Documento de Trabajo 9514 

GEOMETRY OF SELF­

SIMILAR MEASURES 

, 
MANUEL MORAN 

JOSÉ-MANUEL REY 

_w 
d~ 

(~\íL\') 
"",,-

FACULTAD DE CIENCIAS ECONOMICAS Y EMPRESARIALES 

UNIVERSIDAD COMPLUTENSE DE MADRID 

VICEDECANATO 

Campus de Somosaguas. 28223 MADRID. ESPAÑA. 



GEOMETRY OF SELF-SIMILAR MEASURES 

MANUEL MORÁN JOSÉ-MANUEL REY 

Universidad Complutense (Madrid). 

September, 1995. 

Abstract 

Self-similar measures can be obtained by regarding the self-similar set generated 
by a system of similitudes 1J.i = {<P¡}¡eM as the probability space associated with 
an infinite process of Bernouilli trials with state space 1J.i. These measures are 
concentrated in Besicovitch sets, which are those sets composed oí points with given 
asymptotic frequencies in their generating similitudes. In this paper we obtain some 
geometric-size properties of self-similar measures. We generalize the expression of 
the Hausdorff and packing dimensiona of such measures to the case when M is 
countable. We give a precise answer to the problem of determining what packing 
measures are singular viith respect to self-slmilar measures. Both problems are 
solved by means of a technique which allows us to obtain efficient coverings of balls 
by cylinder sets. We also show that Besicovitch sets have infinite packing measure 
in their dimensiono 

Abstract 

Las medidas autosemejantes pueden obtenerse considerando el conjunto autose­
mejante generado por un sistema de semejanzas 1J.i = {<P¡}¡eM, como el espacio 
de' probabilidad natural asociado a un proceso infinito de ensayos de Bernouilli 
con espacio de estados 1J.i. Estas medidas están concentradas en los conjuntos de 
Besicovitch, que son los conjuntos de puntos cuyas secuencias de semejanzas gen­
eradoras tienen frecuencias asintóticas fijadas. En este artículo obtenemos algunas 
propiedades geométricas de las medidas autosemejantes. Por una parte, general­
izamos la fórmula para las dimensiones Hausdorff y packing de las medidas autose­
mejantes al caso en que M es infinito numerable. También damos una clasificación 
muy precisa de las medidas packing que son singulares respecto a las medidas au­
tosemejantes. Ambos problemas se resuelven mediante una técnica que permite 
recubrir de manera eficiente holas mediante cilindros asociados a la construcción 
geométrica, Demostramos además que los conjuntos de Besicovitch tienen medida 
packing infinito en su dimensión. 



M. Morán & J.-M. Rey 1 

1 Introduction 

The purpose of this paper is to investigate certain geometric properties of self-similar 
measures. Self-similar measures were introduced by Hutchinson in 1981 [Hut 81] (see 
also [Ban 91]). They provide the basic theoretical example of multifractal measures, 
a topic which generates active research amongst mathematicians and applied scientists 
[BMP 92, eM 92, Ols]. 

In [MR 94] it was shown how to get round the problem posed by the overlapping set 
whenself-similar measures satisfy the open set condition (see below). The singularity 
of self-similar measures with respect to the Hausdorff me asure was also analyzed there. 
In this paper we do an analogous analysis for the packing geometry, and generalize the 
formula for the Hausdorff and packing dimensions of self-similar measures to self-similar 
constructions with infinitely many similitudes. This requires new ideas, in particular a 
technique for the efficient covering of balla by cylinder sets. Here we develop this method, 
which we call the "travelling ball technique" . 

We now state the main results of the papero We first introduce the self-similar objects 
this paper deals with. Let M denote either theset {1, 2, ... ,m} or the set lN ofthe positive 
integers. A collection \]! = {'Pi: i E M} of similarity mappings in IRN is said to satisfy 
the open set condition (denoted by ose from now onwards) if there exists a nonempty 
bounded open set V e IRN such that 'Pi (V) ~ V for all i E M, and 'Pi(V) n 'Pi(V) = 0 
for i,j E M, i :/= j. We denote by S(N,M) the set of compact contractive systems 
of similarities in IRN, {'Pi: i E M}, satisfying the ose. The compactness of a system 
\]! is relative to the topology of uniform convergence on bounded subsets of IRN • By a 
contractive system we mean that sup{ri : i E M} < 1, where ri stands for the contraction 
ratio of the mapping 'Pi. 
Let Ilf = {'Pi: i E M} E S(N, M), and let S\]! be the set mapping defined by 

SIlf(X) = U 'Pi(X) (1) 
ieM 

for X e IRN . It is well known that there is a unique compact SIlf-invariant set E (that is 
SIlf(E) = E), which is usually called the self-similar set generated by Ilf [Hut 81, Wic 92]. 
The open set satisfying the ose for the system Ilf will be denoted by V and its closure 
c1V by F. Without loss of generality we can Msume that 1 V 1= 1, where 1 . 1 stands for 



2 INTRODUCTION 

the diameter of a subset of lRN • 

There exists a natural coding map 'Ir = 'Irq" from the product (or code) space Moo := 
M x M x ... onto E, given by 

'Ir(i) = n (<pi, o <Pi. O··· o <Pi. (E», 
keN 

(2) 

for i = (i1i 2 ••• ) E Moo. Let p+ = {(pi)iEM : Pi> O for all i, EiEM Pi = 1} e lRm
• Given 

p = (P¡)ieM E P+, Vp denotes the infinite-fold product probability measure on Me>:>, i.e 
00 

vp:= IIpi 
1 

(3) 

and /-Ip stands for the image measure of the measure vp under the mapping 'Ir, i.e. /-Ip = 
vp o '/1"-1. The measure /-Ip is a Borel measure, and is called the self-similar measure 
associated with the pair (1J!,p). Let M+ = M+(1J!) = {/-Ip: pE P+} be the set of 
self-sioúlar measures associated with the system 1J!. It can be seen that sUPPIL = E for 
all /-1 E M+. Self-similar seta and measures, as considered here, were first introduced by 
Hutchinson [Hut 81]. In section 3 we prove 

THEOREM A. Let p = (P¡)iEM E P+, and let 1J! = {<Pi: i E M} be a 
finite or infinite countable compact contractive system of similarities of lRN 

fulfilling the open set condition. Then, the Hausdorff and packing dimensions 
of the self-similar measure /-Ip associated with the pair (1J!, p) are given by the 
formula 

dim/-lp = Dim/-lp = s(p) := EiEM Pi logpi , 
EiEM Pi log ri 

provided that the series EieM Pi log ri converges. 

