Relative multifractal analysis

$: Relative multifractal analysis$
Relative multifractal analysis

Julian Cole *

Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife, Scotland KY16 9SS, UK

Accepted 28 July 1999

Abstract

We introduce a general formalism for the multifractal analysis of one probability measure with respect to another. As an example,

we analyse the multifractal structure of one graph directed self-conformal measure with respect to another. Ó 2000 Elsevier Science

Ltd. All rights reserved.

1. Introduction

Given a metric space X, let us denote the family of Borel probability measures on X by M1�X �. Letl 2M1�X � and x 2 supp l, where supp l denotes the topological support of l, then the lower and upperlocal dimensions of l at x are de®ned to be

al x� � � lim infr!0

log lB x; r� �log r

and al x� � � lim supr!0

log lB x; r� �log r

;

respectively. If al x� � � al�x� then we refer to the common value as the local dimension of l at x and de-note it by al x� �. For aP 0 we de®ne the Hausdor� and packing multifractal spectra of l by

fl�a� � dimH x 2 supp l j al�x�� a

and Fl�a� � dimP x 2 supp l j al x� �� a

;

respectively, where dimH and dimP denote Hausdor� and packing dimension, respectively. In recent yearsthe key idea in multifractal analysis has been to calculate these spectra. In this paper we explore a modi-®cation of this type of analysis. Instead of studying sets of points which have a given local dimension withrespect to Lesbegue measure, we study sets of points which have a given local dimension with respect to anarbitrary probability measure. More speci®cally, we calculate the size of sets

K�c� � x 2 supp l \ suppp m limr&0

log l�B�x; r��log m�B�x; r��

�� c

�;

where l; m 2M1�X �. The analysis of these sets has a long history; in particular, a key idea in Billingsley's1965 book Ergodic Theory and Information [2] was the calculation of the size of these sets, and in [3], Cajarstudied these sets in the setting of symbolic dynamics. In several recent papers on multifractal analysis thistype of multifractal analysis has re-emerged as mathematicians and physicists have begun to discuss the

www.elsevier.nl/locate/chaos

Chaos, Solitons and Fractals 11 (2000) 2233±2250

* Present address: Department of Philosophy, Ohio State University, 230 N. Oval Mall, 350 University Hall, Columbus, OH 43210,

USA. Tel.: +1-614-292-7914; fax; +1-614-292-7502.

E-mail address: [email protected] (J. Cole).

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 4 3 - 5

idea of performing multifractal analysis with respect to an arbitrary reference measure (see [1,6,11,14]). Inthis paper we formalise these ideas by introducing a formalism for the multifractal analysis of onemeasure with respect to another. This formalism is based on the ideas of the `multifractal formalism' asexplicated by Halsey et al. [9], and closely follows the formal treatment of this formalism given byPeyri�ere and Olsen (see [12,13]). As an example of relative multifractal analysis we analyse the multifractalstructure of one graph directed self-conformal measure with respect to another. In particular, for technicalreasons relating to the introduction of the Hausdor� and packing dimensions, for a; c P 0 and u 2 V , weintroduce the sets

Ku�c; a� � x 2 supp lu \ supp mu limr&0

log lu�B�x; r��log mu�B�x; r��

�� c and lim

r&0

log mu�B�x; r��log r

� a

�:

We also de®ne functions amu ; flu; mu, and clu

; mu (see Section 3.2), such that

dimH�Ku�clu;mu�q�; amu�q�� dimP�Ku�clu;mu

�q�; amu�q�� amu�q��qclu;mu�q� � flu;mu�q��:

In addition, we give a counter example to the natural conjecture that,

dimH�Ku�clu;mu�q�� dimP�Ku�clu;mu

�q�� amu�q��qclu;mu�q� � flu;mu�q��:

Before we begin our analysis we must introduce two well known measures; the centred m-Hausdor� measureand the m-packing measure. For q 2 R de®ne uq: �0;1� ! �0;1� by

uq�x�

1 for x � 0

xq for 0 < xfor q < 0;

1 for q � 0;

0 for x � 0

xq for 0 < xfor q < 0;

8>>>>>>>>>><>>>>>>>>>>:With this de®nition in place, given m 2M1�X �, for s; d > 0 set

Hsm;d�E� � inf

Xi

us m B xi; ri� �� ( �� B xi; ri� �� i is a centred d-covering of E

); E 6� ;;

Hsm;d ;� � � 0; Hs

m;0 E� � � supd>0

Hsm;d E� �; Hs

m E� � � supF�E

Hsm;0 F� �:

Psm;d�E� � sup

Xi

us�m�B�xi; ri��( ��xi; ri��i is a centred d-packing of E

); E 6� ;;

Psm;d�;� � 0; Ps

m;0�E� � infd>0

Psm;d�E�; Ps

m�E� � infE�[

iEi

Xi

Psm;0�Ei�:

It is well-known that these set functions are metric outer measures and that these measures de®ne twodimension functions in the usual way, we denote them by dimm and Dimm, respectively. These measures anddimensions are obviously related to the measures and dimensions introduced by Billingsley; the di�erencebetween the two is that Billingsley used centred m-d-coverings rather than centred d-coverings.

2. A relative multifractal formalism

2.1. Relative multifractal measures

We begin our formalism by de®ning two generalised measures.

2234 J. Cole / Chaos, Solitons and Fractals 11 (2000) 2233±2250

De®nition 2.1. Let X be a metric space. For l; m 2M1 X� �, E � X , q, t 2 R, and d > 0, we make the fol-lowing de®nitions:

Hq;tl;m;d�E� � inf

Xi

uq l B xi; ri� �� ut m B xi; ri� �� ( ��B�xi; ri��i is a centred d-covering of E

);

E 6� ;;

Hq;tl;m;d ;� � � 0; Hq;t

l;m;0 E� � � supd>0

Hq;tl;m;d E� �; Hq;t

l;m E� � � supF�E

Hq;tl;m;0 F� �;

Pq;tl;m;d E� � � sup

Xi

uq l B xi; ri� �� ut m B xi; ri� �� ( �� B xi; ri� �� i is a centred d-packing of E

);

E 6� ;;

Pq;tl;m;d ;� � � 0; Pq;t

l;m;0 E� � � infd>0

Pq;tl;m;d E� �; Pq;t

l;m E� � � infE�[

iEi

Xi

Pq;tl;m;0 Ei� �;

where we set 0 � 1 � 1 � 0 � 0.

The reader should note two important features of the pre-measures. First, Hq;tl;m;0 is countably sub-

additive but not necessarily monotone. Second, Pq;tl;m;0 is a monotone set function but not necessarily

countably subadditive.

Proposition 2.2. The set functions Hq;tl;m and Pq;t

l;m are metric outer measures and thus measures on the Borelalgebra.

