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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 34, Num. 2 (2019), 237 – 254

doi:10.17398/2605-5686.34.2.237

Available online October 2, 2019

The µ-topological Hausdorff dimension

Hela Lotfi

Department of Mathematics, Faculty of Sciences of Monastir

5000-Monastir, Tunisia

helalotfi@hotmail.fr

Received May 13, 2019 Presented by M. Mbekhta
Accepted June 18, 2019

Abstract: In 2015, R. Balkaa, Z. Buczolich and M. Elekes introduced the topological Hausdorff
dimension which is a combination of the definitions of the topological dimension and the Hausdorff

dimension. In our manuscript, we propose to generalize the topological Hausdorff dimension by
combining the definitions of the topological dimension and the µ-Hausdorff dimension and we call it

the µ-topological Hausdorff dimension. We will present upper and lower bounds for the µ-topological

Hausdorff dimension of the Sierpiński carpet valid in a general framework. As an application, we
give a large class of measures µ, where the µ-topological Hausdorff dimension of the Sierpiński carpet

coincides with the lower and upper bounds.

Key words: Hausdorff dimension, Topological Hausdorff dimension.

AMS Subject Class. (2010): 28A78, 28A80.

1. Introduction

Different notions of dimensions have been introduced since the appear-
ance of the Hausdorff dimension by F. Hausdorff in 1918 (see e.g. [10], [16]
and [15]), such as the topological dimension, (see e.g. [4] and [7]). In 1975
when Mandelbrot coined the word fractal (see [13]). He did so to denote an
object whose Hausdorff dimension was strictly greater than its topological di-
mension, but he abandoned this definition later, (see e.g. [6], [13] and [14]).
In the Euclidean space Rn, there has been no generally accepted definition of
a fractal, even though fractal sets have been widely used as models for many
physical phenomena (see e.g. [9],[11] and [12]). The idea behind these models
is that of self-similarity (see e.g. [5] and [12]). Then Billingsley defined the
Hausdorff measure in a probability space (see e.g. [2] and [3]). In [1], R. Balka,
Z. Buczolich and M. Elekes introduced a new dimension concept for metric
spaces, called the topological Hausdorff dimension. It was defined by a very
natural combination of the definitions of the topological dimension and the
Hausdorff dimension. The value of the topological Hausdorff dimension was

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.34.2.237
mailto:helalotfi@hotmail.fr
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


238 hela lotfi

always between the topological dimension and the Hausdorff dimension. In
particular, this dimension was a non-trivial lower estimate for the Hausdorff
dimension.

P. Billingsley introduced a dimension defined by a measure µ, (see e.g. [2]
and [3]) called the µ-Hausdorff dimension, which was a generalization of the
Hausdorff dimension. In the same vein, we propose to generalize the topo-
logical Hausdorff dimension by combining the definitions of the topological
dimension and the µ-Hausdorff dimension, so-called the µ-topological Haus-
dorff dimension.

The paper is organized as follows. In Section 2, we recall the topological
Hausdorff dimension. Afterwards, we state the basic properties of this dimen-
sion and we cite some examples. In Section 3, we introduce the µ-topological
Hausdorff dimension. Finally in Section 4, we give an estimation of the µ-
topological Hausdorff dimension of the Sierpiński carpet and then we provide
a class of measures µ for which we compute the later dimension.

2. Topological Hausdorff dimension

Let (X,d) be a metric space. We denote by A the closure of subset A and
∂A its boundary in the metric space X. If B ⊆ A then ∂AB designates the
boundary of subset B in the metric space A with an induced topology.

Let B(x,r) =
{
y ∈ X : d(x,y) < r

}
be the open ball of radius r centered

at point x.
For a bounded subset U of X we denote the diameter by:

|U| = sup
{
d(x,y) : x,y ∈ U

}
.

For two metric spaces (X,d) and (Y,d′), a function f : X → Y is Lipschitz
if there exists constant λ ∈ R+ such that d′

(
f(x),f(y)

)
6 λ d(x,y), for all

x,y ∈ X.
We begin by recalling the definition of the Hausdorff dimension.

Definition 2.1. Let A be a subset of a separable metric space X, and α
be a positive number. For any ε > 0, we define:

Hαε (A) = inf


∑

j

|Uj|α : A ⊂
⋃
j

Uj, |Uj| < ε


 .

We also define:
Hα(A) = lim

ε→0+
Hαε (A).



the µ-topological hausdorff dimension 239

Then the Hausdorff dimension of A is given as follows:

dimH(A) = inf{α > 0 : Hα(A) < +∞}.