(4) 

We denote the Hausdorff and packing dimensions by dim and Dim respectively (see section 
2 for definitions). Formula (4) aboye generalizes the known formula in the finite case 
[DGSH 91]. 

We then prove theorem B below, which embodies a non-trivial syrnmetry between 
the Hausdorff and packing measures in the context of deterministic self-similarity. The 
notations H'" and P'" below stand respectively for the Hausdorff and packing measnres 
associated with an admissible dimension function ¡P (see section 2). 



M. Morán & J.-M. Rey 

THEOREM B. Let IJI = {'Pi: í E M} be a finite contractive system of 
similarities of IRN satisfying the open set condition, and let p = (Pi)¡EM E P+. 
Let 9 = N", }"'EJR be the family oí real variable functions given by 

1/!",(e) = e'(p) exp (a(2Iogec(p) log log log eC(p)j1/2) , 

where s(p) is defined in (4), and c(p) = (¿;iEM Pi logri)-l . Let 

( )
~2 

d(p) = ¿(logp¡ - s(p) logri)2pi 
¡EM 

Then, the self-similar measure J-Lp associated with the pair (lJI,p) satisfies 

i) For a < -d(p), J-Lp is singular with respect to (w.r.t) p"'a (and thus w.r.t. 
H"'·). 

ii) For 1 a 1< d(p), J-Lp is absolutely continuous w.r.t. p"'. and singular w.r.t. 
H"'a. 

iii) For a > d(p), J-Lp is absolutely continuous w.r.t. H"'· (and thus w.r.t 
p"'.). 

iv) J-Lp is not representable as an integral in terms of either the Hausdorff 
measure H I or thepacking measure pt for O ::; t < 8 = dimE. 

v) Let /1-, be the self-similar measure associated with the system IJI and the 
probability vector p, := (rt)iEM' Then J-L, admits an integral representation 
w.r.t. the measures H' and P'. In fact, the measures J-LB) HS, and p s coincide 
up to a constant factor. 

3 

The statements made in the aboye theorem concerning Hausdorff measure properties were 
proven in [MR 94]. In section 4 we prove their counterpart for the packing geometry of 
self-similar measures (see theorem 4.5). In theorem B we gather both results in order 
to illustrate the syrnmetric role of the Hausdorff and packing geometries in self-similar 
constructions. 

Notice that SIJI-invariant sets may be properly named 'self-similar' sets. From the J-Lp­
measure-theoretic point of view, only those sets with positive J-Lp-measure are relevant. 

The Besicovitch sets Bp and B~oo) introduced in [MR 94] (see the definition (9) below) 
are thus (non-compact) 'self-similar' sets oí full /1-p-measure. They can be regarded as 



4 NOTATION AND PRELIMINARY RESULTS 

the set-theoretic dual counterpart of self-similar measures and thus provide a natural 
alternative approach to self-similar geometry. In [MR 94] several Hausdorff measure and 
dimension properties of the Besicovitch sets were obtained. In particular, they were shown 
to have Hausdorff dimension given by s(p), and an s(p)-dimensional Hausdorff measure 
that is either zero or infinity under the hypothesis that dim(Bp n e) < s(p), where e is 
the overlapping set associated with the geometric construction (see (6) for the definition). 
We prove in section 5 that the Hausdorff and packing dimensions of Besicovitch sets 
associated with (w, p) are given by s(p), and are thus 'fractal sets' in the sense of Taylor 
[Tay 86]. Moreover, using the results obtained in section 4, we are able to prove that 
the Besicovitch sets have infinite packing measure. Our results are collected in theorem 
5.1. Sorne results concerning measure and dimension properties of other ¡.tp-full measure 
geometric point sets in self-similar constructions, characterized by the frequencies of their 
generating similarities, can be seen in [Rey 95]. 

The paper is organized in the following way. In section 2 we introduce notation and 
prove sorne preliminary results. In section 3 we prove theorem A aboye. The statements 
of theorem B concerning packing geometry are proved in section 4. Finally, section 5 
gives an account of the packing geometric size of Besicovitch sets. 

2 Notation and Preliminary Results. 

We first give sorne basic definitions and notation from geometric measure theory. 
Given A e IRN and 's > 0, a collection of balls {Bi : i E lN} is a S-covering of the set 
A if Ui B¡ J A with 1 B¡ 1:-:; ti for all i. A ti-packing of A is a collection of closed balls 
{B(x¡, T¡) : Xi E A }¡eJN satisfying 2Ti < Hor all i, and B(x¡, T¡)nB(xj, Tj) = 0 for al! i =J j. 
Let :F denote the set of dimension functions, that is, the set of those increasing continuous 
functions q, defined in sorne nonempty interval (O,e), satisfying lime-+o+ q,(e) = O and 
limsup(-+0(q,(2e)!4>(e» = q,' < +00. For q, E :F, the spherical q,-Hausdorff measure of 
A e IRN is given by 

H"'(A) = sup inf{ ¿: q,0 Bd): {B¡} is a S - covering of A by balls}. 
6>0 ieJN 

It is a standard fact that H'" is comparable to the standard Hausdorff measure [Fal 85]. 
In particular, if Ht denotes the Hausdorff measure associated with the dimension function 



M. Morán & J.-M. Rey 

"'( e) = e, then the Hausdorff dimension oí the set A is given by the threshold value 

dimA = sup{t: HI(A) > O} = iní{t: HI(A) < +oo}. 

5 

The "'-packing measure is defined in two steps. First the <p-packing pre-measure oí 
A is defined as 

piCA) = infsup{1: "'(1 B¡ 1): {B;}¡ is a8-packingofA}. 
6>0 ¡EN 

Then the "'-packing meastire of A is given by 

P~(A) = inf{ 1: PiCA;) : U A¡ J A}. 
iE'.N i 

The packing dimension of A is the value given by 

DimA = sup{t : pl(A) > O} = inf{t : pl(A) < +oo}, 

pt denoting, for each t 2:: O, the packing measure defined from the potential dimension 
function e. 