Proof. Follows by standard arguments (see, for example [12, Propositions 2.2 and 2.3]). �

Proposition 2.3. There exist unique extended real valued numbers dimql;m�E� 2 �ÿ1;1�, Dimq

l;m�E� 2�ÿ1;1� and Dq

l;m�E� 2 �ÿ1;1� such that:

Hq;tl;m�E� � 1 t < dimq

l;m�E�;0 dimq

l;m E� � < t;

(

Pq;tl;m�E� � 1 t < Dimq

l;m�E�;0 Dimq

l;m�E� < t;

(

Pq;tl;m;0�E� � 1 t < Dq

l;m�E�;0 Dq

l;m�E� < t:

(

Proof. This follows by elementary arguments. �

The following properties of Hq;tl;m, P

q;tl;m, and Pq;t

l;m;0 are easily seen from the de®nitions: for t P 0

H0;tl;m �Ht

m; P0;tl;m � Pt

m and P0;tl;m;0 � Pt

m;0:

Hence if, as before, we denote m-Hausdor�, m-packing and m-pre-packing dimension by dimm, Dimm and Dm

respectively, then for E � suppl \ suppm we have

dimm E� � � dim0l;m E� �; Dimm E� � � Dim0

l;m E� � and Dm�E� � D0l;m�E�:

An important feature of the Hausdor� and packing measures is that for t > 0 they satisfy Ht6Pt. Ournext step is to show a similar relationship between the generalised measures. We start by de®ning a subclass

J. Cole / Chaos, Solitons and Fractals 11 (2000) 2233±2250 2235

of measures. For l 2M1 X� � and a > 1 write Ta l� � � lim supr&0 supx2supp l

�lB x; ar� ��=�lB x; r� ��

and de®nethe family M1

D X� � of doubling probability measures on X by

M1D X� � � l 2M1 X� � j Ta l� ��

<1 for some a > 1:

It is easily seen that the de®nition of M1D�X � is independent of a (see [12]).

Theorem 2.4. Let l; m 2M1 Rdÿ �

and q; t 2 R. Then1. Pq;t

l;m6Pq;tl;m;0;

2. there exists an integer f 2 N, such that Hq;tl;m6 fPq;t

l;m;3. dimq

l;m 6Dimql;m6Dq

l;m;4. for l; m 2M1

D�Rd�, Hq;tl;m6Pq;t

l;m.

Note. In fact, for q6 0 the condition in �4� that l 2M1D�Rd� can be relaxed; similarly for t6 0 the condition

that m 2M1D�Rd� can be relaxed.

Proof.

1. Follows immediately from the de®nitions.2. Follows by a standard application of the Besicovitch covering theorem (see ([12, Proposition 2.4]).3. Follows immediately from �2�.4. Follows by a standard application of the Vitali covering theorem (see [12, Proposition 2.4] for the key

idea). �

2.2. Dimension functions

Our next step is to de®ne three multifractal dimension functions bl;m;Bl;m and Kl;m : R! �ÿ1;1� bysetting

bl;m�q� � dimql;m suppl \ suppm� �; Bl;m�q� � Dimq

l;m suppl \ suppm� �and

Kl;m�q� � Dql;m�supp �l� \ supp �m��:

The de®nition of these dimension functions makes it clear that they are counterparts of the s function whichappears in the `multifractal formalism'. This being the case it is important that they have the propertieswhich physicists ascribe to them. The next theorem shows that these functions do indeed have some of theseproperties.

Theorem 2.5. Let X be a metric space and l; m 2M1�X �, then the following hold.1. Pq;t

l;m;0 PPp;tl;m;0 for q6 p and Pq;s

l;m;0 PPq;tl;m;0 for s6 t.

2. Kl;m is decreasing and convex.3. The map �q; t� ! Pq;t

l;m;0 is logarithmically convex i.e., for all a 2 �0; 1�, p; q; s; t 2 R and E � X ,

Pap��1ÿa�q;at��1ÿa�sl;m;0 �E�6 �Pp;t

l;m;0�E��a�Pq;sl;m;0�E��1ÿa

:

4. Pq;tl;m PPp;t

l;m for q6 p and Pq;sl;m PPq;t

l;m for s6 t.5. Bl;m is decreasing and convex.6. Hq;t

l;m PHp;tl;m for q6 p and Hq;s

l;m PHq;tl;m for s6 t.

7. bl;m is decreasing.

Proof. We refer the reader to [12, Proposition 2.10] for details of how to prove this theorem. �


Corollary 2.6. For l; m 2M1�X � be such that suppl � suppm, then we have:1. for q < 1, 06 bl;m�q�6Bl;m�q�6Kl;m�q�;2. bl;m�1� � Bl;m�1� � Kl;m�1� � 0;3. for q > 1, bl;m�q�6Bl;m�q�6Kl;m�q�6 0.

Proof. This follows immediately from the above theorem and de®nitions. �

2.3. The m-multifractal spectrum

Having de®ned the generalised multifractal m-Hausdor� and m-packing measures and the m-Hausdor�,m-packing and m-pre-packing dimension functions we wish to demonstrate their usefulness by showing theirconnection to the m-multifractal spectra.

Given l; m 2M1�X � the upper, respectively lower, local dimension of l with respect to m at x 2 X is de®nedby

cl;m�x� � lim supr&0

log l�B�x; r��log m B�x; r�� ; respectively c

l;m�x� � lim inf

r&0

log l�B�x; r��log m�B�x; r�� :

If cl;m�x� � cl;m�x�, then the common value, known as the local dimension of l with respect to m at x, is

denoted by cl;m�x�. Given l; m 2M1�X � for cP 0 set

Kc � x 2 suppl \ suppm j cl;m x� ��

6 c

; Kc � x 2 suppl \ suppm j c�6 cl;m x� �;

Kc � x 2 suppl \ suppm j cl;m

x� �n

6 co

; Kc � x 2 suppl \ suppm j cn

6 cl;m

x� �o:

Also, let

K�c� � Kc \ Kc � x 2 suppl \ suppm j cl;m�x�

� � c:

With these de®nitions we have the following theorem.

Theorem 2.7. Let X be a metric space and l; m 2M1�X �. Also fix cP 0, q; t 2 R and d > 0 such thatd6 cq� t. Then we have the following:1.

(a) Hcq�t�dm �Kc�6Hq;t

l;m�Kc� for 06 q.

(b) Hcq�t�dm �Kc�6Hq;t

l;m�Kc� for q6 0.(c) If 06 cq� bl;m�q�, then

dimm�Kc�6 cq� bl;m�q� for 06 q; dimm�Kc�6 cq� bl;m�q� for q6 0:

In particular, dimm�Kc�6 c.2.

(a) Pcq�t�dm �Kc�6Pq;t

l;m�Kc� for 06 q.

(b) Pcq�t�dm �Kc�6Pq;t

l;m�Kc� for q6 0.(c) If 06 cq� Bl;m�q�, then

Dimm �Kc�6 cq� Bl;m�q� for 06 q; Dimm �Kc�6 cq� Bl;m�q� for q6 0:

In particular, Dimm �Kc�6 c.3.

(a) If A � Kc

is Borel, then Hq;tl;m�A�6Hcq�tÿd

m �A� for q6 0.(b) If A � Kc is Borel, then Hq;t

l;m�A�6Hcq�tÿdm �A� for 06 q. In particular, if A � Kc is Borel and

l�A� > 0, then c6 dimm�A�.4.

