Advanced Studies in Pure Mathematics 57, 2010
Probabilistic Approach to Geometry
pp. 231–241

Remarks on non-diagonality conditions for
Sierpinski carpets
Naotaka Kajino
Abstract.
We prove the equivalence of three diﬀerent formulations of nondiagonality for Sierpinski carpets given by Barlow, Bass, Kumagai and
Teplyaev [5], by Hino [10] and by Kigami [12]. We also derive somegeometric property of Sierpinski carpets from the non-diagonality. Asan application, we give a simpler treatment of a criterion stated in
Kigami [12, Section 3.4] for the volume doubling property of self-similar
measures on Sierpinski carpets.

§1.

Introduction

The notion of fractal sets goes back to B. B. Mandelbrot [16]. He
has pointed out there that lots of objects in nature can be regarded
as consisting of ‘completely non-smooth’ parts like unrectiﬁable curves
and surfaces with inﬁnite area, and he has called such non-smooth sets‘fractals’. Since then fractal-like structure of objects in nature has been
discovered in various ﬁelds of science such as physics, chemistry andbiology, and study of fractals has grown to be an important researcharea of science.
Mathematical analysis on fractals has originated from the need for
rigorous methods of analysis of physical phenomena, like propagation ofheat and wave, on fractals. Mathematically, propagation of heat andwave is described by solutions of heat and wave equations(1.1)∂t u(t, x) = ∆x u(t, x),∂t2 u(t, x) = ∆x u(t, x),Received January 15, 2009.
Revised June 11, 2009.
2000 Mathematics Subject Classiﬁcation. 28A80, 31C25, 60J45.
Key words and phrases. generalized Sierpinski carpet, non-diagonality, volume doubling property, self-similar measure.
JSPS Research Fellows DC (20·6088): The author is supported by Japan
Society for the Promotion of Science.

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Naotaka Kajino(a) The Sierpinski gasket(b) The pentagasket(c) The Sierpinski carpet

Figure 1. Examples of self-similar fractals

where ∆ denotes the ‘Laplacian’ on the space, which is typically deﬁned as a diﬀerential operator. Deﬁnition of the Laplacian on smooth
spaces like Riemannian manifolds is more or less trivial, although it getscomplicated when we consider more singular spaces such as Alexandrov
spaces. On the contrary, the usual notion of diﬀerentiation does notwork on fractal spaces, and therefore even existence of a (canonical)
Laplacian on a given fractal space is highly non-trivial.
To establish theory of Laplacians on fractals, we need to choose anappropriate class of fractal spaces to work on. One handy choice isthe class of self-similar fractals like the Sierpinski gasket (Figure 1(a)),
the pentagasket (Figure 1(b)) and the Sierpinski carpet (Figure 1(c)).
A self-similar fractal is deﬁned as the unique non-empty compact set
that is invariant with respect to a given ﬁnite collection of injective
contraction maps. Study of analysis on fractals so far has concentrated
mainly on self-similar fractals due to their mathematical accessibility,
but even on them the analysis is diﬃcult and lots of fundamental orinteresting questions are still open.
The ﬁrst signiﬁcant step for achievement of theory of analysis on
fractals is the result of Goldstein [9] and Kusuoka [13]. They haveconstructed a canonical ‘Brownian motion on the Sierpinski gasket’ by
taking the scaling limit of a sequence of random walks on graphs, andhence a canonical Laplacian has been obtained as the inﬁnitesimal generator of this Brownian motion. Barlow and Perkins [7] have studied
the properties of this Brownian motion in detail, and have proved thatits transition density (the heat kernel corresponding to the Laplacian)pt (x, y) is subject to the following two-sided sub-Gaussian estimate:(1.2)(( |x − y|dw ) d 1−1 )wpt (x, y)  c1 t−df /dw exp −c2,tt ∈ (0, 1],