Finally, the Hausdorff dimension of a Borel measure J1. is defined by 

dimJ1. = inf{dimA: J1.(A) > O}, (5) 

where the infimum is taken over the class of Borel sets. The packing dimension of J1. is 
defined in the same way, and will be denoted by DimJ1.. 

We now introduce notation and previous results which will be used in the remaining 
sections. These results guarantee in particular that the 'travelling ball' technique works 
for a set of full J1.-measure on E. 

The overlapping set El of the system 1JI is the set given by 

El = {x E E : card{1I"-1(x)} > 1}, (6) 

Le. the geometric set where 11" fans to be an injection. The notation M* stands for the set 
UkEJN Mio, which is the set of finite sequences with terms in M. For j = (jd2 ... j¡,) E Mio, 
i E Moo, and n E 1N, n > k, let 

8j(i,n) = ~card{q: i q = j¡, i q+1 = j2, ... ,iq+k-l = j¡" 1 ~ q ~ n-k}, (7) 
n 



6 NOTATION AND PRELlMINARY RESULTS 

and write Dj(i) = limn .... +oo Dj(i, n), whenever such a limit exists. 

Given p E p+ and k E lN, consider the sets of codes 

B~) = n {i E Moo : Dj(i) = Pj}, (8) 
jeM· 

where Pj = PhPi> " . Pi. for j = (hh ... ik)¡ and the set 

B~oo) = n B~) = n {i E M oo : Dj(i) = Pj}. 
ke:tl jeM* 

We define the geometric sets 

Bp := 1l"(B~») and B~oo) := 1l"(B~oo»), (9) 

which in [MR 94) were respectively called the Besicovitch set and the thin Besicovitch set 
associated with the pair (1l1, p). 

Given i E Moo and k E lN, i(k) stands for the curtailed sequence (i1i2 ... ik)' For 
j E Mk, jj) = {i E Moo : i(k) = j} is called a cylinder set (the one whose heading 
sequen ce is j). We write 'Pj for the composite similitude 'Ph o 'Pi> o ... o 'Pi. ¡ and Ej 
(respectively lij) for the image sets 'Pj(E) (resp. !pj(F)), which we call geometric cylinder 
sets. Reeall that we abbreviated the product Pi,Pj, ... Pi. by Pj aboye: We also denote 
Ti, Tj, ... Ti. by Tj. We eall the projeeted sets 1r([i(k)]) = 'Pi(k)(E), geometrie cylinders oí 
the k-th generation. 
We now state a result concerning the JLp-sizes of the overlapping set e and of the thin 
Besicoviteh sets. 

Theorem 2.1 Let ll1 E S(N,M), pE P+, and let JLp be the self-similaT meaSUTe associ­
ated with (1l1, p). Then 

i) B~oo) n e = 0 
ii) (e-lemma) JLp(e) = o 
iii) Jlp(Ej) = vp(U)) = Pj fOT all j E M*. 

The 0-lernma aboye is a consequence of part i) together with Birl<hoff's ergo die theorem. 
Part iii) follows from the e-lemma. For a prooí of theorem 2.1 see [MR 94). 



M. Morán & J.-M. Rey 7 

Remark 2.2 Let e denote the O'-algebra generated by the class of eylinder sets and 
let C" denote the O'-algebra indueed on E by 'Ir. Notice that the 8-lemma implies that 
(E, e", Jlp) and (MOO, e, vp) are isomorphic measure spaces. Remark 2.3 below strengthens 
the equivalence of those spaees from the dynamieal viewpoint. 

Let r : Moo H MOO be the Bernouilli shift in the code space, i.e. r(i1i2ia ... ) = (i2ia ... ). 
A Bernouilli shift, which will be denoted by T, is defined in the geometric space E by 
means of 

T IE\0= 7l' o r o 7l'-1. 

The geometrie shift mapping T defined aboye will be fixed throughout the papero 

Remark 2.3 The 8-lemma implies that T is Jlp-preserving for all p E P+, and fur­
thermore that (T,Jlp) and (r, vp) are isomorphic measure-preserving mappings. Observe 
that T is also ergo die for any measure Jlp E M+, since ergodicity is preserved under 
isomorphisms. The definitions from ergo die theory used aboye can be seen in [Wal 82]. 

We will need the following auxiliary resulto Recall that the sets E¡ = 'Pi(E) are geometric 
cylinders of the first generation for the system W. 

Lemma 2.4 Let W E S(N,M). For k E lN, let Ak be the class of sets 

(Sift° is the identity map), and let u(Ak) denote the O'-algebra generated by Ak. Then, 
for each p E lN and JI E M+ 

P 

JI(A1 n A2 n ... n Ap) = II JI(A;), (10) 
j=l 

whenever Aj E u(A;) for each j = 1,2, ... ,p. 

Proof: For eaeh k E lN, let Jh denote the class of sets in Ak plus all sets obtained 
as finite interseetion of sets in Ak' Let p E lN. Observe that the 8-lemma implies that 



8 NOTATION AND PRELIMINARY RESULTS 

equality 
P 

¡.t(Eil n SiJ!(Ei2) n ... n SiJ!P-1(Eip)) = II ¡.t(Eij) , 
j=l 

holds for any i = (i 1 , i2 , ••• , ip) E MI>. Thus (10) holds if Aj E A¡ for 1::; j ::; p. Using the 
0-lernma again, it is easy to show that (10) actually holds when Aj E Ah j = 1, ... ,p. 
That is to say, the dasses Al> A2 , • •• ,Ap are independent [Bil 78J. Since the dass Aj 

is a 1I"-system for each j, a result by Billingsley [Bil 78, Theorem 4.2) implies that the 
(1-algebras (1(A1 ), (1(A2 ) , ••• ,(1(Ap ) are also independent, and the lernma follows. O 

For i E M, let x¡ : E H lN U {O, +oo} be the random variable defined by 

X¡(x) = sup{p: x E (E \ Ei) n T-1(E \ E¡) n ... n T-V+l(E \ E¡)} (11) 

for x E E \ Ei, and X¡(x) == O for x E E¡. The following typicallimit property for X¡ will 
be used in subsequent sections. 

Proposition 2.5 Let iJ! E SeN, M). For i E M and a > O, let 

(12) 

Then, ¡.t(G(i, a)) = 1 for any It E M+. 