(a) If A � Kc

is Borel, then Pq;tl;m�A�6Pcq�tÿd

m �A� for q6 0.(b) If A � Kc is Borel, then Pq;t

l;m�A�6Pcq�tÿdm �A� for 06 q. In particular, if A � Kc is Borel and l�A� > 0,

then c6Dimm �A�.


Proof. An exhaustive proof of this theorem would require considerable repetition. All of the ideas needed toprove this theorem can be found in the proofs of Propositions 2.5 through 2.8 in [12]. To aid the reader inusing these ideas we prove 1�b�.

1�b� It is well-known that the statement is true for q � 0. For m 2 N let us set

Tm � x 2 Kc c

�� d

q6 log�l�B�x; r��

log�m�B�x; r�� for 0 < r <1

m

�:

Given m 2 N let us choose g such that 0 < g < 1m and let �Bi :� B�xi; ri��i be a centred g-covering of E � Tm.

Then we have

log�l�B�xi; ri��log�m�B�xi; ri�� P c� d

q) l�B�xi; ri��6 m�B�xi; ri��c�

dq

) l�B�xi; ri��q P m�B�xi; ri��cq�d

) l�B�xi; ri��qm�B�xi; ri��t P m�B�xi; ri��cq�d�t:

Hence,

Hcq�d�tm;g �E�6

Xi

m�B�xi; ri��cq�d�t6X

i

l�B�xi; ri��qm�B�xi; ri��t:

Now from this we can deduce that for g < 1m, Hcq�d�t

m;g �E�6Hq;tl;m;g�E�, and letting g& 0 gives that for all

E � Tm,

Hcq�d�tm;0 �E�6Hq;t

l;m;0�E�6Hq;tl;m�E�6Hq;t

l;m�Tm�:Hence,

Hcq�d�tm �Tm�6Hq;t

l;m�Tm�:The result follows since Tm % Kc. �

Theorem 2.7 allows us to consider the relationship between the dimension functions bl;m and Bl;m and them-multifractal spectra. We start by giving an upper bound theorem. For l; m 2M1�X � set

al;m :� sup0<qÿ bl;m�q�

q; al;m :� inf

q<0ÿ bl;m�q�

q; Al;m :� sup

0<qÿBl;m�q�

qand Al;m :� inf

q<0ÿBl;m�q�

q;

then Al;m6 al;m and al;m6Al;m. With these de®nitions we have the following theorem.

Theorem 2.8. Let X be a metric space, l; m 2M1�X � and cP 0. Then the following hold:1. al;m6 inf cl;m�x�6 sup cl;m�x�6Al;m and Al;m6 inf c

l;m�x�6 sup c

l;m�x�6 al;m;

2.

dimm�K�c�� 6 b�l;m�c�; c 2 al;m; al;m

ÿ �;

� 0; c 2 0;1� � n �al;m; al;m�;�

3.

Dimm �K�c�� 6 B�l;m�c�; c 2 �al;m; al;m�;� 0; c 2 0;1� � n �al;m; al;m�:

�

Proof. This theorem follows immediately from Theorem 2.7 and the following lemma. �

Lemma 2.9. If X is a metric space, l; m 2M1�X � and c P 0, then1. Kc � ; for c < Al;m.2. Kc � ; for al;m < c.


3. Kc � ; for Al;m < c.4. K

c � ; for c < al;m.

Proof. See [12, Lemma 4.4] for the key ideas needed to prove this lemma. �

Theorem 2.10. Let X be a metric space and l; m 2M1�X �. If A � K�c� is a Borel set such that Hq;tl;m�A� > 0,

where q; t 2 R are such that cq� t P 0, then

dimm�A�P cq� t:

In particular, if bl;m is differentiable at q and we set c�q� � ÿb0l;m�q�, then, provided that b�l;m�c�q��P 0 andHq;bl;m�q�

l;m �K�c�q�� > 0, we have

dimm�K�c�q�� b�l;m�c�q��:

Proof. Follows immediately from Theorem 2.7. �

Theorem 2.11. Let X be a metric space and l; m 2M1�X �. If A � K�c� is a Borel set such that Pq;tl;m�A� > 0,

where q; t 2 R are such that cq� t P 0, then

Dimm �A�P cq� t:

In particular, if Bl;m is differentiable at q and we set c�q� � ÿB0l;m�q�, then, provided that B�l;m�c�q��P 0 andPq;Bl;m�q�

l �K�c�q�� > 0, we have

Dimm �K�c�q�� B�l;m�c�q��:


2.4. The relative multifractal spectrum

While the m-multifractal spectra are of theoretical interest it is more natural for us to want to know whatthe actual multifractal spectra are i.e., to know the Hausdor� and packing dimensions of the sets K�c�. Inthe general case the best that we can do is to decompose the sets K�c� according to the m-local dimension oftheir points and then calculate the size of the subset of K�c� whose points have m-local dimension a (this ideacan ®rst be found in [14]).

Given m 2M1�X � recall that the upper, respectively lower, local dimension of m at x 2 X is de®ned by,

am�x� � lim supr&0

log m�B�x; r��log r

; respectively am�x� � lim infr&0

log m�B�x; r��log r

;

and that if am�x� � am�x�, then the common value is known as the local dimension of m at x and denoted byam�x�. Given l; m 2M1�X �, for c; aP 0, set

K�c; a� � x 2 suppl \ suppm j cl;m�x��

6 c and am�x�6 ag;K�c; a� � x 2 suppl \ suppm j c

n6 c

l;m�x� and am x� �6 a

o;

K�c; a� � x 2 suppl \ suppm j cl;m xÿ ��6 c and a6 am�x�g;

K�c; a� � x 2 suppl \ suppm j cn

6 cl;m

x� � and a6 am�x�o:

Also, let

K�c; a� � K�c; a� \ K�c; a� \ K�c; a� \ K�c; a� � x 2 suppl \ suppm j cl;m�x�� c; am�x� � a

:


Finally, set

fl;m�c; a� � dimH K�c; a� and Fl;m�c; a� � dimP K�c; a�:

Theorem 2.12. Let X be a metric space and l; m 2M1�X �. Also fix c; aP 0, q; t 2 R and d1; d2 > 0 such thatd16 cq� t and d26 a�cq� t ÿ d1�. Then we have the following:1.

(a) Ha�cq�t�d1��d2�K�c; a��6 2a�cq�d1��d2Hq;tl;m�K�c; a�� for 06 q.

(b) Ha�cq�t�d1��d2�K�c; a��6 2a�cq�d1��d2Hq;tl;m�K�c; a�� for q6 0.

2.(a) Pa�cq�t�d1��d2�K�c; a��6 2a�cq�d1��d2Pq;t

l;m�K�c; a�� for 06 q.

(b) Pa�cq�t�d1��d2�K�c; a��6 2a�cq�d1��d2Pq;tl;m�K�c; a�� for q6 0.