Remarks on non-diagonality conditions for Sierpinski carpets

233

where  means ≤ and ≥ with diﬀerent c1 , c2 ∈ (0, ∞), df := log2 3
is the Hausdorﬀ dimension of the Sierpinski gasket with respect to the
Euclidean metric and dw := log2 5 is called the walk dimension of the
Sierpinski gasket. Note that the Sierpinski gasket is ﬁnitely ramiﬁed, i.e.
we can disconnect it by removing ﬁnitely many points. This property of
the gasket plays essential roles in the above results. The above results
have been generalized to a class of ﬁnitely ramiﬁed self-similar fractals
called (aﬃne) nested fractals; see [15, 8]. The Sierpinski gasket andthe pentagasket are typical examples of nested fractals. A lot of otherinteresting results are also known in the case of aﬃne nested fractals;
See [1, 11, 17] and the references therein.
On the other hand, the Sierpinski carpet is one of the simplest examples of inﬁnitely ramiﬁed self-similar fractals: The Sierpinski carpet
consists of eight smaller copies of itself and each pair of adjacent smaller
copies intersects on a one-dimensional interval, which is an inﬁnite set.
This fact makes every stage of analysis much more diﬃcult, and it seems
natural to try to extend the results in the ﬁnitely ramiﬁed case to inﬁnitely ramiﬁed self-similar sets. Study of the Sierpinski carpet and its
analogues may be considered as the ﬁrst step of this trial.
In [2], Barlow and Bass have constructed a ‘Brownian motion on the
Sierpinski carpet’ as the scaling limit of a sequence of reﬂected Brownian motions on Euclidean domains approximating the carpet. Later in[3] they have established the two-sided sub-Gaussian bound (1.2) of thecorresponding heat kernel. On the other hand, Kusuoka and Zhou [14]used graph approximations of the carpet and Dirichlet form techniques
to provide an alternative construction of a Brownian motion. Uniqueness of such a Brownian motion had been an open problem in this ﬁeld
for about two decades, but has recently been proved by Barlow, Bass,
Kumagai and Teplyaev [5]. In particular, two Brownian motions given
by [2] and [14] are the same.
By [4, 5], it is known that these results are valid for generalized Sierpinski carpets (GSCs for short), a class of inﬁnitely ramiﬁed self-similar
fractals constructed from the d-dimensional unit cube [0, 1]d , d ≥ 2, in asimilar way to the Sierpinski carpet. Barlow and Bass [4] have assumed
that GSCs satisfy a geometric condition called non-diagonality. Thiscondition is not so essential, but without it the details of the argumentin [4] would get signiﬁcantly complicated and the Brownian motion constructed on the GSC would exhibit strange behaviors. Therefore thenon-diagonality condition is indispensable in keeping the class of GSCs
reasonable for our analysis. Barlow, Bass, Kumagai and Teplyaev [5]have recently realized, however, that the non-diagonality condition given
in [4] is too weak for some part of the arguments of [4], and in [5] they

234

Naotaka Kajino

have given a stronger formulation of non-diagonality. On the other hand,
Hino [10] has also introduced a simpler reformulation of non-diagonality
to discuss some properties of the domain of the Dirichlet form on GSCs.
In fact, before [5] and [10], Kigami [12, Section 3.4] has introduced
another stronger formulation of non-diagonality to give a criterion for thevolume doubling property of self-similar measures on generalized Sierpinski carpets. The volume doubling property is known to be closely relatedwith the validity of the sub-Gaussian heat kernel bounds; see [6] and thereferences therein. By [12, Theorem 3.2.3], in the case of reasonable selfsimilar fractals like GSCs and aﬃne nested fractals, the volume doublingproperty with respect to a certain distance d is equivalent to the subGaussian heat kernel upper bound with respect to the same d, where dneeds to be ‘adapted’ to the measure and the Dirichlet form in a suitablesense.
Consequently, there are at least three diﬀerent reformulations of
non-diagonality for GSCs. The purpose of this article is to guarantee
the equivalence of the three formulations and derive some consequenceof the non-diagonality.
The organization of this paper is as follows. In the beginning ofSection 2 we introduce our framework and deﬁne generalized Sierpinski carpets (GSCs) as well as the three formulations of non-diagonality
for GSCs. Then in the rest of Section 2, we ﬁrst prove the equivalence of the three reformulations of non-diagonality, and secondly, wederive some geometric property of GSCs as an easy consequence of thenon-diagonality. As an application, in Section 3 we provide a simpler
description of the results of Kigami [12, Section 3.4] on the volume doubling property of self-similar measures on Sierpinski carpets.
Convention. In this article, we follow the convention that 0 6∈ N.
We abbreviate ‘if and only if ’ to ‘iﬀ ’.
§2. Non-diagonality for Sierpinski carpets and its consequence
The following framework will be ﬁxed throughout this article.
Framework 2.1. Let d ∈ N and set Q0 := [0, 1]d . Let L ∈ N, L ≥ 2
{∏d [
]
}
−m
and set Qm :=, ki L−m | k1 , . . . , kd ∈ {1, . . . , Lm }i=1 (ki − 1)L]∏d [for each m ∈ N. For
each q = i=1 (ki − 1)L−m ,)ki L−m ∈ Qm , set(z q = (z1q , . . . , zdq ) := (k1 − 1)L−m , . . . , (kd − 1)L−m .
Let S ⊂ Q1 be non-empty, and for each q ∈ S we deﬁne Fq :Rd → Rd by Fq (x) := L−1 x + z q , so that Fq (Q0 ) = q ⊂ Q0 . Forq = q1 . . . qm ∈ S m , m ∈ N, we set Fq := Fq1 ◦ · · · ◦ Fqm , Qq := Fq (Q0 )Remarks on non-diagonality conditions for Sierpinski carpets