Proof: Let i E M,a > O, and ¡.t E M+. We will write A for the set E \ Ei. For 
positive integers k < j consider the set 

(13) 
. . 

Notice that, using the notation of lemma 2.4, T-q(A) == SiJ!q(A) E (1(Aq) for all q E lN. 
Thus, the T-invariance of ¡.t (see remark 2.3) together with lemma 2.4 imply that 

(14) 

Let n E lN. For each k E lN, we define the set 

G~(i,a,k) = {x: Xi(Tk(x)) > k"}. 
n 



M. Morán & J.-M. Rey 9 

For each k E lN, consider the integer p(k) = min{p E lN: pn > ka}. From (11) and (13), 

p(G~(i, a, k)) ~ p(Ak,p(k)) 

because the sequence {Ak,;}; is non-increasing. The choice of p(k) and (14) then give 

p(G~(i, a, k)) ::; p(A)p(k) ~ (p(A)l /n)kQ

• 

Writing r = p(A)l/n, the estímate (15) and a change of variable give 

00 00 

L:JL(G~(i,a,k))::; L:e-kQllogrl ~ a-1 IlogT 1-11a r(a-1 ), 

k=l k=l 

(15) 

(16) 

where ro denotes the eulerian integral. Therefore the series in (16) converges, and the 
first Borel-Cantelli leroma implies that p(limsuPk~oo G~(i, a, k)) = O. Since n E lN is 
arbitrary, E \ G(i, a) is a p-null set, which proves the resulto O 

3 Dimension of self-similar measures. 

In this section we assume that the set M is infinite and find the Hausdorff and packing 
dimensions of a self-simBar measure. Let (w,p) E S(N,M) X P+. Assume that 

L: Pi logri > -oo. 
iEM 

and let s(p) be the real number defined in (4). The formula 

dimpp = Dimpp = s(p) 

(17) 

(18) 

is known to hold when M is finite[DGSH 91]. The prQof.deP€;lndscrucially o_n theel,{i~tel).ce 
of a positive minimum contraction ratio u := min{ri : i E M} > O, which does no longer 
hold for the infinite case. We prove in this section that formula (18) still holds for the 
infinite case. Using (18) for M finite it can be proved that the convergence hypothesis 
(17) implies that the series in the numerator of s(p) also converges, namely 

Lemma 3.1 Let \]i = {'Pi: i E M} E SeN, M), and p = (Pi)iEM E P+. Jf the series 
L"EM Pi log Ti converges, then the series LieM Pi log Pi also converges. 



10 DIMENSION OF SELF-SIMILAR MEASURES 

Proof: Suppose ¿iEM Pi logpi = -oo. Let 1 < O be the sum of the series ¿iEM Pi logri, 
andlet {e,.}nEI'I bethesequencedefined by e,. = ¿i=1Pi, n E IN. Chooseapositiveinteger 
k satisfying 

k 

LPi logpi < NI + Ck log Ck. (19) 
i=l 

Now consider the finite probability distribution p = (Pi)iEK on the set K == {1, 2, ... , k} 
defined by Pi = cl/Pi> i E K. The system ~ == {epi: i E K} belongs to the set S(N, K). 
Thus the dimension formula given in (18) implies that the self-similar measure ¡'¡p asso­
ciated with the pair (~,p) has Hausdorff and packing dimensions given by s(p) (see (4». 
From (19) it then fol!ows that 

( ~) ¿~=l Pi logpi - Ck log Ck N 
sp= k >, 

¿i=1 Pi logri 

which is a contradiction, since the measure ¡'¡p is defined in IRN. O 

Remark 3.2 In terms of ergodic theory, lemma 3.1 asserts that the dynamical system 
(E,T,¡.¡p) has finite Kolmogorov-Sinai entropy provided the Liapunov exponent of the 
system is bounded aboye. An interpretation of (18) from this viewpoint can be seen in 
[MR 94J. 

We will often use the fol!owing notation. Let p E p+ and let (MOO, vp) be the 
corresponding product probability space (see (3». For a random variable Z : M H IR, 
we cal! {Zj};EI'I the associated independent process in Moo, Le. for each j E lN, Zj : 
MOO H IR is the r.v. defined by 

Z Z j-1 ; = o prl o r , 

where pr1 : MOO H Mis the projection pr1(i1i2 ••• ) = i l , and r is the Bernouilh-shlfí ih 
Meo. For i E Moo we write 

k 

Sl(i) = L Zj(i). (20) 
;=1 

The expression &[ZJ means the expectation of Z with respect to the probability p in M. 



M. Morán & J.-M. Rey 11 

Let ¡.¿ be a Borel measure. The upper and lower spherical logarithmic densities of l' 
at x E E are defined by 

- () l' log I'(B(x, 1'» 
a~ x = lmsup 1 ' 

r .... O og l' 
() l

. . f log I'(B( x, 1')) 
a~ x = lmlll 1 . 

r .... O og l' 

We first prove that, for a given p E P+, the value s(p) in (4) is an upper bound for 
the packing dimension of I'p' This is a consequence of the following 

Lemma 3.3 Let W E S(N,M), p E P+, and let l' denote the self-similar measure 
associated with (111, p). Assume that (17) holds. Then i'i~(x) < s(p) ¡.¿-a.e. 

Proof: Consider the random variables W1(i) = logp¡, W2(i) = logr¡, i E Mi and let 

(21) 

Let e > O. For i E N, take ka such that 

for all k > ko. Let O < l' < ri(ko), and take k1 = min{k : ri(k) < r}. From the choice of k1 
and part iii) of theorem 2.1, it follows that 

log I'(B(7I'(i), r» < log¡.¿(EI(k,) = 
logr - logrl(k'_l) 

= (1+ 1 ) ki'lSr,'l~. ~ (1 + e) e[W¡) + e. 
k1 - 1- (k1 -l)-lSki_i(l) .e[W21- e 

Since s(p) = e[W1]/í'[W2J, we get ap(7I'(i» ~ s(p) for all i E.N. Hypothesis (17) and 
lemma 3.1 together imply.. that tite strong law of large.numbets holds for both .. W1JlJl.d. ... 
W2 • Hence ¡'¿p(7I'(.N» = 1 and the lemma follows. O 

We introduce now the 'travelling ball' idea which allows us to obtain a geometric 
cylinder covering any small given ball, and then to show that s(p) is a lower bound for 
the Hausdorff dimension of I'p. 