3.(a) If A � K�c; a� is Borel then; for q6 0;Hq;t

l;m�A�6 2ÿ�a�cq�tÿd1�ÿd2�Ha�cq�tÿd1�ÿd2�A�.(b) If A � K�c; a� is Borel then; for 06 q, Hq;t

l;m�A�6 2ÿ�a�cq�tÿd1t�ÿd2�Ha�cq�tÿd1�ÿd2�A�.4.

(a) If A � K�c; a� is Borel then; for q6 0, Pq;tl;m�A�6 2ÿ�a�cq�tÿd1�ÿd2�Pa�cq�tÿd1�ÿd2�A�.

(b) If A � K�c; a� is Borel then; for 06 q,Pq;tl;m�A�6 2ÿ�a�cq�tÿd1�ÿd2�Pa�cq�tÿd1�ÿd2�A�.

Proof. An exhaustive proof of this theorem would require considerable repetition. Thus we only prove 4(b).4(b) The statement is well-known for q � 0. For m 2 N let us set

Tm � x 2 A c

��ÿ d1

q6 log l B x; r� ��

log m B x; r� �� and aÿ d2

cq� t ÿ d2

6 log�m�B�x; r��log r

for 0 < r <1

m

�:

Now given m 2 N, E � Tm and 0 < g < 1m let B xi; ri� �� i2N be a centred d-packing of E. Then we have

log l B xi; ri� �� log m B xi; ri� �� P cÿ d1

q) l B xi; ri� �� 6 m B xi; ri� �� cÿ

d1q

) l B xi; ri� �� q6 m B xi; ri� �� cqÿd1

) l B xi; ri� �� qm B xi; ri� �� t6 m B xi; ri� �� cq�tÿd1 :

Also, we have,

log m B x; r� �� log r

P aÿ d2

cq� t ÿ d1

) m B xi; ri� �� 6 raÿ d2

cq�tÿd1i

) m B xi; ri� �� cq�tÿd1 6 ra cq�tÿd1� �ÿd2

i :

Putting these together we see that l B xi; ri� �� qm B xi; ri� �� t6 ra cq�tÿd1� �ÿd2

i . HenceXi

l B xi; ri� �� qm B xi; ri� �� t6 2ÿ a cq�tÿd1� �ÿd2� �Xi

2ri� �a cq�tÿd1� �ÿd2 6 2ÿ a cq�tÿd1� �ÿd2� �Pa cq�tÿd1� �ÿd2g �E�:

From this we can deduce that for g < 1m, Pq;t

l;m;g�E�6 2ÿ a cq�tÿd1� �ÿd2� �Pa cq�tÿd1� �ÿd2g �E�. Thus letting g& 0 gives

that for all E � Tm

Pq;tl;m;0�E�6 2ÿ a cq�tÿd1� �ÿd2� �Pÿa cq�tÿd1� �ÿd2

0 �E�:Finally, let �Ei�i2N be a covering of Tm. Then we have

Pq;tl;m�Tm�6Pq;t

l;m

[i

Tm \ Ei� � !

6X

i

Pq;tl;m Tm \ Ei� �6

Xi

Pq;tl;m;0 Tm \ Ei� �

6 2ÿ a cq�tÿd1� �ÿd2� �Xi

Pa cq�tÿd1� �ÿd2

0 Tm \ Ei� �6 2ÿ a cq�tÿd1� �ÿd2� �Xi

Pa cq�tÿd1� �ÿd2

0 �Ei�:


Hence

Pq;tl;m�Tm�6 2ÿa cq�tÿd1� �ÿd2Pa cq�tÿd1� �ÿd2�Tm�;

and the result follows since A � Sm Tm. �

Theorem 2.12 allows us to consider the relationship between the dimension functions bl;m and Bl;m andthe spectral functions fl;m and Fl;m. We start by giving an upper bound theorem.

Theorem 2.13. Let X be a metric space, l; m 2M1�X � and c; aP 0. Then the following hold:1.

fl;m c; a� � � 6 a � b�l;m c� �; c 2 al;m; al;m

ÿ �;

� 0; c 2 0;1� � n �al;m; al;m�;

(2.

Fl;m c; a� � � 6 a � B�l;m�c�; c 2 al;m; al;m

ÿ �;

� 0; c 2 0;1� � n �al;m; al;m�:

(

Proof. Follows immediately from Theorem 2.12 and Lemma 2.9. �

Theorem 2.14. Let X be a metric space, c; a P 0 and l; m 2M1�X �. If A � K�c; a� is a Borel set such thatHq;t

l;m�A� > 0, where q; t 2 R are such that cq� t P 0, then

dimH�A�P a�cq� t�:

In particular, if bl;m is differentiable at q and we set c�q� � ÿb0l;m�q�, then, provided that b�l;m c q� �� P 0 andHq;bl;m�q�

l;m K c�q�; a� �� > 0, we have

fl;m c q� �; a� � � a � b�l;m�c�q��:


Theorem 2.15. Let X be a metric space, c; a P 0 and l; m 2M1�X �. If A � K c; a� � is a Borel set such thatPq;t

l;m�A� > 0, where q; t 2 R are such that cq� t P 0, then

dimP�A�P a�cq� t�:In particular, if Bl;m is differentiable at q and we set c�q� � ÿB0l;m�q�, then, provided that B�l;m�c�q��P 0 andPq;Bl;m�q�

l K c�q�; a� �� > 0, we have

Fl;m c�q�; a� � � a � B�l;m c�q�� :


3. Graph directed self-conformal measures

In this section we give an example to illustrate the general theory which we have been developing. Weanalyse the relative multifractal structure of one graph-directed self-conformal measure with respect toanother in the case where the two measures are based on the same iterated function scheme. For detailsconcerning the preliminaries in Section 2.1 we recommend that the reader refer to [4].


3.1. The setting

We begin by introducing graph directed self-conformal iterated function schemes (GCIFSs). Given�V ;E�, a ®nite directed connected graph, a GCIFS is a way of constructing a vector of non-empty compactsubsets of

Qu2V Rd . Let G � �V ;E� be a ®nite directed connected graph, for u; v 2 V let Eu;v denote the set of

edges from u to v and set Eu �S

v2V Eu;v i.e., the set of all edges leaving u. For e 2 E let us denote the initialvertex of e by i�e� and the terminal vertex of e by t�e�. With each vertex u 2 V let us associate three sets, Uu,Ju and Wu. We choose these sets in the following way, for each u 2 V let Uu be an open and connected subsetof Rd , Ju be a regular compact subset of Uu, and Wu be an open connected set such that W u is compact andJu � Wu � W u � Uu. Also, with each edge e 2 E let us associate a map Te and a number pe 2 �0; 1� such that:1. for some c 2 �0; 1� and all e 2 E, Te : Ut�e� ! Ui�e� is a conformal C1�c di�eomorphism such that

Te Jt�e�ÿ � � Ji�e�;

2. for all e 2 E and x 2 Ut�e�; 0 < jT 0e�x�j < 1;3. for all u 2 V ,

Pv2V

Pe2Eu;v

pe � 1.A collection G � �V ;E; �Te�e2E; �pe�e2E� where the above conditions are satis®ed is called a GCIFS with

probabilities. Given G � �V ;E; �Te�e2E; �pe�e2E�, a GCIFS with probabilities, the triple �V ;E; �Te�e2E� iscalled a GCIFS. The vector of sets �Ju�u2V is known as the vector of seed sets of G.