The topological Hausdorff dimension of a non-empty separable metric
space X, introduced in [1], is defined as:

dimtH(X) = inf
{
d : X has basis U such that dimH(∂U) 6 d− 1

for every U ∈U
}
,

with convention dimH(∅) = −1.
In [1], the authors proved that dimt(X) 6 dimtH(X) 6 dimH(X) where

dimt denotes the topological dimension of a non-empty separable metric space
defined by:

dimt(X) = inf
{
d : X has basis U such that dimt(∂U) 6 d− 1

for every U ∈U
}

where convention dimt(∅) = −1, (see [4] or [7]).
Moreover, the authors gave an alternative recursive definition of the topo-

logical dimension as follows:

dimt(X) = min
{
d : there is A ⊆ X such that dimt(A) 6 d− 1

and dimt(X\A) 6 0
}
.

Notice that Balka et al. defined the topological Hausdorff dimension of a subset
A of X by considering A as a metric space and equipping it with a dimension
induced from that of X.

Next, we recall some properties of the topological Hausdorff dimension of
X given in [1].

Proposition 2.2. Let X be a separable metric space.

(i) If A ⊆ B ⊆ X, then dimtH(A) 6 dimtH(B).

(ii) If X =
⋃
n∈N

Xn, where Xn (n ∈ N) are closed subsets of X, then:

dimtH(X) = sup
n∈N

dimtH(Xn).



240 hela lotfi

(iii) Let Y be a separable metric space and f : X → Y be a Lipschitz
homeomorphism, so:

dimtH(Y ) 6 dimtH(X).

Particularly, if f is bi-Lipschitz, then dimtH(Y ) = dimtH(X).

Example 2.3. (a) Let X = R, so we have:

dimt(R) = dimtH(R) = dimH(R) = 1.

In addition, we have:

dimtH(Q) = dimtH(R\Q) = 0.

It is noticeable that:

dimtH(R) 6= sup (dimtH(Q), dimtH(R\Q)) .

Indeed, Q is not a closed set of R.

(b) Let X = R2.
Let D be the von Koch snowflake curve. Then:

dimt(D) = dimtH(D) = 1 < dimH(D) =
ln 4

ln 3
.

Let S be the Sierpiński triangle. Thereby:

dimt(S) = dimtH(S) = 1 < dimH(S) =
ln 3

ln 2
.

Let T be the Sierpiński carpet. Thus:

dimt(T) = 1 < dimtH(T) =
ln 6

ln 3
< dimH(T) =

ln 8

ln 3
.

Remark. The topological Hausdorff dimension is not a topological notion.
Indeed, the following property was established in [1]:

dimtH(X × [0, 1]) = 1 + dimH(X). (2.1)



the µ-topological hausdorff dimension 241

Based on property (2.1), we can build two homeomorphic spaces such that
their topological Hausdorff dimensions are different as follows:

Consider X,Y ⊆ [0, 1] two Cantor sets such that:

dimH(X) 6= dimH(Y ).

Since these two sets are totally discontinuous then X and Y are homeomorphic
to the middle-thirds Cantor set (see [8]). Therefore, there exists homeomor-
phism ϕ : X → Y , so:

X × [0, 1] −→ Y × [0, 1]
(x,t) 7−→ (ϕ(x), t)

is a homeomorphism. Thus, using property (2.1), we obtain:

dimtH(X × [0, 1]) 6= dimtH(Y × [0, 1]).

3. µ-Topological Hausdorff dimension

In the following, we propose to generalize the topological Hausdorff di-
mension. We begin by recalling the dimension defined by Billingsley in [2]
and [3]. Let X be a metric space, F a countable set of subsets of X, and µ a
non-negative function defined on F and satisfying the following property:

For each x ∈ X and ε > 0, there is U ∈F
such that x ∈ U and µ(U) < ε.

(3.1)

Let A be a non-empty subset of X and α be a positive number. For any ε > 0,
we define:

Hαµ,ε(A) = inf


∑

j

µ(Uj)
α : A ⊂

⋃
j

Uj, Uj ∈F, µ(Uj) < ε


 (3.2)

with convention 0α = 0.
As ε decreases, the class of permissible covers of A in (3.2) is reduced.

Then the infimum Hαµ,ε(A) increases, and so approaches a limit as ε → 0. We
write the following:

Hαµ(A) = lim
ε→0
Hαµ,ε(A).

Therefore, the µ-Hausdorff dimension of a non-empty subset A of X relative
to F, as defined by Billingsley, is given by:

dimµ(A) = inf
{
α > 0 : Hαµ(A) < +∞

}
.



242 hela lotfi

It is noted that the µ-Hausdorff dimension and the Hausdorff dimension have
the same properties.