12 DIMENSION OF SELF-SIMILAR MEASURES 

Lemma 3.4 ('Travelling Ball Lernma', logarithmic version) 
Let 111 E S(N,M), P E P+, and let ¡.t be the self-similar measure associated with (1I1,p). 
Assume hypothesis (17). Then s(p) :s al'(x) ¡t - a.e., where s(p) is given in (4). 

Proof: Recall that V denotes the open set satisfying the OSC for the system 111. It 
is known [Sch 94] that, for M finite, the OSC implies that the intersection V nEis not 
void. It can be easily seen [MR 94] that this is also true for M = JN. Therefore, under 
the OSC, there exist e> O and y = 11"(i) E E n V such that B(y,e) e V. We choose a 
positive integer ko large enough so that 

El(ko) e B(y, e). (22) 

The set EI(ko) is a cylinder set of the first generation for the system 

1110 := 111 o 1110 ~~o1 0111 = {<pj : j E Mko} E S(N, MkO), 

which also has the set E as the unique compact S1I10-invariant seto Furthermore, if the 

product probability Po := pko = px ~k.o1 xp is considered in Mko, Vo denotes the infinite­
fold product measure nf Po, and 11"0 is the natural coding map associated with the system 
1110 (see (2», it can be seen that the induced measure Vo o 11"01 coincides with the self­
similar measure ¡.tp. This follows from a uniqueness argument, since both rneasures can be 
shown to be fixed points of the same contractive operator (see [Hut 81] for this theory). 

Since there exists a p > O such that d(B(y,é), aV) > p, we have from (22) that 

d(B,av) > p, (23) 

where B := Ei(ko) is a cylinder of the first generation for the system 1110. Notice that 
the system 1110 satisfies the ose with open set V. Therefore, given an arbitrary system 
111 E S(N,M), it can be assumed that there exists 1 E M and a cylinder B := E, of the 
first generation for the system 111 satisfying (23). The geometric cylinder B will be fixed 
throughout the proof. 

Recall the definitions of the sets BLoo), G(i,a), and N given in (9), (12) and (21) 
respectively. Let G = BLoo) n G(l, 1) n 11"(N) and take x E G. Notice that theorem 2.1 i) 
implies that 11"-l(X) ~" i is a singleton. Let q = min{n : in = l}, and let r > O be such 
that r < pri(q), where p > O is the constant in (23). Consider the integer 

k, = max{ k : r < pri(k)}' (24) 



M. Morán & J.-M. Rey 13 

Hence the ('travelling') ball B(Tk,(x),p) satisfies 

'Pi(k,)(B(Tk,(x),p)) = B(x,p1'i(k,)):::> B(x,r). (25) 

Since liminfk .... ood(Tk(x),8V) = O for x E B~oo), we do not have that B(Tk,(x),p) e V 
in general. That is why we consider the integer p, defined by 

p, := p(k,) =: TIÚn{j : Tkr-i(x) E B}. (26) 

Notice that this choice guarantees that B(Tk,-P,(x),p) e V because of (23). Thus, from 
(24), we get 

B(x,r) e B(x,prl(k,_p,») = 'f'1(k,_p,)(B(Tk,-P,(x),p)) e 'f'i(k,-p,)(cl(V)) = Fl(k,-p,), 

so that ¡t(B(x, r)) :::; ¡t(F¡(k,-pr»)' Since pri(kr+1) < r, we obtain 

10g¡t(B(x,r)) > log¡t(Fl(k,-v,») 
log r - 10g(pri(k,+1») . 

(27) 

Using the random variables W; (i = 1,2) and part iii) of theorem 2.1, as we did in the 
proof of lemma 3.3, inequality (27) can be written as 

(28) 

where the notation is that in (20). From the choice of p, (see (26)), and the definition 
(11) it follows that X¡(Tk,-P,+I(x)) ~ PT for an x E E. Thus, if PT 2: 1, we have 

X¡(Tk,-p,+I(x)) > p'. 
kr -p,+1 -kT 

(29) 

It can be seen that kT - p, -> +00 as r -> O for x E B~oo), because otherwise we would 
obtain that liminfk .... +oo 6¡(71'-1(X), k) = O whicl! is a contradiction (see the notation in 
(7)). Using (29), definition (12), and taking liminf as r -> O, we obtain from (28) 

l .. flog¡t(B(x,r)) > &[W11_ ( ) 
1mm l -"[W.l- sP . ' .... 0 og r c. 2 

This proves the theorem, since the E>-Iemma, proposition 2.5, lemma 3.1, and the strong 
law of large numbers imply that the set G has fun ¡t-measure. O 



14 CONTINUITY WITH RESPECT TO THE PACKING MEAsuRE 

Proof of theorem A: It íol!ows as a consequence oí standard results [Tri 82, You 81] 
(see also [Cut 86, Cut 90]) connecting the logarithmic densities ªI" 0/1' oí a Borel measure 
p. with the Hausdorff and packing dimensions of p.. More precisely, the equality ªI'(x) = O/. 

p.-a.e. implies that dimp. = O/., whereas O/I'( x) = O/. p.-a.e implies that Dimp. = O/. (see e.g. 
[Tri 82, Theorem 1]. Proposition 3.3 and theorem 3.4 together imply that O/. = O/. = s(p) 
in our case. O 

4 Absolute continuity of self-similar measures with 
respect to the packing measure. 

We assume that M = {1,2, ... ,m} throughout this section. Let 1Ji' E S(N,M), and 
p E P+. We now address the problem oí determining the behaviour oí the selí-similar 
measure p.p with respect to the packing measure in its dimension, Le. deciding whether 
p.p is either singular or absolutely continuous w.r.t. p.(p). 

We approach the problem by means oí local techniques. Given a Borel measure p. and 
1/J E :F, recal! that the standard lower spherícal 1/J-density of p. at x is defined by 

8"'( ) -l' . f p.(B(x,r)) 
-1' x - 1I?.1J1 1/J(2r) . 

Given p. E M+ and 1/J E :F, we define the fol!owing lower cy/indrical 1/J-density oí p. at x 

4"'(x) = sup{liminf P.(Ei(k») : i E 1T-1(X)}. 
l' k __ +oo 1/J(ri(k») 

N otice that part iii) in theorem 2.1 implies that 

slf.(x) = liminf vp([i(k)])f.,p(ri(k») = liminf Pi(k)f.,p(ri(k») 
,- k-++oo k-++oo 

for aH x = 1T(i) belonging to the set of p.-ful! measure E \ e, where p E p+ is the 
probability p. is associated with. We write f!.~(-) and sl~(-) when 1/J(e) = e. 