Given G � �V ;E; �Te�e2E; �pe�e2E�, a GCIFS with probabilities, it is well-known that there exists a uniquevector K � �Ku�u2V of non-empty compact subsets of Rd satisfying

Ku �[v2V

[e2Eu;v

Te�Kv�:

We call these sets the graph directed self-conformal sets associated with G. Similarly, it is well-known thatthere exists a unique vector �lu�u2V of probability measures satisfying

�lu�u2V �Xe2Eu

pe:lt�e� � Tÿ1e

!u2V

:

We call these measures the graph directed self-conformal measures associated with G.We now consider three well-known separation conditions. Given �V ;E; �Te�e2E�, a GCIFS, set

D � min dist Te Jt e� �ÿ �

; Te0 Jt e0� ��

j e; e0 2 E; en

6� e0 and i�e� � i�e0�o:

We say that G satis®es the strong separation condition (SSC) if D > 0. If for all e 2 E; Te�int Jt�e�� int Ji�e�and for all u 2 V and e; e0 2 Eu such that e 6� e0; Te�intJt�e�� \ Te0 �intJt e0� �� ;, then we say that G satis®esthe open set condition (OSC). We note that in e�ect we are specifying our seed sets to be the closure of theopen sets �Vu�u2V in the OSC. Finally, if G satis®es the OSC and for each u 2 V , int Ju \ Ku 6� ;, then we saythat G satis®es the strong open set condition (SOSC).

For the remainder of these preliminaries, let G � V ;E; �Te�e2E; �pe�e2E

ÿ �be a GCIFS with probabilities

coded by a strongly connected graph i.e., a graph such that there exists a directed path between each pair ofvertices, and satisfying the strong open set condition. Also let �Uu�u2V , �Ju�u2V , �Wu�u2V , �Ku�u2V , �lu�u2V , and�pu�u2V be, respectively, the sets, measures, and maps associated with G that appear in the above de®nition.

Our analysis will be aided by the following notation. Let �V ;E� be a ®nite directed strongly connectedmultigraph. We call a ®nite string e1e2 . . . en of edges a ®nite path if for i � f1; . . . nÿ 1g, t�ei� � i�ei�1�.Similarly we call an in®nite string e1e2 . . . of edges an in®nite path if for all i 2 N, t�ei� � i�ei�1�. In addition,for u; v 2 V and n 2 N, we introduce the following notation:

E n� �u;v � e1 . . . en j paths of length n such that i e1� �f � u and t en� � � vg;

E �� u;v �

[n2N

E n� �u;v ; E n� �

u �[v2V

E n� �u;v ;E

�� u �

[v2V

E �� u;v ; E n� � �

[u2V

E n� �u ; E ��

[u2V

E �� u ;

ENu � e1e2 . . . j infinite paths such that i�e1�f � ug; EN �

[u2V

ENu :


If a � a1 . . . an 2 E��, b � b1b2 . . . 2 E�� [ EN and t�an� � i�b1� then ab � a1 . . . anb1b2 . . . Also if a 2 E�n�,then let us write j a j� n. Now if a 2 E�n�, b 2 E�m�, n6m and for j � f1; . . . ; ng; aj � bj, then we say a6b.Similarly if a 2 E�n�, x 2 E�N� and for j � f1; . . . ; ng; aj � xj then we say a6x. Also given b 2 E�n� orb 2 EN and m6 n or m 2 N, respectively, then we write b j m for b1 . . . bm. For a 2 E�� let us write t�a� fort�ajaj� and for a 2 EN [ E �� , i�a� for i�a1�. In addition, for s 2 E �� we call �s� � fx 2 EN j xjjsj � sg thecylinder associated with s.

It follows from our de®nitions that there exist numbers rmin and rmax such that 0 < rmin6jT 0e�x�j6 rmax < 1 for all e 2 E and x 2 Wt�e�. Let pmin � mine2E pe and pmax � maxe2E pe. Finally, givens 2 E �� , set Ts � Ts1

� � � � � Tsjsj , ps � ps1. . . psjsj and, given A � Xt�s�, set As � Ts�A�.

In this paper, we will frequently use symbolic dynamics. It is well-known that for each u 2 V we are ableto de®ne a `code map' or `projection' pu : EN

u ! Rd by setting fpu�x�g �T

n�Txjn�Kt�xn��.An important property of the code maps is that for each e 2 E and x 2 EN

t�e�, pi�e��ex� � Te�pt�e��x��. Ifwe introduce r : EN ! EN given by r�x1x2 . . .� � x2x3 . . . and, for s 2 E �� , rs : EN ! EN given byrs�x� � sx, then we see that the dynamical systems �Su2V Ku; Te� and �EN; re� are conjugate. We note thatfor u 2 V , we have Ku � pu�EN

u �.We now equip the code space EN with the following metric which induces the product topology. With

rmax and c equal to the values appearing in the de®nition of our GCIFS, if we de®ne cmax � rcmax then

cmax < 1. For s;x 2 EN set d�s;x� � cnmax, where n � min i j si 6� xif g. Given this metric, it is easy to see that

for each u 2 V , the map pu is Lipschitz.Kolmogorov's consistency theorem tells us that we can de®ne a probability measure lu on EN

u such thatlu��s�� ps for all s 2 E ��

u . With lu de®ned in this way we have the following lemma.

Lemma 3.1. For each u 2 V ,1. lu � lu � pÿ1

u ,2. supplu � Ku.

An important implication of this lemma is that, provided the GCIFS satis®es the SSC, for each s 2 E �� u ,

lu�Ks� � ps. In fact, if a GCIFS satis®es the strong open set condition, then this is still true.A concept which we will use later in this paper is that of a maximal anti-chain. Let F be a subset of E �� ,

we call a subset C of F a maximal anti-chain of F if for each x 2 F there exists a unique s 2 C such thats6x.

We now de®ne the metric scale function of G. This is the map w : EN ! R given by

w�x� � log jT 0x1�p�r�x��j:

The metric scale function w is c-H�older continuous and represents the local change in scale as one moves,under the map Tx1

, from p�r�x�� 2 Kt�x1� to p�x� 2 Ki�x�.The measure scale function of G is the map / : EN ! R given by

/ x� � � log px1:

We note that / is c-H�older continuous and represents the change in measure between a cylinder �s� and thecylinder �sx1�.