Similar to the definition of the topological Hausdorff dimension, we intro-
duce the µ-topological Hausdorff dimension of X relative to F.

Definition 3.1. Let X be a metric space, F a countable set of subsets
of X, and µ a non-negative function defined on F and satisfying (3.1). Then
the µ-topological Hausdorff dimension of X relative to F is given by:

dimtµ(X) = inf
{
α : X has basis U such that, for every U ∈U,

dimµ(∂U) 6 α− 1
}

with convention dimµ(∅) = −1.

Notice that the µ-topological Hausdorff dimension of a subset A of X is
defined by considering A as a metric space and equipping it with a dimension
induced from that of X. Moreover, the µ-topological Hausdorff dimension is
monotonous in the sense of inclusion.

Remark. We find the topological Hausdorff dimension in the following
particular case: Let Q+\{0} be the set of all positive rational numbers. If
X is a separable metric space, F = {B(xn,r) : n ∈ N,r ∈ Q+\{0}} where
{xn : n ∈ N} is dense in X, and µ is the function such that µ

(
B(xn,r)

)
= 2r.

Then:
dimtµ(X) = dimtH(X). (3.3)

Indeed, let ε > 0 and x ∈ X. We choose r ∈ Q+\{0} where r < ε2 .
Then there exists n ∈ N such that d(xn,x) < r. Hence, x ∈ B(xn,r) and
µ
(
B(xn,r)

)
= 2r < ε. As a consequence, µ satisfies (3.1).

Now, to find (3.3), it must be showed that for all subset A of X:

dimµ(A) = dimH(A).

Firstly, it is clear to see that:

dimµ(A) > dimH(A).

For the second inequality, let α > 0 and ε > 0. Consider {Uj}j, a ε-cover of
A. For all j ∈ N we pick λj ∈ Q+\{0} such that:

|Uj|α < λαj < |Uj|
α +

ε

2j
.



the µ-topological hausdorff dimension 243

Let yj ∈ Uj, then there exists nj ∈ N such that d(yj,xnj ) < λj. Whence:

Uj ⊂B(xnj, 2λj).

Since A ⊂
⋃
j
Uj, A ⊂

⋃
j
B(xnj, 2λj). Thus:

Hα
µ,4(εα+ε)

1
α

(A) 6
∑
j

µ
(
B(xnj, 2λj)

)α
= 4α

∑
j

λαj

< 4α
∑
j

|Uj|α + 2 · 4αε.

As a result, Hα
µ,4(εα+ε)

1
α

(A) 6 4αHαε (A) + 2 ·4αε. When ε approaches to zero,
we obtain Hαµ(A) 6 4αHα(A). Finally, dimµ(A) 6 dimH(A).

4. Calculating µ-topological Hausdorff
dimension of Sierpiński carpet

In this section, we give an estimation of the µ-topological Hausdorff di-
mension of the Sierpiński carpet T. Let X = R2, F =

⋃
n>1 Fn where Fn is

the triadic squares set of the n-th generation, and µ is a non-negative function
defined on F and satisfying (3.1). Let us recall that a triadic square of the
n-th generation is defined by:

C = I ×J ⊆ R2,

where I and J are two triadic intervals of the n-th generation.

4.1. Lower bound of µ-topological Hausdorff dimension of
Sierpiński carpet T. Now we establish a lower estimation of the µ-
topological Hausdorff dimension of the Sierpiński carpet T. For µ, we as-
sociate functions W1 and W2, defined on the set of triadic intervals I in R
by:

If I is a triadic interval of the n-th generation, contained in [0, 1[, then:

W1(I) = inf
J
µ(I ×J) and W2(I) = inf

J
µ(J × I) (4.1)



244 hela lotfi

where the lower bound is taken on all the triadic intervals J of the n-th
generation contained in [0, 1[. Else:

W1(I) = W2(I) = 0.

Seeing that µ satisfies (3.1), then the W1(respectively W2)-Hausdorff dimen-
sion is well-defined. Indeed, it must be proved that W1 and W2 satisfy (3.1).
It is clear that W1 satisfy (3.1) when x 6∈ [0, 1[. Let x ∈ [0, 1[ and ε > 0.
As µ satisfies (3.1), then for all y ∈ [0, 1[ there exists C ∈ F such that
(x,y) ∈ C = I ×J and µ(C) < ε. Therefore, x ∈ I and W1(I) < ε. Conse-
quently, W1 satisfies (3.1), and similarly we prove that W2 satisfies (3.1).

Thereby, we have the following result.

Theorem 4.1. We have

dimtµ(T) > 1 + sup
(

dimW1 (K), dimW2 (K)
)

where K is the middle-thirds Cantor set.