To obtain the main result of this section, the information supplied by the cylindrical 
densities, which are well-fitted to self-similar constructions, must be translated into geo­
metrical information about spherical densities. We recall that u := min{r¡ : i E M}. Let 
:F+ = {1/J E :F : 1/J( x )1/J(y) ::; 1/J( xy) for all x, y > O smal! enough} 



M. Morán & J.-M. Rey 15 

Lemma 4.1 Let W E S(N,M), pE P+, and let JL be the self-similar measure associated 
with (W, p). Let 1/J E :¡:+. Then the inequality 

(30) 

holds for all x E E. 

Proof: The idea of the proof is similar to the one used in the proof of lernma 3.3. For 
x E E and T > O, we take i E ?r-

1 (x) and ko == min{j : Ti(j) < r}. Since 1 V 1= 1, we get 
that B(x, r) ::) Ei(ko), so that 

JL(B(x, r)) > 1/J(U/2)JL(Ei(ko»), 
1/J(2r) - 1/J(ri(ko») 

because ri(k.) ~ ur and 1/J E :¡:+. Letting r -t O and taking the infimum over the codes 
i E ?r-1(x) we obtain (30). O 

Let 9 == Q(W, p) == N", : a E lR} be the one-parameter family of real functions 
defined by 

1/J",(e) == es(p) exp{ a(2log ec(p) log log log eC(p»)1/2}, (31) 

where s(p) is given in (4), and c(p) is the negative real number 

c(p) == (L: p;log ri)-l 
ieM 

(32) 

It can be seen that 1/J", E :¡:+ for aH a S O [Rey 95]. We recall that the definition of the 
Besicovitch set Bp was given in (8). We will need the following lemma, which was proven 
in [MR 94] .. 

Lemma 4.2 For a < O, let fa be the real variable function 

(33) 

Then, for every x E Bp there is an i E ?r-1 (x) such that 

lim fc(p) (rl(k») == 1 
k_+oo (2k log log k )1/2 ' 

(34) 

where c(p) is the constant defined in (32). 



16 CONTINUITY WITH RESPECT TO THE PACKING MEASURE 

Lemma 4.3 ('Travelling Ball' lernma, non-Iogarithmic version) 
Let W E SeN, M), p E P+, and let '" denote the self-símílar meaSUTe assocíated with 

(w, p). Let g be the famíly defined in (31). For 1/>" E g with Ct < O, the following inequality 
holds ",-a. e. 

(35) 

where K = K(w,a) > O is a constant which depends only on the system W and the 
constant Ct. 

Proof: The proof follows Hnes similar to those used in the proof of theorem 3.4, even 
though the 'travelling hall' technique rnust he used here in a slightly different way. 

Let E = B U (E \ B) he the decomposition of the self-similar set E considered in the 
proof of theorern 3.4, Le. B = El for sorne I E M with d(B, 8V) > p, where Vis the open 
set satisfying the OSC for the system W. The set B will he fixed throughout this proof. 

Let G = B~oo) n G(I,1/2), where B~oo) is the thin Besicovitch set defined in (9), and 
G(l, 1/2) is defined in (12). Take any x = '/r(i) E G. Let q = min{j : ij = l}, k > q, and 
choose sorne ek > O such that ek < prl(k) hut eh ;:: pr¡(kt1)' Consider now, as in the proof 
of theorem 3.4, the 'travelling ball' B(Tk(x),p) so that 

B(a:,ek) e 'Pj(k)(B(Tk(x),p)). 

The travelling hall will not in general be contained in the open set V. We thus define 

p = p(k) = rnin{j : TH(x) E B}. 

Since d(B,8V) > p, we have 

Fl(k-p) J 'Pl(k_p)(B(Tk-P(x),p)) = B(a:,prj(k_v») J B(x,ek), 

and thus part iii) in theorem 2.1 gives 

G(k )",(Ej(k») = ",(Ej(k_p») > "'( B( x, ek)), 

where C( k) = (Pi'_P+1 Pi._P+2 ••• Pi. ti. 
Let Ct < O. Since purl(k) :::; ek, and 1/>" E ;¡::+, we obtain from (36) that 

p.(B(x,ek)) :::; (1/>,,(2pU))-IC(k) P.(Ei(k») . 
v>,,(2ck) v>,,(rl(k») 

(36) 

(37) 



M. Morán & J.-M. Rey 17 

We will prove now that 

(38) 

so that taking Iiminfk .... +oo in (37) will give (35). Let X : M ...... lR be the null-expectation 
ralldom variable defined by 

X(i) = logpi - s(p) log ri. (39) 

Observe that ,p", can be written as ,p",(e) = e'(p) exp(afc(p) (e)), where f. is the function 
defined in (33). The left-hand side of (38) can be rewritten as 

{l' . f( ~ ( ) ( Sr (i) EJ=k-P+1 1og Pi;))} exp 1mm Jc(p) ri(k) ~ ( ) - a - ~ () , 
k->+oo J c(p) ri(k) J c(p) rl(k) 

(40) 

where the notation in (20) has been used. From lemma 4.2 it follows that showing 

l· EJ=k-p+l10g Pi; o' 1m = 
k->+oo (2k log log k )1/2 

(41) 

proves (38). It can be assumed that P ~ 1. The reasoning used in theorem 3.4 to obtain 
(29) also applies here, so that (41) follows from inequality 

X¡(Tk-p+l(a:)) > L 
(k-p+1)1/2 - kI/2 ' 

taking into account the estimate Ej=k_p+l10g Pi ~ p(rniniEM log Pi) and definition (12). 
Therefore inequality (35) holds for all a: E G with the constant J( = (,p",(2pu))-1, which 
only depends on the pair (w, a)). The S-lemma and proposition 2.5 together complete 
the proof. O 

To prove the main result in this section we need theorem 4.4 below, which provides 
a local characterization of both the singularity and the absolute continuity of a Borel 
measure w.r.t. ,p-packing measures. The theorem below stems from the work of C.A. 
Rogers and S.J. Taylor [RT 61) along with the density theorem for ,p-packing measures 
of S.J. Taylor and C. Tricot [TT 85, Theorem 5.4). 