The Gibbs state of a H�older continuous function u is the unique r-invariant probability measure lu onEN such that there exist positive and ®nite constants a0 and P satisfying

aÿ10 exp�Snu�x� ÿ nP �6 lu��xjn��6 a0 exp�Snu�x� ÿ nP�

for all x 2 EN and n 2 N, where Sn/�x� �Pnÿ1

i�0 /�ri�x��.We note that P � P �u� is a constant depending on u and is known as the (topological) pressure of u. The

condition that lu be a probability measure implies that

P �u� � limn!1

1

nlog

Xs2E n� �

exp�Snu�s��:


Important to the analysis of GCIFSs is that we are able to ®nd an explicit expression for the measures�lu�u2V as Gibbs states. From now on let l/ denote the Gibbs state of the measure scale function /. If we setcu � l/�EN

u � and de®ne lu;/ � l/bENu =cu then since both lu;/ and lu are r-invariant ergodic probability

measures on ENu they coincide and for E � Ku, lu�E� � �1=cu��l/ � pÿ1

u �E��.An important property of GCIFSs is that they satisfy the principle of bounded distortion i.e., the

composition of ®nitely many of the maps Te in a GCIFS does not distort the geometry of the seed sets ofthat GCIFS too much.

Lemma 3.2. There exists a1 such that for all x 2 EN; n P 1 and x; y 2 Wt xn� � we have

aÿ11 jxÿ yj exp Snw xjn� �� 6 jTxjn x� � ÿ Txjn y� �j6 a1jxÿ yj exp Snw xjn� �� ;

where j � j denotes Euclidean distance.

The following bounds on the diameter of images of the seed sets and the invariant sets are an immediateconsequence of G satisfying the property of bounded distortion.

Corollary 3.3. For x 2 EN and n P 1 we have that if B � Wt�xn� then

aÿ11 exp Snw xjn� �� diam B6 diam Txjn B� �6 a1 exp Snw xjn� �� diam B;

in particular

aÿ11 exp Snw xjn� �� diam Jt xn� �6 diam Jxjn6 a1 exp Snw xjn� �� diam Jt xn� �;

and

aÿ11 exp Snw xjn� �� diam Kt xn� �6 diam Kxjn6 a1 exp Snw xjn� �� diam Kt xn� �:

We end our preliminary remarks about GCIFSs by introducing Graf's lemma. Let us begin by intro-ducing an auxiliary function b q� �. For q; b 2 R let lq;b be the Gibbs state of q/� bw, where / and w denotethe measure and metric scale functions. Also de®ne P : R2 ! R by P �q; b� � P �q/� bw�, where P �f�denotes the topological pressure of f.

Lemma 3.4. The function P q; b� � is real analytic, and if Di denotes partial differentiation with respect to theith variable, then D1P q; b� � � R / dlq;b and D2P�q; b� � R w dlq;b.

Proof. See [15]. �

The variational principle gives us that D2P < 0, so the implicit function theorem tells us that there existsa real analytic function b�q� such that P �q;b�q�� 0 for all q 2 R.

For q 2 R let lq be the Gibbs state of q/� b�q�w. We call these measures the q-equilibrium measures.For q 2 R and u 2 V , let lq

u denote the projection of the measure lq onto the set Ku under pu i.e., forE � Ku; lq

u�E� � lq � pÿ1u �E�. We now state a generalisation of Lemma 3.4 in [8].

Lemma 3.5. For q 2 R and u 2 V we haveR j log dist�x; oJu�j dlq

u�x� <1.

Proof. See [4]. �

Corollary 3.6. If q 2 R and u 2 V then for all a; s 2 E �� u such that a 6� 6 s and s 6� 6 a we have

lqu�Ja \ Js� � 0. Also for all s 2 E ��

u we have lqu�Js� � lq��s��.

3.2. Relative multifractal analysis

Having covered the preliminary material we are now in a position to calculate the relative multifractalspectrum of one graph directed self-conformal measure with respect to another.


For the remainder of this paper let G � V ;E; �Te�e2E; �pe�e2E

ÿ �and G0 � V ;E; �Te�e2E; �me�e2E

ÿ �be two

GCIFSs with probabilities based on the same GCIFS and satisfying the SOSC. Let us adopt the notationgiven in the previous section for G. For G0, let mu; mu; mq

u; mq; mmin; mmax and ms play the respective roles oflu; lu; lq

u; lq, pmin, pmax and ps. In addition, let v�x� � log mx1.

Our calculation of the relative multifractal spectrum of the measures lu with respect to mu is performed intwo steps. First we calculate the m-multifractal spectra and then the spectra flu;mu�c; a� and Flu;mu�c; a�.

In order to analyse the sets

Ku�c� � x 2 supplu \ suppmu j limr&0

log lu B x; r� �� log mu B�x; r��

�� c

�it is natural to consider the Gibbs state of the function q/� fv. Using a suitable generalisation of Lemma3.4 we are able to introduce a di�erentiable function flu;mu such that P �q; flu;mu�q�� 0, whereP �q; f� � P �q/� fv�. This leads us to consider the Gibbs state of the function q/� flu;mu�q�v, which wedenote by qq. By de®nition qq satis®es the following, there exists c 2 0;1� � such that for all s 2 E �� ,

cÿ1pqsm

flu ;mu �q�s 6 qq s� �� 6 cpq

smflu ;mu �q�s :

This being the case we hope that the reader will see that the calculation of the m-multifractal spectra isequivalent to the calculation of the multifractal spectra of a graph directed self-similar measure. In par-ticular, it is equivalent to calculating the multifractal spectra of the invariant measures of the GCIFSM � �V ;E; �Se�e2E; �pe�e2E�, where the maps Se are chosen to satisfy the following conditions; they havecontraction ratio me and map 0; 1� � into 0; 1� � such that for all e; f 2 Eu, Se� 0; 1� �� \ Sf � 0; 1� �� is either asingleton or empty. For in analysing this measures associated with M, one would introduce auxiliarymeasures lq satisfying,

lq s� �� pqsmf�q�

s for all s 2 E �� ;

where f�q� is de®ned to be the number which makes the spectral radius of the matrix with elementsPe2Eu;v

pqe mf�q�

e equal to 1, see [7].The above considerations serve to justify the following theorem. Let clu;mu

�q� � ÿf0lu;mu�q� and set

a � infq2R clu;mu�q�

n oand a � supq2R clu;mu

�q�n o

.

Theorem 3.7. If a < a, then for all u 2 V ,1. dimm Ku�c� � Dimm Ku�c� � f�lu;mu

�c� for c 2 a; a� �, where f�lu;mudenotes the Legendre transform of flu;mu .

2. Ku�c� � ; for c 6�2 a; a� �.

We now prove the main result of this example i.e., that for two pairs of graph directed self-conformalmeasures we have equality in Theorem 2.13.

Theorem 3.8. If a < a, then for each c 2 �a; a� there exist q 2 R such that clu;mu�q� � c and we have that for all

u 2 V ,1. flu;mu clu;mu

�q�; amu�q��

� Flu;mu clu;mu�q�; amu�q�

� �� amu�q� f�lu;mu

clu;mu�q�

� �� 2. Ku c; a� � � ; for c 6�2 a; a� �.

Proof.