235

qand z q = (z1∪, . . . , zdq ) := z Qq . We also set QSm := {Qq | q ∈ S m }(⊂ Qm )S
and Qm := q∈S m Qq for m ∈ N.

The non-diagonality conditions are deﬁned in terms of S, as follows.
Definition 2.2 (Non-diagonality, [5, 10, 12]). We deﬁne three nondiagonality conditions (ND)U , (ND)H and (ND)K on S as follows.
(ND)U For any m ∈ N and any d-dimensional cube B ⊂ Q0 of side
length 2L−m which is the union of elements of Qm , intRd (B ∩ QS1 ) iseither empty or connected.
(ND)H For any d-dimensional rectangle B ⊂ Q0 with each side length
L−1 or 2L−1 which is the union of elements of Q1 , intRd (B ∩ QS1 ) iseither empty or connected.
(ND)K For any x ∈ QS1 , there exists η ∈ (0, ∞) such that for any r ∈(0, η), intRd (QS1 ∩ Br (x)) is non-empty and connected, where Br (x) :={y ∈ Rd | |y − x| ≤ r} with | · | the Euclidean norm.
(ND)U is due to [5, Section 2.2] and it is a modiﬁed version of theoriginal non-diagonality condition given in [4]. (ND)H is due to Hino[10], and (ND)K is provided in [12, Deﬁnition 3.4.1].
Definition 2.3 (Generalized Sierpinski carpets). Let GSC(d, L, S)
betheself-similar set associated with {Fq }q∈S , i.e. GSC(d, L, S) :=∩dSQm∈N∪ m , which is the unique non-empty compact K ⊂ R such that
K = q∈S Fq (K). We call GSC(d, L, S) a generalized Sierpinski carpet,
GSC for short, iﬀ S satisﬁes (ND)H and the following two conditions:(GSC1) If q ∈ S, k ∈ {1, . . . , d} and {(x1 , . . . , xd ) ∈ Rd | xk ∈ {0, 1}} ∩q 6= ∅ then Rk (q) ∈ S, where Rk : Rd → Rd denotes the reﬂection in thehyperplane {(x1 , . . . , xd ) ∈ Rd | xk = 1/2}.
d∪(GSC2) (Borders included){(0, . . . , 0, xj , 0, . . . , 0) | xj ∈ [0, 1]} ⊂ QS1 .
j=1

Note that the above deﬁnition of generalized Sierpinski carpets is broader
than the usual one in [4, 5]. We have omitted the conditions which are
not required in the arguments below.
The following is the ﬁrst main theorem of this article.
Theorem 2.4. The three conditions (ND)U , (ND)H and (ND)K
are equivalent to each other.
Proof. We set I := {1, . . . , d} in this proof.

