C. R. Acad. Sci. Paris, t. 325, Sbrie I, p. 1163-1168, 1997

Analyse mathCmatique/Mathematical Analysis

On the influence of coverings in multifrac t al analysis

Abstract. We study the influence of coverings on the local dimensions 01‘ positive measures. Open balls and I/-adic intervals are considered. We show that both coverings yield the same local dimension. except possibly on a set of Hausdorff dimension 0. As an application. we study a measure whose multifractal spectrum is covering dependent.

Sur 1 ‘influence des recouvrenzents en nnnlyse nzultifractale

Version frangaise abrdgt!e

Soit ,U une mesure positive dkfinie sur [O. l[. La dimension locale de 1’ au point I peut &tre dkfinie

de diffkrentes faGon”. Les thkoriciens considbent gCnCralement les dkfinitions suivantes :

a(t) := lim log //z(B(t, ;)) ~ - log~“(r~(l>-‘)) (I( I) := lim

c+o - log(2E) ’ c-0 log(2c)

oti B(t. C) est une boule ouverte centrke en t et de diamktre 2~. Lorsque fk(t) = i?(l), la valeur commune est appelke ditnension locale de /I au point f (voir [2] et [3] pour une justification de cette dkfinition). Cependant, en pratique, on utilise plus gCnCralement d’autres dkfinitions :

(Y(t) := b log p( I” (f )) - log ,/(1”(f))

(Y(f) := lirri --, I! log k” . ,l log k”

Note prksent6e par Paul DEHELWLS.

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Ici 1’1 est un entier, 1) > 2 et I”(t) I’unique intervalle b-adique semi-ouvert B droite de longueur h-” contenant t.

Une question naturelle concerne 1’Cquivalence entre ces definitions (les rCsultats obtenus ont permis de construire des mesures extirieures de spectre de Hausdorff prescrit, voir [4]). On constate saris peine que la masse prise en compte dans le calcul d’une dimension locale d6pend de la nature du recouvrement (notamment aux points b-adiques).

Dans ce qui suit, nous montrons que les dtfinitions sont tquivalentes pour tous les reels t. sauf peut-&tre sur un ensemble de dimension de Hausdorff nulle. Cet ensemble est composC des reels de [O. l[ dont le dCveloppement b-adique contient une proportion de 0 et de h - 1 6gale ?I 1 (on ne considi?e que les dCveloppements qui se terminent).

Nous donnons un exemple de mesure de probabiliti dont les dimensions locales ne sent pas identiques sur tout le segment [O. l[, et montrons les cons6quences que cela entraine sur les spectres de Hausdorff.

Le but principal ici est de partir d’une boule ouverte et de I’encadrer par deux intervalles h-adiques, la difficult6 risultant dans la determination de tels intervalles dont les diamktres sont proches de ceux de la boule. A I’inverse, partant d’un intervalle b-adique, on cherche & encadrer celui-ci par des boules dont les diamktres sont proches de celui de I’intervalle (c’est par exemple impossible si t est un rCel h-adique, car aucune boule ne sera contenue dans I”(f) B partir d’un certain rang 1)).

Soient f := c t,b-’ (si 1 est h-adique, on Mectionne le dCveloppement qui se termine) et />O

p,+(f) := lini,, r/,-l card{,; < 71; f, = X:} la proportion de X: dans le developpement de I (lorsque celle-ci existe). Un rtsultat dfi h Billingsley (voir [I]) affirme que tlim(l E [O. I[ : VA.(~) existe et vaut l} = 0 pour tout 0 < k < h (dim designe la dimension de Hausdorff).

Notons 13 et Z,] les classes formCes. respectivement, des boules ouvertes et des intervalles b-adiques. DCfinissons les dimensions locales iuf et snp de la mesure /I, relativement au recouvrement par boules ouvertes et par intervalles O-adiques :

oti I”(t) est, rappelons-le, l’intervalle dyadique de diamktre h-“, contenant t. Les spectres de Hausdorff de I/ sont d6finis par .f’(ck, a) := dim E(rr, a) et f((.y,Z,) := dim E(~L,&,).