(1) The upper bound follows from Theorem 2.13; the lower bound follows from Theorem 3.11 below.(2) Follows from Lemma 2.9. �

Our ®nal task in this section is to introduce the material needed to prove Theorem 3.11 which we used inthe above proof. The above section indicates that the measure qq which satis®es

qq s� �� pqsm

flu ;mu �q�s for all s 2 E ��


is important in this task. We note that we can choose c equal to one because we know by uniqueness thatthis Bernoulli measure must coincide with the Gibbs state. We also require qq

u, its projection under pu ontoKu. At this stage we note the following lemma which follows as a corollary to Lemma 3.5.

Lemma 3.9. For q 2 R and u 2 V we haveR j log dist�x; oJu�j dqq

u�x� <1.

Set k�q� � R w dqq, g�q� � R / dqq, j�q� � R v dqq, clu;mu�q� � g�q�=j�q�, hlu;mu�q� � g�q�=k�q� and

amu�q� � j�q�=k�q�. With these de®nitions in place let us note that hlu;mu�q� � clu;mu�q�amu�q� and that the clu;mu

which we have de®ned here coincides with the clu;muwhich we de®ned in the last section. Now the Ergodic

theorem gives us the following lemma.

Lemma 3.10. With k�q�, g�q� and j�q� defined as above:1. limn!1 1

n Snw xjn� � � k�q� for qq-a:a: x 2 EN;

2. limn!1 1n log pxjn � g�q� for qq-a:a: x 2 EN;

3. limn!1 1n log mxjn � j�q� for qq-a:a: x 2 EN:

We now use this lemma to show that qqu is a measure with local dimension almost surely equal to

amu�q�f�lu;mu�clu;mu

�q�� supported on Ku�clu;mu�q�; amu�q�� :� Ku \ K�clu;mu

�q�; amu�q��.The following maximal anti-chains will be useful in the proof of the next theorem. For u 2 V and

r 2 �0; 1� set

Cu;r � s 2 E �� u j exp Sjsjw s� �ÿ ��

<r

a1 diam Ju6 exp�Sjsjÿ1w�sksj ÿ 1��

�:

Theorem 3.11. Given u 2 V and q 2 R, for qqu-a:a: x 2 Ku

1.

limr!0

log lu�B�x; r��log r

� hlu;mu�q�;

2.

limr!0


� amu�q�;

3.

limr!0

log lu�B�x; r��log mu�B�x; r�� clu;mu

�q�;

4.

limr!0

log qqu�B�x; r��log r

� amu�q��qclu;mu�q� � flu;mu�q��;

5.

dimH Ku clu;mu�q�; amu�q�

� �P amu�q� qclu;mu

�q��

� flu;mu�q��:

Proof. (1) Given r > 0 and x 2 ENu choose kr x� � 2 N such that xjkr x� � 2 Cu;r. Then Jxjkr x� � � B p x� �; r� �.

Therefore with kr � kr x� �log lu B p x� �; r� ��

log r6

log lu Jxjkr

ÿ �log r

6 log pxjkr

Skrÿ1w xjkr ÿ 1� � � log diam J � log a1

� log pxjkr

kr

Skrÿ1w xjkr ÿ 1� � � log diam J � log a1

kr

�


and hence

lim supr!0

log lu B p x� �; rÿ �ÿ �log r

6 hlu;mu�q� for qq-a:a: x 2 ENu :

Now, since qqu � qq � pÿ1

u ;

lim supr!0

log lu B x; r� �� log r

6 hlu;mu�q� for qqu-a:a: x 2 Ku:

To get the opposite inequality we de®ne the following functions: for u 2 V and m 2 N, let du;m : ENu ! R

be given by, du;m x� � � dist�pu x� �; oJxjm�. Lemma 3.2 gives us that du;m x� �P aÿ11 exp�Smw�xjm��

di�xm�;0�rm x� ��. For m � 0; 1; . . ., let us set Rm � x 2 EN j di x� �;m x� � > 0�

. It follows from Lemma 3.9 thatqq�EN n Rm� � 0 for m � 0; 1; . . . Also if we set

R � x 2 EN j di x� �;m x� ��> 0; for m � 0; 1; . . .

;

then qq EN n R� � � 0, since R � \mRm. Now for 0 < r < 1 and x 2 R we are able to choose mr x� � 2 N suchthat du;mr x� ��1 x� �6 r < du;mr x� �. Then by the de®nition of du;m; B�pu x� �; r� � Jxjmr x� �, thus with mr � mr x� �we have

log lu�B�p x� �; r��log r

Plog lu Jxjmr

ÿ �log r

Plog pxjmr

Smr�1w xjmr � 1� � � log di xmr�1� �;0 rmr�1 x� �� ÿ log a1

� log pxjmr

mr

Smr�1w xjmr � 1� � � log di xmr�1� �;0 rmr�1 x� �� ÿ log a1

mr:

,Since for all u 2 V , log du;0 is integrable, the Ergodic theorem gives us that

limk!1

log di�xk�1�;0�rk�1 x� ��k

� 0:

Hence

lim infr!0

log lu�B�p x� �; r��log r

P hlu;mu�q� for qq-a:a: x 2 ENu :

Thus,

lim infr!0

log lu�B�x; r��log r

P hlu;mu�q� for qqu-a:a: x 2 Ku:

(2) This follows by the same argument as that used in (1) with lu replaced by mu and p replaced by m.(3) Follows since

log lu B x; r� �� log mu B x; r� ��

log lu B x;r� �� log r

log mu B x;r� �� log r

;

and clu;mu�q� � hlu;mu�q�=amu�q�.

(4) follows by similar arguments to (1) if we use the following equality

qq s� �� pqsm

flu ;mu �q�s :

(5) follows immediately from (2), (3) and (4). �

3.3. Counter example and conjecture

Having calculated the Hausdor� and packing dimensions of the sets Ku�clu;mu�q�; amu�q�� we would hope

that we could use this calculation to calculate the Hausdor� and packing dimensions of Ku�clu;mu�q��. A


natural conjecture is that dimH�Ku�clu;mu�q�� dimP�Ku�clu;mu

�q�� alu;mu�q��q clu;mu�q� � flu;mu�q��. We

now give an example to show that this is not the case for all pairs of invariant measures based on the sameGCIFS.

Binomial measures are examples of measures that can be de®ned using GCIFSs. Let l be the binomialmeasure supported on 0; 1� � based on the probability vector �p; 1ÿ p�, where p 2 �0; 1

2� i.e., l is the unique

probability on 0; 1� � satisfying l � p � l � Tÿ11 � �1ÿ p�l � Tÿ1

2 , where T1�x� � x=2 and T2�x� � x=2� 12. Set

m � l. Then by de®nition we have that cl;m�q� � 1 for all q. This implies that fl;m�q� � 1ÿ q. Thusam�q��qcl;m�q� � fl;m�q�� am�q��q� 1ÿ q� � am�q�. Also, for all x 2 �0; 1�,

limr&0

l�B�x; r��m�B�x; r�� 1:

Thus for all q we have that K�clu;mu�q�� 0; 1� �. Hence dimH�K�cl;m�q�� dimH� 0; 1� �� 1.