236

Naotaka Kajino(ND)U ⇒ (ND)H : Let B ⊂ Q0 be a d-dimensional rectangle witheach side length L−1 or 2L−1 which is the union of elements of Q1 . Then
∏ [ ki − 1 ki + 1 ] ∏ [ ki − 1 ki ]
(2.1)
B=
,
×
,
,
L
L
L
L
i∈I0

i∈I\I0for(ki )i∈I ∈ {1,]. . ∏. , L − 1}[I0 ×{1, . . . , L}I\I0 . Let] C :=[ I0−1⊂ I and∏ some
−1
−2
−1
−2
−1
−2
(⊂ B)
×
k
L
−
2L
,
k
L
k
L
−
L
,
k
L
+
L
i
i
i∈I\I0 ii∈I0 ( i)−1−1−1
and f (xi + ki L )i∈I := (xi L + ki L )i∈I0 ×(xi L/2 + ki L )i∈I\I0 for(xi )i∈I ∈ Rd . Then f : Rd → Rd is a bijective aﬃne map and f (C) = B.
We easily see that f (q ∩ intRd C) = q ∩ intRd B for any q ∈ Q1 . Therefore()∪∪
f QS1 ∩ intRd C =f (q ∩ intRd C) =(q ∩ intRd B) = QS1 ∩ intRd B.
q∈S

q∈S

(
)
Hence f maps intRd (C ∩QS1 ) = intRd QS1 ∩intRd C , which)( is either empty
or connected by (ND)U , homeomorphically onto intRd QS1 ∩ intRd B =intRd (B ∩ QS1 ). Thus (ND)H follows.
(ND)H ⇒ (ND)U : Let m ∈ N and let C ⊂ Q0 be a d-dimensional
cube of side length 2L−m which is the union of 2d distinct elements of
Qm . If m = 1 then (ND)H immediately implies that intRd (C ∩ QS1 ) iseither empty or connected. Therefore we may assume that m ≥ 2. Then] ∏ []∏ [ kiki
1 ki
1`i + 2 ki`iC=− m, + m ×
−
,
−
L
L
L
L
L
Lm L
Lm
i∈I0

i∈I\I0for some I0 ⊂ I, (ki )i∈I ∈ {1, . . . , L − 1}I0 × {0, . . . , L − 1}I\I0 and(`i )i∈I\I0 ∈ {0, . . . , Lm−1 − 2}I\I0 . Deﬁne B (⊃ C) by (2.1) and
((
( 2x
( x
ki ) )
ki )
ki
`i )
i
i
f xi +
+
×
+
−
:=
L i∈I
Lm−1
L i∈I0
Lm−1
L
Lm i∈I\I0
for (xi )i∈I ∈ Rd . Then again we easily see that f : Rd → Rd is a bijective
aﬃne map satisfying)( f (q ∩ intR)d B) = q ∩ intRd C for( any q ∈ S andf (B) = C. Hence f QS1 ∩intRd B = QS1 ∩intRd C and f intRd (B∩QS1 ) =intRd (C ∩ QS1 ). Now using (ND)H for B yields (ND)U .
(ND)H ⇔ (ND)K : This is easily proved by replacing the rectangle C in the above proof with Br (x) and constructing an appropriate
homeomorphism f : Rd → Rd satisfying f (Br (x)) = B.
Q.E.D.
Remark. (ND)U ⇒ (ND)H of the above proof immediately showsthat (ND)U , (ND)H and (ND)K are also equivalent to the following:(ND)0U Any d-dimensional cube B ⊂ Q0 of side length 2L−2 which is
the union of elements of Q2 , intRd (B ∩ QS1 ) is either empty or connected.