Theorem 4.4 (Rogers-Taylor theorem) 
Let ¡t be a finite Bore! measure in lRN , and,p E :F. Then 



18 CONTINUITY WITH RESPEOT TO THE PACKING MEASURE 

(a) ¡.t is singular w.r.t. P'" il and only ifD1;(x) == +00 ¡.t-a.e. 
(b) ¡.t is absolutely continuous w.r.t. P'" if and only il!it(x) < +00 ¡.t-a.e. 
(e) ¡.t has an integral representation w.r.t. P'" if and only ifO < !it(x) < +00 ¡.t-a.e. 

Now we are ready to give a proof of 

Theorem 4.5 Let W E SeN, M), p == (P¡)¡EM E P+, and let 9 == {tP"},,ER be the fam­
i/y defined in (31). Let d(p) == (l::¡EM(logp¡ - s(p)logr¡)2p¡)1/2. Then, the self-similar 
measure ¡.tp induced by the paír (w, p) satisfies 

i) ¡.tp is singular w. r.t. p"'o il a < -d(p). 

ii) ¡.tp is absolutely contínuous w. r. t. p"'a il a > -d(p). 

iii) ¡.tp has an integral representation w. r. t. pt lor Bome t > O il and only il p = p" 
i.e. p¡ == r[ lor i E M (and thus s = t). 

Proof: We drop the subindex p from ¡.tp' Given a < O, it follows from lemma 4.1 
together with lemma 4.3 that there exist positive finite constants I<l and I<2 (depending 
only on (W, a» such that inequalities 

(42) 

hold for a set of ¡.t-full measure. Therefore theorem 4.4 and (42) reduce the problem to 
the computation of the ¡.t-typical value of the cylindrical density d.ta instead of that oí 
!ita. Denote by D the geometric set where (42) holds. Let C be the set oí codes 

" {. M oo l' . f Sr (i) d( )} '- = 1 E : ll!;~';, (2k loglog k)l/2 = - P , 

where X is the r.v. defined in (39) and the notation is that of (20). Since e(X] == O and 
d(p) is the standard deviation of X, it íollows from the law of the iterated logarithm that 
¡.t('1l'(C)) = 1. 



',;' 

M. Morán & J.-M. Rey 19 

Let x E B},oo) n D n 71'(.c). Notice that the lower cylindrical 1/>a-density of JL at x can be 
written as 

(43) 

where f. and a = c(p) are defined in (33) and (32) respectively. Since 71'-1 (x) E e, taking 
into account the asymptotic identity (34) oflernma 4.2, we obtain from (43) 

!lf,a(x) = +00 if a < -d(p), 

and 
(44) 

Since x E D, this implies 

(45) 

(the case IX > O is also included aboye since 1/>" is an increasing íunction oí a). Because 
D, 71'(.c), and B~oo) are all sets oí íull JL-measure, parts i) and ii) of the theorem follow 
írom (45) aboye and from parts (a) and (b) of theorem 4.4. 

To prove part in), we need the equivalence relationship (42) to hold JL-a.e. íor the 
clasa oí potential dimension functions e. It is straightíorward to check, along the lines of 
the proof oí lernma 4.3, that this result also holds for 1/>(e) = e, t ~ O. This, together with 
inequality (30), implies that (42) holds for the class of potential íunctions, and thus part 
(e) in theorem 4.4 implies that the statement in iii) about the integral representability 
of JL is equivalent to the fact that 

O < ~(x) < +00 JL - a.e. (46) 

Using the strong law of large numbers and the law of the iterated logarithm (as in the 
prooí oí theorem 2 part iii) in [MR 94]), the boundedness condition (46) can be shown 
to be equivalent to the choice p = Ps' This proves the theorem. O 

As a corollary of theorem 4.5 notice that the choice a = O gives 1/>0 = es(p), and thus 
ii) implies that the self-similar measure JLp is absolutely continuous w.r.t. the packing 
measure ps(p). 



20 PACKING GEOMETRY OF BF,srcoVITCH SETS 

5 Packing Geometry of Besicovitch Sets. 

In this section we use results from sections 3 and 4 to study the packing geometry of 
Besicovitch sets. Let I]i E S(N, M), and P E P+. We assume in this section that 
M = {1, 2, ... ,m}, except when otherwise stated. Recall that p, denotes the probability 
vector (rf)iEM E P+. 

Theorem 5.1 Let I]i E S(N, M) and P E P+. Let Bp and B~oo) be the Besicovitch sets 
defined in (9). Then 

i) DimB~oo) = DimBp = s(p). 

ii) 0< p'(Bb":'» = P'(Bp ,) = PS(E) < +00. 

iii) 11 p :1 PSI every set with positive ¡tp-measure has infinite s(p )-packing measure. 
In particular 

Proof: The inequaJity DimB~oo) ;::: s(p) follows from (5), (18) and the fact that 
Jlp(Bboo» = 1. Let x E Bp. From the definition of Bp there exists an iz E 1I'-I(X) such 
that Sj(iz ) = Pj for aH j E M. Taking kr := min{j : r¡.(j) < r} and proceeding as in the 
proof of lemma 3.3, we get that 

since limk~+ook-lS1"j(i",) = t:[Wj ) for j = 1,2. The inequality DimBp :5 s(p) then 
follows from the work of C. Tricot [Tri 82, Theorem 1). This proves i). 

Part ii) is a consequence of the uniqueness of the invariant measure associated with 
the pair (I]i,p) [Hut 81). Since the measures H' and P' coincide up to a constant factor 
in the finite case [Haa, Spe 92), ii) foHows from theorem 3 part ii) in [MR 94). 



M. Morán & J.-M. Rey 21 

We omit p from p.p. Let A be a borelian set of positive p.-measure. From theorem 4.3 
and the fact that d.~(P) = O p.-a.e. (see (44)), it follows that there is a set A* e A such 
that p.(A) = p.(A") and !!.~(P)(re) = O for all re E A". Since p.(A*) > O, the Taylor-Tricot 
density theorem [TT 85, Theorem 5.4) implies that P8(p)(A*) = +00. This proves part 
iii) because the Besicovitch sets are Borel sets of full p.-measure. O 

The finiteness oí M is essential to obtain the result in part iii) of theorem 5.1. We are 
able, however, to obtain the same result for infinite M in sorne cases. This follows from 
the following proposition, which is a packing version oí proposition 3.7 in [MR 94). 