Now, it is well-known (see, for example [5]) that am�q� is a decreasing function of q such thatlimq!1 am�q� � log�1ÿp�

log�1=2� < 1. Thus there exists q0 such that for all q > q0, am�q� < 1. Hence there exists a qsuch that am�q� 6� 1. This gives us a counter example to the conjecture that dimH�Ku�clu;mu

�q�� dimP�Ku�clu;mu

�q�� amu�q��qclu;mu�q� � flu;mu�q�� for all pairs of of invariant measures based on the same

GCIFS.While the above example is a counter example to our initial conjecture we feel that it is more than likely

that the situation where we set l � m is a degenerate case. Our conjecture is as follows:

Conjecture 1.

1. If mu � lu, then there exist q such that dimH Ku clu;mu�q�

� �� 6� amu�q� qclu;mu

�q� � flu;mu�q��

and there

exist q such that dimP Ku clu;mu�q�

� �� 6� amu�q� qclu;mu

�q� � flu;mu�q��

.

2. If lu ? mu, then dimH Ku clu;mu�q�

� �� dimP Ku clu;mu

�q��

� amu�q� qclu;mu�q� � flu;mu�q�

� �for all q.

For q 2 R set

E�q� � x 2 supplu \ suppmu limr&0

log lu B�x; r�� log mu�B�x; r��

��(

� clu;mu�q�; lim sup

r&0


< amu�q�);

and

F �q� � x 2 supplu \ suppmu limr&0

log lu�B�x; r��log mu�B�x; r��

��(

� clu;mu�q�; lim inf

r&0


> amu�q�):

Then

Ku clu;mu�q�

� �� K clu;mu

�q�; amu�q��

[ E�q� [ F �q�:

So far what we have been able to prove the following:

Theorem 3.12. If the GCIFS that we are considering satisfies the strong separation condition, then for06 q6 1,

dimH�E�q��6 dimP�E�q��6 amu�q� qclu;mu�q�

�� flu;mu�q�

�:

Proof. We start our proof by quoting the following lemma which can be proved by using standard argu-ments from the literature (see, for example [12]).

Lemma 3.13. Let the GCIFS that we are considering satisfy the strong separation condition and let lu; mu andqq

u be defined as above. There exists a constant c such that for all x 2 supplu and r > 0


cÿ1qqu�B�x; r��6 lu�B�x; r��qmu�B�x; r��flu ;mu �q�6 cqq

u�B�x; r��:

Fix q 2 R, and for m; n 2 N de®ne �m; fn and gm;n using the following equations:

amu�q�amu�q� ÿ 1=m

:� 1� �m;clu;mu�q�

clu;mu�q� � 1=n

:� 1ÿ fn and �1� �m��1ÿ fn� :� 1� gm;n:

Suppose, by way of contradiction, that

dimP�E�q�� > amu�q� qclu;mu�q�

�� flu;mu

�q��

:� f �q�:

For m 2 N set

Em � x 2 E�q� ramu �q�ÿ1=m

��6 mu�B�x; r�� for r <

1

m

�:

Then Em % E�q� and thus there exists an M 2 N such that for m P M , dimP�Em� > f �q�. Fix m P M andchoose n 2 N such that gm;n P 0. For l 2 N set

Fl � x 2 Em mu�B�x; r��clu ;mu �q��1=n

��6 lu�B�x; r��6 mu�B�x; r��clu ;mu �q�ÿ1=n

for r <1

l

�:

Then since Fl % Em we can choose an L 2 N such that for all l P L, dimP�Fl� > f �q�. Fix l P L, then usingresults in [10] we are able to ®nd a compact set C such that C � Fl and dimP�C� > f �q�. Let g > 0 be givenand let U be an open d-neighbourhood of C such that qq

u�U�6 qqu�C� � g6 qq

u�E�q�� g. Now choose � suchthat 0 < � < dist C;Rd n U

� .

The condition that dimP�C� > f �q� implies that there exists a d such that 0 < d < � and a centredd-packing �Bi :� B�xi; ri��i of C such that

16X

i

�2ri�f �q�:

Thus

16 2f �q�Xi

rf �q�i � 2f �q�X

i

ramu �q� qclu ;mu �q��flu ;mu �q�� i 6 2f �q�X

i

mu�Bi��1��m� qclu ;mu �q��flu ;mu �q��

6 2f �q�Xi

mu�Bi�q 1��m� �clu ;mu �q�mu�Bi�flu ;mu �q� 1��m� �6 2f �q�Xi

lu�Bi�q 1��m� � 1ÿfn� �mu�Bi�flu ;mu �q��1��m�

6 c2f �q�Xi

qqu�Bi�lu�Bi�qgm;nmu�Bi��mflu ;mu �q�6 c2f �q�X

i

qqu�Bi�6 c2f �q�qq

u

[i

Bi

!6 c2f �q�qq

u�U�

6 c2f �q� qqu�E�q��

ÿ � g�:

Letting g& 0 yields a contradiction since qqu�E�q�� 0: �

Acknowledgements

We would like to thank Dr. Lars Olsen for his assistance in preparing this manuscript.

References

[1] Barreira L, Schmeling J. Sets of non-typical points have full topological entropy and full Hausdor� dimension, preprint 1998.

[2] Billingsley P. Ergodic theory and information, Wiley, New York, 1965.


[3] Cajar H. Billingsley dimension in probability spaces, Lecture notes in mathematics, vol. 892, Springer, New York, 1981.

[4] Cole J. The multifractal geometry of graph directed self-conformal measures, preprint 1998.

[5] Cawley R, Mauldin RD. Multifractal decomposition of moran fractals. Adv. Math. 1992;92:196±236.

[6] Das M. Local properties of self-similar measures. Illinois J. Math. 1998;42:313±32.

[7] Edgar GA, Mauldin RD. Multifractal decomposition of digraph recursive fractals. Proc. London Math. Soc. 1992;65(3):604±28.

[8] Graf S. On Bandt's tangential distribution for self-similar measures. Monatshefte fur Mathematik 1995;120:223±46.

[9] Halsey TC, Jensen MH, Kadano� LP, Procaccia I, Shraiman BJ. Fractal measures and their singularities: the characterization of

strange sets. Phys. Rev. A 1986;33:1141±51.

[10] Joyce H, Preiss D. On the existence of subsets of ®nite positive packing measure. Mathematika 1995;42:1995.

[11] Levy Vehel J, Vojak R. Multifractal analysis of Choquet capacities. Adv. Appl. Math. 1998;20:1±43.

[12] Olsen L. A multifractal formalism. Adv. Math. 1955;116:82±196.

[13] Peyri�ere J. Multifractal measures, In: Proceedings of the NATO Advanced Study Institute on Probabilistic and Stochastic

Methods in Analysis with Applications, Il Ciocco, NATO ASI Series, Series C: Mathematical and physical sciences, vol. 372,

Kluwer Academic Press, Dordrecht, 1992, pp. 175±186.

[14] Riedi R, Scheuring I. Conditional and relative multifractal spectra. Fractals: An Interdisciplinary J. 1997;5:153±68.

[15] Ruelle D. Thermodynamic formalism, Addison-Wesley, Reading, MA, 1978.


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