Posons N(h) :=’ {t := C t,b-’ ; pa(t) < 1 et pt,+-] (t) < l}. On a tlimN(b)” = 0 (le symbole ’ ;>1

dCsigne le complCmentaire). Les rCsultats principaux sont les suivants :

THI~OR~MI: 1. - Soient b, c deu.x entiers, b, c 2 2.

I) Sic = l~~‘pourp E N*. alors g(t.Z,,) = ck(t,B) et (Y(t,&,) = C(t.8) / 7our tout i E [O. l[. Sinew, pour f E N(b) fl N(c), on u fk(Z,Z[]) = g(1:Z,.) et qt,&,) = qt>Z,.).

2) s; t E N(h). on a g(t,Z[>) = (y(f,B) et qt.&) = n(t,B).

Ce rtsultat a une influence directe sur les spectres de Hausdorff de p (spectw multifi-uctal) :

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On the influence of coverings in multifractal analysis

Voici un exemple de mesure dont les dimensions locales varient selon le choix du recouvrement. Definissons les points t~.,~’ := 2Fk’ + 22-r’ pour (X:, p) E N* x N* et la mesure de probabilite 11, := C 2~"{ C 2-11b+, ,!} sur [O, l[.

A,> 1 112 1

Posons D := {2-“’ ; X: E N*} et 7’ := {tb,,, ; (k.p) E N* x N*}. On montre facilement que :

dim T = 0 si f1: = 0. .f(fk> B) = diniD = 0 si 0 = 1, et

--x sinon,

Interessons-nous a present a f(f~.Z:~). Si t $! T Un. alors o(t) n’existe pas. Si t E T, o(t) = 0. Soient f = 2-r;+1 E I) et 71 asset grand tel que J”(t) = [tili , tl,, + X”[ c [2-“.fk+l,l [

(tl,, := 2 t;Y'). Si p. est le plus petit entier p tel que th.,,, > tl,,. on a j,(.]“(j)) = 2-“‘-/‘“+’ r=l

et t - tl,, < ,~(.I”(~)) < 2(t - ti,$). Par exemple, si t = l/2 = c X’ , on obtient I>1

2-'r" < p(P(t)) 5 zr", ce qui ajoute une nouvelle dimension locale au spectre multifractal de 1” : (x(1/2) = 1 Ainsi. E(1.Z3) # ti et E(1.3Y3) c D, ce qui donne ,f(l.Zz3) = 0.

Given a (positive) measure I-/, defined on [O. l), the local dimension of 1~ at point I is defined in various ways in the literature. Theorists usually consider the following detinitions:

c(t) := b ‘(X /“(wt. c)) qt) := E lwP(l3(t. cl)

r-0 log(2c) ' F+o log(2c) -

where B(t. C) is the open ball centered at t and of diameter 2~. When a(t) = ~(1). the common value is called the local dimension of //, at f (see [2] and [3] for instance). Unfortunately, these definitions are not well suited to computational purposes, and the following definitions are used instead:

Here b is an integer, h > 2, and I”(t) the unique b-adic interval semi-open on the right of sidelength b-” which contains t.

The natural question which arises concerns equivalence between these definitions (the answer to this question allows to construct outer measures whose Hausdorff spectrum is prescribed, SPCJ [4]). In the case of balls, all the mass concentrated at the right and at the left of I/, is taken into account, whatever the value of t may be. In the case of h-adic intervals, the proportion of mass taken into account varies with the value of f, the worst case being when t is a b-adic real, where only the mass located at the right of t determines the local dimension of 1” at this particular point.

In the sequel, we show that in fact, the only points where problems may arise are those whose relative frequency of O’s or b - l’s in their h-adic expansion is equal to one. Therefore, both definitions of lower (resp. upper) local dimensions agree except possibly on a set of Hausdorff dimension 0.

We close the paper by looking at the local dimensions of a probability measure at some points of the exceptional set aforementioned, and see how it changes the behavior of the Hausdorff spectrum.

