad.

Sci. Paris 318 (1994), 105–109.

[3] —, On the distribution of digits in triadic expansions, preprint.

[4] A. B i s b a s and C. K a r a n i k a s, On the Hausdorff dimension of Rademacher Riesz products, Monatsh. Math. 110 (1990), 15–21.

[5] —, —, On the continuity of measures, Appl. Anal. 48 (1993), 23–35.

[6] J. R. B l u m and B. E p s t e i n, On the Fourier transforms of an interesting class of measures, Israel J. Math. 10 (1971), 302–305.

[7] H. G. E g g l e s t o n, Sets of fractional dimensions which occur in some problems of number theory , Proc. London Math. Soc. (2) 54 (1952), 42–93.

[8] A. H. F a n, Quelques propri´et´es des produits de Riesz , Bull. Sci. Math. 117 (1993), 421–439.

[9] C. C. G r a h a m and O. C. M c G e h e e, Essays in Commutative Harmonic Analysis, Springer, Berlin, 1979.

[10] J.-P. K a h a n e, Fractals and random measures, Bull. Sci. Math. 117 (1993), 153–159.

[11] G. M a r s a g l i a, Random variables with independent binary digits, Ann. Math.

Statist. 42 (1971), 1922–1929.

[12] R. S a l e m, On singular monotonic functions which are strictly increasing , Trans.

Amer. Math. Soc. 53 (1943), 427–439.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF CRETE

IRAKLION 71409, GREECE

E-mail: BISBAS@TALOS.CC.UCH.GR

Re¸cu par la R´edaction le 7.6.1994