Proof. We will establish that dimtµ(T) > 1 + dimW1 (K). The other in-
equality dimtµ(T) > 1 + dimW2 (K) can be proved in a similar way. For this
purpose, we need the following intermediate result.

Lemma 4.2. Let s < dimW1 (K). Then there exists xs ∈ K satisfying

dimW1
(
]xs −r,xs + r[∩K

)
> s for each r > 0. (4.2)

Proof of Lemma. Assume, on the contrary, that for all x ∈ K, there exists
rx > 0 such that dimW1 (]x−rx,x + rx[∩K) 6 s. It is clear to see that
K ⊂

⋃
x∈K

]x−rx,x + rx[. As K is compact and according to the compactness

of subsets, we have K ⊂
p⋃
i=1

]xi−rxi,xi + rxi[. Then the middle-thirds Cantor

set can be written as:

K =

p⋃
i=1

(]xi −rxi,xi + rxi[∩K) .

Hence:
dimW1 (K) = sup

16i6p
dimW1 (]xi −rxi,xi + rxi[∩K) 6 s.

This contradicts the fact that dimW1 (K) > s.



the µ-topological hausdorff dimension 245

Now, we return to the proof of Theorem 4.1. For a fixed s < dimW1 (K),
from Lemma 4.2, there exists xs ∈ K such that for all r > 0 we have
dimW1

(
]xs −r,xs + r[∩K

)
> s.

Let U be an open basis of T. Since K × [0, 1] ⊂ T and as we remark that:

dimtµ(X) = 1 + inf
U

sup
U∈U

dimµ(∂U) (4.3)

where the lower bound is taken on all basis U of X, then it sufficient to
demonstrate that there exists U ∈ U such that dimµ(∂TU ∩ K × [0, 1]) > s,
where ∂T is the boundary in the Sierpiński carpet T.

1. First case: xs ∈ K\{0}.
The point (xs, 1) ∈

]
0, 7

6

[
×
]
0, 7

6

[
∩T =

⋃
i
Ui = U with Ui ∈U. Consider

i such that Ui contains point (xs, 1). Note U instead of Ui.

We put ys = inf {y 6 1 : (xs,y) ∈ U}, then (xs,ys) ∈ ∂TU ∩K × [0, 1].
On the other hand, there exists rs > 0 such that:

]xs −rs,xs + rs[×]1 −rs, 1 + rs[∩T ⊂ U and xs −rs > 0.

Therefore:

]xs −rs,xs + rs[∩K ⊂ P (∂TU ∩K × [0, 1]) ,

with P : R2 → R, (x,y) 7→ x.
Fix ε > 0 and let

⋃
i Ci be a covering of ∂TU ∩ K × [0, 1] by triadic

squares, where Ci = Ii × Ji while satisfying µ(Ci) < ε. It follows that⋃
i Ii is a covering of ]xs −rs,xs + rs[∩K satisfying W1(Ii) < ε. Thus:∑

i

µ (Ci)
s >

∑
i

W1(Ii)
s > HsW1,ε (]xs −rs,xs + rs[∩K) .

As a consequence:

Hsµ,ε (∂TU ∩K × [0, 1]) > H
s
W1,ε

(]xs −rs,xs + rs[∩K) .

Accordingly, when ε approaches to zero we obtain:

Hsµ (∂TU ∩K × [0, 1]) > H
s
W1

(]xs −rs,xs + rs[∩K) .

Based on dimW1 (]xs −rs,xs + rs[∩K) > s, we have:

dimµ (∂TU ∩K × [0, 1]) > s.



246 hela lotfi

2. Second case: xs = 0.

The point (0, 0) ∈
]
−1

6
, 1
[
×
]
−1

6
, 1
[
∩ T =

⋃
i
Ui = U with Ui ∈ U.

Consider i such that Ui contains point (0, 0). Note U instead of Ui. Next,
we put ys = sup{y > 0 : (0,y) ∈ U}, then (0,ys) ∈ ∂TU ∩ K × [0, 1].
Moreover, there exists rs > 0 such that:

] −rs,rs[× ]ys −rs,ys + rs[∩T ⊂ U.

Hence,
] −rs,rs[∩K ⊂ P (∂TU ∩ K × [0, 1])

with P : R2 → R, (x,y) 7→ x. Indeed, if t = 0 ∈ ] − rs,rs[∩K, then
according to the above reasoning, we have (0, 0) ∈ U, and there exists
y0 ∈ [0, 1] such that:

(0,y0) ∈ ∂TU ∩K × [0, 1]

where y0 = sup{y > 0 : (0,y) ∈ U}. Therefore:

0 = P (0,y0) ∈ P (∂TU ∩K × [0, 1]) .