1. Notations and preliminary results

Let b be an integer, 1) > 2. Detine, for all t := c f,F (only terminating expansion are considered), i>O

yk(t) := lirn C1 #{,i < 71,: t; = ,C} (when the limits exists). II

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A result due to Billingsley (.w [I]) asserts that dim {t E [O: 1) ; c+I~-(/) exists and equals 1 } = 0 for all 0 5 k < b.

In order to compare multifractal analysis performed with respect to different coverings, we need come up with results relating open balls and b-adic intervals. Given an open ball. it is easy to tind two h-adic intervals such that one of them is contained in the ball, whereas the second one contains the ball. The main difficulty lies in finding two such intervals whose diameter is close enough to that of the ball. The same problem arises if. given a b-adic interval, we seek for two open balls. one included in the interval and the other one containing it.

The following lemmas are the tools yielding to the main results.

2. Open balls versus b-adic intervals

The capital letters I, .J, and B, respectively, refer to h-adic, c.-adic intervals, and open balls.

LEMh4A 1. - Let ic > 0 und f E [O. 1) nrch thtrt p,,(t) < 1 und pl,- 1( t) < 1. There exist integecs 71,(c) ur7d l(c) such that I”“‘(t) 2 I%(~.F) 5 I@‘(t) cztrd

log(2c) &:, ]()g(j,-uk) = liili

log( 2E) ) c-0 ]()g([J-e)) = l.

LEMMA 2. - La t E [O: 1) and I) E N. Then l”(f) C B(t, !J-“). lf’ir? addition, p()(f) < 1 trnd

~b-~(f) < 1, there exist.s E,, > 0 such that B(t. c,,) & I”(t) with the property lirn h(2c,,) I$ log k”

= 1.

3. b-adic intervals versus r:-adic intervals

Let 0, c be two integers, b, c > 2. The following result is a consequence of Lemma 1 :

LEMMA 3. - Let 71 E N* Cd / E [(I. 1) such thut q+)(t) < 1 m7d 91,~ 1 (t) < 1. Tj7ere exist ir7tegrr.s p,, rrr7d q,, such that .J”” (t) C I” (t) C .I”,# (t) md

4. Multifractal analysis and the influence of the coverings

Let us first set up various definitions of (lower and upper) local dimensions. The sets D and It, denote the classes of open balls and /J-adic intervals respectively.

~(f. a) := ]irrl log t’(B(t’ “’ i E’O log( 2f)

Y(f. l?) := liln ‘““;:g;y, c+o

cv(/,Z,,) := ! ig ‘“yp. - log ,,&

Z(l.&,) := 11n1 II log b-11

Define E(cp.13) := {t; “(1.‘;7) =rY(f.D) = CY} and E(tr:&,) := {t; CP(/,&) = ~(f,&) = CY} and the Hausdorff (or multifractal) spectra j’(~. I3) := (lint E(tu. a) and .f((k. I,,) := (lim E(tu. 2,) (dirrr denotes the Hausdorff dimension).

Finally, define the set I”\’ := {” := c /;/I-‘; PO(~) < 1 and pr,-r(f) < 1). Recall that 121

tlirn N(b)’ = 0 (N(h)’ is the complementary of N(h)). Let t E N(C)) and 0 < E < l/2. Using Lemma 1. we find CY(~,&,) < (t((t. I!) and ii(t.&,) 2 i?(l: n).

along with CV(~.&,) > e(t.D) and E(t.&,) 5 (Y(t. a). Thus we conclude:

LEMMA 4. - Let b. 1’ he two integer-s, b. c 2 2. I) For 011 t E N(h), ~‘e hmv cr(t,Z,,) = cu(t. B) and cu(t.Z,,) = z(t. 23).