Remarks on non-diagonality conditions for Sierpinski carpets

237

Next, we establish an easy consequence of (ND)H . The following
proposition is due to Hino [10].
Notation. Let m ∈ N. For w, v ∈ S m , we deﬁne δm (w, v) :=∑dwv
Li=1 |zi − zi |.
mNote that δm (w, v) ∈ {0, . . . , d} if Qw ∩ Qv 6= ∅, and that δm (w, v) = 1
iﬀ Qw and Qv intersect on a (d − 1)-dimensional face.
Proposition 2.5 (Hino [10]). (ND)H holds iﬀ for any i, j ∈ S with
δ1 (i,j)Qi ∩ Qj 6= ∅ there exists a sequence {n(`)}`=0⊂ S such that n(0) = i,n(δ1 (i, j)) = j and δ1 (n(` − 1), n(`)) = 1 for any ` ∈ {1, . . . , δ1 (i, j)}.
∩δ1 (i,j)
Remark. For i, j, n(`) ∈ S as above, Qi ∩Qj = `=0Qn(`) , which
is a (d − δ1 (i, j))-dimensional cube of side length L−1 .
Proof. The ‘only if’ part is easily proved. To show the converse,
assume (ND)H , let i, j ∈ S satisfy Qi ∩ Qj 6= ∅ and let J := δ1 (i, j)(∈{0, . . . , d}). The proof is by induction in J.
∪ The statement is clear if
J ≤ 1, so suppose J ≥ 2 and let B := {q ∈ Q1 | q ⊃ Qi ∩ Qj },which is a d-dimensional rectangle with each side length L−1 or L−2 .
Then since intRd (B ∩ QS1 ) ⊃ intRd Qi ∪ intRd Qj 6= ∅, (ND)H implies
that intRd (B ∩ QS1 ) is connected. Therefore we may choose q ∈ S sothat Qq ⊃ Qi ∩ Qj and δ1 (q, j) = 1. Then L zkq − zkj = 1 for a uniquek ∈ {1, . . . , d} and zki = zkq . Hence δ1 (i, q) = J − 1. Now the induction
hypothesis implies that there exists {n(`)}J−1`=0 ⊂ S such that n(0) = i,n(J −1) = q and δ1 (n(`−1), n(`)) = 1 for any ` ∈ {1, . . . , J −1}. Settingn(J) := j completes the induction procedure.
Q.E.D.
Furthermore, (ND)H together with (GSC1) yields the following
stronger statement, which is our second main result.
Theorem 2.6. Assume (ND)H and (GSC1). Let m ∈ N, let w, v ∈S m satisfy Qw ∩Qv 6= ∅ and set J := δm (w, v)(∈ {0, . . . , d}). Then there
exists a sequence {n(`)}J`=0 ⊂ S m such that n(0) = w, n(J) = v andδm (n(` − 1), n(`)) = 1 for any ` ∈ {1, . . . , J}.
Remark. (1) The remark after Proposition 2.5 applies: Qw ∩ Qv =−m.
`=0 Qn(`) , which is a (d − J)-dimensional cube of side length L(2) (GSC1) cannot
beomittedin
Theorem
2.6.
Indeed,
if
d
=
2,
L
=
3
{∏2 [ ki −1 ki ]}and S = Q1 \| (k1 , k2 ) ∈ {(1, 2), (3, 1), (3, 3)} , then
i=1
3 , 3
(ND)H holds but both (GSC1) and the conclusion of Theorem 2.6 fail.
∩J

Proof. First, by (GSC1) we easily have the following (GSC1)m :(GSC1)m If q ∈ S m , k ∈ {1, . . . , d} and Qq ∩ {(x1 , . . . , xd ) ∈ Rd | xk ∈{0, 1}} 6= ∅ then Rk (Qq ) ∈ QSm , where Rk is the same as in (GSC1).