Proposition 5.2 Let Ili E S(N, M), with M infinite countable. Assume that 'LeMrf = 
1. Let O < t < s, and .,p(0 = eg(e) E :F with 9 non-increasing in some nonempty 
interval (O,e). Alternatively, let.,p E :F+ be such that limsup~ .... olog.,p(e)/ loge < s. Then 
the .,p-packing measure of every SIli--invariant set with DimB > Dim(Bn 0) is either zero 
or infinity. 

The proof of proposition 5.2 is similar to that of proposition 3.7 in [MR 94]. From the 
result aboye it follows that the thick Besicovitch sets Bp associated with systems in 
S(N, lN) which intersect the overlapping set 0 in a set oí packing dimension stricly less 
than s(p), have either zero or infinite packing measure in their dimensiono In particular, 
from part i) oí theorem 2.1 it follows that the s(p )-packing rneasure of thin Besicovitch 
sets is either null or infinite. 

Remark 5.3 Notice that theorem 5.1 applies to the classical case of the Besicovitch­
Egglestone sets [Bes 34, Egg 49]. These are subsets oí the unit interval composed of 
points with given asymptotic frequencies in the figures oí their m-base expansiono In 
particular, these sets (as well as the thin Besicovith-Egglestone sets) have infinite packing 
measure in their dimensiono A classical problem remains open, that is to decide whether 
the Hausdorff measure of the Besicovitch sets in their dimension is either zero or infinity, 
see [MR 94]. 



22 

References 

[Ban 91] 

[Bes 34) 

[Bil 78] 

[BMP 92] 

[CM 92] 

[Cut 86] 

[Cut 90] 

[DGSH 91] 

[Egg 49] 

[Fal 85] 

[Haa] 

[Hut 81] 

[MU] 

[Mor] 

[MR 94] 

REFERENCES 

C. Bandt, Deterministic Fractals and Fractal Measures, Lecturo Notes 
01 the School on Measuro Theory and Real Ana/ysis, Grado, Italy, 1991. 

A.S. Besicovitch, On the Sum of Digits of Real Numbers Represented 
in the Dyadic System, Math. Anna/en, 110 (1934), 321-30. 

P. BiIlingsley, Probability and Measuro, Wiley, New York, 1978. 

G. Brown, G. Michon and J. Peyriere, On the Mulfifractal Analysis 
of Measures, J. Stat. Physics 66 (1992), 775·790. 

R. Cawley and R.D. Mauldin, Multifractal Decomposition of Moran 
Fractals, Adtl. in Math. 92 (1992), 196·236. 

C.D. Cutler, The HausdorffDimension Distribution ofFinite Measures 
in Euclidean Spaces, Canad. J. Math. (1986) 38 (6), 1459-1484. 

C.D. Cutler, Connecting Ergodicity and Dimension in Dynamical Sys­
tems, Ergodic Theory Dyn. Syst. (1990) 10 451·462. 

A. DeUu, J.S. Geronimo, R. Shonkwiler and D. Hardin, Dimen­
sions AS80ciated with Recurrent Self-similar Sets, Math. Proc. Cambo 
Phi/o Soco 110 (1991), 327-36. 

H.G. Egglestone, The Fractional Dimension of a Set Defined by Dec­
imal Properties, Quart. J. 01 Math. Oxford Ser. 20 (1949), 31-6. 

K.J. Falconer, The Geometry 01 Fracta/ Sets, Cambridge University 
Press, 1985. 

H. Haase, Densities of Hausdorff Measures on Generalized Self-Similar 
Sets, preprint. 

J .E. Hutchinson, Fractals and self similarity, Indiana Unitl. Math. J. 
30 (1981), 719-47. 

R.D. Mauldin and M. UrbalÍski, Dimensions and Measures in Infi· 
nite Iterated Function Systems, Proc. London Math. Soco (to appear) 

M. Morán, Hausdorff Measure of Infinitely Generated Self-Similar 
Sets, Monast. jtJr Math. (forthcoming) 

M. Morán and J.-M. Rey, Singularity of Self-Similar Measures with 
respect to Hausdorff Measures, preprint. 



[Ols] 

[Rey 95] 

[RT 61] 

[Sch 94] 

[Spe 92] 

[Tay 86] 

[TT 85] 

[Tri 82] 

[Wal82] 

[Wic 92] 

[You 81] 

M. Morán & J.-M. Rey 23 

L. OIBen, A Multifractal FormaJism, Adv. in Math. (to appear). 

J.-M. Rey, Geometría de Medidas y Conjuntos Autosemejantes, Ph.D. 
thesis, Universidad Complutense de Madrid, 1995. 

C.A. Rogers and S.J. Taylor, Functions Continuous and Singular 
with respect to a Hausdorff Measure, Mathematika, 8 (1961), 1-31. 

A. Schief, Separation Properties for Self-Similar Seta, Proc. Amer. 
Math. Soco 122 (1994), 111-115. 

D.W. Spear, Measures and Self-Similarity, Adv. in Math. 91(2) 
(1992), 143-157. 

S.J. Taylor, The Measure Theory of Random Fractals, Math. Proc. 
Cambo Phi!. Soco 100 (1986),383-406. 

S.J. Taylor and C. Tricot, Packing Measure, and its Evaluation for a 
Brownian Path, lhms. Amer. Math. SOCo 288(2) (1985), 679-699. 

C. Tricot, Two Definitions ofFractional Dimension, Math. Proc. Cambo 
Phi/o SOCo 91 (1982), 57-74. 

P. Walters, An Introductíon to Ergodic Theory, Springer-Verlag, 1982. 

K.R. Wicks, Fractals and Hyperspaces, Lecture Notes in Math. 1492, 
Springer-Verlag, 1992. 

L.-S. Young, Dimension, Entropy and Liapunov Exponents, Ergodic 
Theory & Dynamical Systems 2 (1982), 109-24. 

Manuel Morán & José--Manuel Rey 

Departamento de Análisis Económico 
Universidad Complutense. 
Campus de Somosaguas. 
28223 Madrid. 
Spain. 

ececo06@sis.ucm.es, ececo07@sis.ucm.es 


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