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On the influence of coverings in multifractal analysis

5. An example of a measure with two different spectra

Lemma 4 shows that the values of lower and upper local dimensions do not change if they are determined with respect to open balls instead of b-adic intervals. except maybe on a set of Hausdorff dimension 0. The same remark remains valid if we compare local dimensions determined with respect to b-adic intervals and r.-adic intervals (h # c.). In the following example, we compute the local dimensions of a probability measure at some points of :V(2) and i’V(3), and we show that the Hausdorff spectra with respect to balls. Sadie, and 3-adic coverings do not agree.

Define tk.~,, := 22” + 2-“‘-J’ for (X:.;cl) E N* x N* and the probability measure:

/I := c 2-“{ c 2-“hf, ,,}. h being the Dirac mass. A.2 1 1121

on [O. 1). Set U := {2-“‘: X: E N*} and T := {/A~.!,: (k,p) E N* x N*}.

5.1. Computation of f(0. B)

(i) For t $ T U II, th e open ball B(f.,-) does not contain any point of 7 for c small enough, and its measure is null. Thus t?(t) does not exist.

(ii) For t E 7’, the open ball B(t. c) does not contain any point of 7’ other than 1 itself for c small enough. Thus (r(t) = 0.

(iii) For 1 E [I, there exists a (unique) integer X, such that I = 22”+‘. Choose f > 0 such that cc < 22”‘+’ ~ ‘)Y” = 2”‘. Now let boo be the smallest integer p such that f - c < fk.,,,,,, i.6,: %-‘. pI’fl <-

- r < 2p”‘-P1~+‘. We have ,,,(B(t,)) = 2-” c 2-i’ = 2ph.pl)~~~’ and

I’>l’ll ; < j/,(II(f,. c)) 5 2r. which yields o(t) = 1.

The Hausdorff spectrum is given by:

{

dim T = 0 if o = 0. .f(lV. B) = dim D = 0 if 0 = 1.

--;x; otherwise

5.2. Computation of f( CY, IL)

(i) For t E 7 and ‘/t large enough, the interval I”(t) has lower bound t, and does not contain any point of T other than t. Let (X:.1,) be the unique element of N* x N* such that f = tl.,l,. We have ,1(1”(t)) = 22(“+/‘) and (r(t) = 0.

(ii) For t $ T, the interval I”(t) does not contain any point of 7’ for n large enough, and its measure is null. Thus o(t) does not exist.

We conclude:

if II’ # 0. if (Y = 0.

Hence, the spectra .f’(rr.B) and j(ct.&) do not agree.

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5.3. Computation of f(rr,&)

(i) If t $ T U D, then (r(l) does not exist. (ii) If t E T, t?(l) = 0.

(iii) If t E D. there exists a (unique) integer k such that f = 2- ‘+I Choose 71 large enough so that . -J”(t) = [ti,, . tl,, + 3-“[ c [“-“‘, tk.+l., [. Th us, if IQ is the least integer such that TV.,,, 2 tj,,

(tl,, := 2 t,3-‘) , we have ,/,(.J”(t)) = 22”+“~1+‘, which yields t-t/,, < /(,(.7”(f)) < 2(t-tl,,). ikl

For instance, if t = l/2 = c :3-‘. we obtain: i>l

which brings up a new local dimension O( l/2) = 1. Thus, E(l.Z,) # ti and E( 1.1~) c D. which gives ,f( 13Z3) = 0 since dim D = 0 (U is countable).

Hence, the spectra f(-. &) and .f(.,Z,) are not identical.

Note remise le 20 juillct 1YY7, accept&e le 22 septembrc 1997.

References

[ I] Billingsley P., lY78. Euprlic t/wot~ rind i~fimmrtion. Krieger.

121 Cutler C. I)., 1986. The Hausdorff dimension distribution of finite measures in Euclidean space. Ctrntr~l. ./. Morl~. 38.

p. 14.59-1484.

131 Cutler C. D. et Dawson D. A., lY8Y. Estimation of dimension for spatially disributed data and related limit theorctna.

J. Multi\wricctc, Ad.. 28, p. I 15-148.

141 Vo,jak R., 1997. Construction of outer measures with prescribed (Hausdorff) multifractal spectrum. Prq~rint.

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