238

Naotaka Kajino

The proof of Theorem 2.6 is by induction in m. The statement for m = 1
follows by Proposition 2.5, so suppose m ≥ 2. Choose i, j ∈ S so thatQw ⊂ Qi and Qv ⊂ Qj . Then we have J0 := δ1 (i, j) ∈ {0, . . . , J}. By0⊂ S such that α(0) = i, α(J0 ) = j
Proposition 2.5, there exists {α(`)}J`=0and δ1 (α(` − 1), α(`)) = 1 for any ` ∈ {1, . . . , J0 }. Then for any ` ∈{0, . . . , J0 }, Qα(`) ⊃ Qi ∩ Qj ⊃ Qw ∩ Qj , which is a (d − J0 )-dimensional
0cube of side length L−m . We deﬁne {n(`)}J`=0⊂ S m inductively as
follows. First let n(0) := w. Next, let ` ∈ {1, . . . , J0 } and suppose wehave n(` − 1) ∈ S m satisfying Qw ∩ Qj ⊂ Qn(`−1) ⊂ Qα(`−1) . Then weeasily see that Qn(`−1) ∩ (Qα(`−1) ∩ Qα(`) ) 6= ∅. Therefore (GSC1)m−1
implies that the reﬂection of Qn(`−1) in the (d − 1)-dimensional faceQα(`−1) ∩Qα(`) belongs to QSm , which we deﬁne as Qn(`) with n(`) ∈ S m .
This deﬁnition of n(`) immediately yields δm (n(` − 1), n(`)) = 1 and0
Qw ∩ Qj ⊂ Qn(`) ⊂ Qα(`) . Thus we get {n(`)}J`=0⊂ S m satisfyingn(0) = w and δm (n(`−1), n(`)) = 1 for any ` ∈ {1, . . . , J0 }. We also have
Qw ∩ Qj ⊂ Qn(J0 ) ⊂ Qα(J0 ) = Qj , which easily yields δm (n(J0 ), v) =J − J0 and Qn(J0 ) ∩ Qv ⊃ Qw ∩ Qv 6= ∅. Since Qn(J0 ) ∪ Qv ⊂ Qj , anapplication of the induction hypothesis to elements of QSm included in Qjm0yields the existence of {β(`)}J−J
such that β(0) = n(J0 ), β(J −`=0 ⊂ S
J0 ) = v and δm (β(` − 1), β(`)) = 1 for any ` ∈ {1, . . . , J − J0 }. Settingn(`) := β(` − J0 ) for ` ∈ {J0 + 1, . . . , J} completes the proof. Q.E.D.
§3.

Application: A criterion for the volume doubling property

Now we use Theorem 2.6 to provide a simpler description of theresult of Kigami [12, Theorem 3.4.3]. Throughout this section, we put
the following framework and notations in addition to Framework 2.1.
See [11, Chapter 1] and [12, Section 1.2] for basics on self-similar sets.
∪
Framework 3.1. Set W# (S) := m∈N S m . Let K := GSC(d, L, S)
be a GSC and let L := (K, S, {Fq }q∈S ) denote the self-similar structureassociated with {Fq }q∈S . For q = q1 . . . qm ∈ S m , m ∈ N, set Kq :=Fq (K), |q| := m and q[−1] := q1 . . . qm−1 (:= ∅ if m = 1, where ∅ is anelement called the empty word ).
For s ∈ [0, 1] and k ∈ {1, . . . , d}, set Hk,s := {(x1 , . . . , xd ) ∈ Rd |xk = s} and Sk,s := {q ∈ S | q ∩ Hk,s 6= ∅}. Note that, by (GSC1),
the reﬂection Rk : Rd → Rd in the hyperplane Hk,1/2 induces naturalbijections Sk,0 → Sk,1 and Sk,1 → Sk,0 given by q 7→ Rk (q).
Definition 3.2 (Self-similar scales and related notions). Let a =(aq )q∈S ∈ (0, 1)S . We deﬁne the following notions:(0) a is called weakly symmetric iﬀ aq = aRk (q) for any k ∈ {1, . . . , d}and any q ∈ Sk,0 .

Remarks on non-diagonality conditions for Sierpinski carpets

239m(1) a∅ := {1 and aq1 ...qm := aq1 . . . aqm for} q1 . . . qm ∈ S , m ∈ N.
Λs (a) := q ∈ W# (S) | aq[−1] > s ≥ aq for s ∈ (0, 1) and S(a) :={Λs (a)}s∈(0,1) . We call S(a) the self-similar scale on S N with weight a.
(2) Λs,q (a) := {w ∈ Λs (a) | Kw ∩ Kq 6= ∅} for s ∈ (0, 1), q ∈ W# (S). We
call (L, S(a)) locally ﬁnite iﬀ sup{#Λs,q (a) | s ∈ (0, 1), q ∈ Λs (a)} < ∞.
(3) IP(L, a) :=∪{(w, v) | w, v (∈∪Λs (a) ∃ s ∈ (0,) 1), w 6= v, Kw ∩ Kv 6= ∅}.
(4) Us (x, a) := q∈Λs (a),x∈Kq w∈Λs,q (a) Kw for (s, x) ∈ (0, 1)×K.

Clearly, {Us (x, a)}s∈(0,1) is decreasing as s ↓ 0 and is a fundamental
system of neighborhoods of x in K. Now as an application of Theorem
2.6, we have the following propositions.
Proposition 3.3 ([12, Theorem 3.4.4 and Proof of Lemma 3.5.16]).
Let a = (ai )i∈S ∈ (0, 1)S . Then (L, S(a)) is locally ﬁnite iﬀ a is weakly
symmetric, which also implies that the sets {|w|−|v| | (w, v) ∈ IP(L, a)}and {aw /av | (w, v) ∈ IP(L, a)} are both ﬁnite.
Proposition 3.4 ([12, Theorems 1.6.6 and 3.4.3]). Let a = (ai )i∈S ,b = (bi )i∈S ∈ (0, 1)S . Then the following conditions are equivalent.
(1) b is gentle with respect to a, i.e. sup{bw /bv | (w, v) ∈ IP(L, a)} < ∞.
(2) For each k ∈ {1, . . . , d}, either aq = aRk (q) and bq = bRk (q) for any
q ∈ Sk,0 , or bq = aηq k for any q ∈ Sk,0 ∪ Sk,1 for some ηk ∈ (0, ∞).
These two propositions together with [12, Theorem 1.3.5] immediately yield the following theorem.
Theorem 3.5 (Volume doubling property, [12, Theorem
3.4.5]).
∑
Let a = (ai )i∈S ∈ (0, 1)S . Let (µi )i∈S ∈ (0, 1)S satisfy i∈S µi = 1 andlet µ be the self-similar measure on K with weight (µi )i∈S , i.e. µ be theunique Borel probability measure on K satisfying µ(Kq ) = µq for any
q ∈ W# (S). Then the following conditions are equivalent.
(1) a and (µi )i∈S are both weakly symmetric.
(2) (L, S(a), µ) is volume doubling (VD), that is, there exist α ∈ (0, 1)
and cV such that µ(Us (x, a)) ≤ cV µ(Uαs (x, a)) for any (s, x) ∈ (0, 1).
In the rest of this section, we brieﬂy present the idea of the proof of
Propositions 3.3 and 3.4. The key of the proof is the following lemma.
Notation. For w, v ∈ W# (S) with Qw ∩ Qv 6= ∅, deﬁne δ# (w, v) ∈{0, . . . , d} so that Qw ∩ Qv is a (d − δ# (w, v))-dimensional cube of side
length L− max{|w|,|v|} .
Note that δ# (w, v) = 0 iﬀ Qw ⊂ Qv or Qw ⊃ Qv in the above situation.
Lemma 3.6. Let a = (ai )i∈S ∈ (0, 1)S . Let s ∈ (0, 1) and w, v ∈Λs (a) and assume Qw ∩ Qv 6= ∅. Then there exist J ∈ {0, . . . , δ# (w, v)}

240

Naotaka Kajinoand {n(`)}J`=0 ⊂ Λs (a) such that n(0) = w, n(J) = v and for any` ∈ {1, . . . , J}, Qn(`−1) ∩ Qn(`) 6= ∅ and δ# (n(` − 1), n(`)) = 1.
Remark. (GSC2) is not needed in Lemma 3.6 and its proof below.
Proof. Noting that intRd (Qα ∩Qβ ) = ∅ for any distinct α, β ∈ Λs (a)and using (GSC1)m for m ∈ N, this lemma follows by an induction in
k := δ# (w, v), similarly to Theorem 2.6. Note that |α| and |β| may notequal for α, β ∈ Λs (a), which is why J 6= δ# (w, v) in general. Q.E.D.
Proof of Propositions 3.3 and 3.4. By Lemma 3.6, we easily seethat IP(L, a) may be replaced with {(w, v) ∈ IP(L, a) | δ# (w, v) = 1}
in the statements of Propositions 3.3 and 3.4. Then some direct calculations as in [12, Section 1.6] show the assertions. (The arguments
in [12, Section 1.6] are quite general and use lots of notations and lemmas, whereas for the case of self-similar scales on a GSC, Lemma 3.6
suﬃces.)Q.E.D.
Acknowledgements. The author would like to thank Professor
Masanori Hino for fruitful discussions and for having allowed the author
to include Proposition 2.5, a result of [10], in this article.

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Graduate school of Informatics
Kyoto University
Kyoto 606-8501
Japan
E-mail address: kajino.n@acs.i.kyoto-u.ac.jp
URL: http://www-an.acs.i.kyoto-u.ac.jp/~kajino.n/