Furthermore, if t ∈ ] − rs,rs[∩K and t 6= 0, i.e. t ∈ ]0,rs[∩K, then
according to the above reasoning, we have (t, 0) ∈ U, and there exists
y1 ∈ [0, 1] such that (t,y1) ∈ ∂TU ∩K × [0, 1], where

y1 = sup{y > 0 : (t,y) ∈ U} ,

so t ∈ P (∂TU ∩K × [0, 1]).
Let ε > 0 and let

⋃
i Ci be a covering of ∂TU ∩ K × [0, 1] by triadic

squares where Ci = Ii × Ji, while satisfying µ(Ci) < ε. It follows that⋃
i Ii is a covering of ] −rs,rs[∩K satisfying W1(Ii) < ε. Thus:∑

i

µ (Ci)
s >

∑
i

W1(Ii)
s > HsW1,ε (] −rs,rs[∩K) .

Hence:
Hsµ,ε (∂TU ∩K × [0, 1]) > H

s
W1,ε

(] −rs,rs[∩K) .

Then when ε approaches to zero, and since dimW1 (] −rs,rs[∩K) > s,
we have:

dimµ (∂TU ∩K × [0, 1]) > s.



the µ-topological hausdorff dimension 247

4.2. Upper bound of µ-topological Hausdorff dimension of
Sierpiński carpet T. Now we establish an upper bound of the µ-topo-
logical Hausdorff dimension of T. In the following, we assume that µ satisfies
this condition:

For each (x,y) ∈ R2, ε > 0, and n ∈ N\{0}, there exists
p > n and C ∈Fp verifiying (x,y) ∈ C with µ(C) < ε.

(4.4)

It is easy to see that µ satisfies (3.1). We can observe that a finite, non-atomic
and Borelian measure µ, defined on R2, satisfies (4.4).

In what follows, we will essentially consider triadic squares contained in
[0, 1[2. These squares are usually coded as follows:

Let An be the n-fold Cartesian product of A = {0, 1, 2} and A∗ =
⋃
n>1
An.

To concatenate two words a and b in A∗, we put b at the end of a. The
resulting word is denoted by ab.

For an element a in A∗ we denote by |a| the length of a where |a| = n,
such that a ∈An.

Let i = i1i2 . . . in ∈An, so we associate a triadic interval as follows:

Ii =

[
n∑
k=1

ik
3k
,
n∑
k=1

ik
3k

+
1

3n

[
⊂
[
0, 1
[
.

For all i,j ∈An, we consider a triadic square of Fn defined by:

Ci,j = Ii × Ij ⊂ [0, 1[2.

For all i,j ∈ A∗, such that |i| = |j|, we consider the following functions
ν1i,j and ν

2
i,j associated to µ and defined on the set of triadic intervals I in R

as follows: If I 6⊂ [0, 1[, then ν1i,j(I) = ν
2
i,j(I) = 0, and for all I = Ik ⊂ [0, 1[,

where k ∈An:

ν1i,j(Ik) = µ(Ci1,jk)

ν2i,j(Ik) = µ(Cik,j1)
(4.5)

with 1 = 11 . . . 1 ∈An.
We observe that ν1i,j and ν

2
i,j satisfy (3.1). Indeed, let i,j ∈A

∗, which are
written as i = i1 . . . in0 and j = j1 . . .jn0 , so case x 6∈ [0, 1[ is trivial. We fix
x ∈ [0, 1[, and we consider the triadic development of x given by:

x =
+∞∑
i=1

xi
3i

where x ∈ Ix1...xn for all n ∈ N
∗.



248 hela lotfi

Put:

x′ =

n0∑
k=1

ik
3k

+
1

2 · 3n0
∈ [0, 1[ and y′ =

n0∑
k=1

jk
3k

+

∞∑
k=n0+1

xk−n0
3k

∈ [0, 1[.

Let ε > 0, seeing that µ satisfies (4.4), then there exists a positive integer
p > n0 such that (x

′,y′) ∈ Ci1,jx1...xp−n0 and µ(Ci1,jx1...xp−n0 ) < ε. Thus,
ν1i,j(Ix1...xp−n0 ) = µ(Ci1,jx1...xp−n0

) < ε. Therefore, ν1i,j satisfies (3.1) since

x ∈ Ix1...xp−n0 . In the same way, we prove that ν
2
i,j satisfies (3.1).

Consequently, the ν1i,j (respectively ν
2
i,j)-Hausdorff dimension is well-

defined. Hence, we have the following result.

Theorem 4.3. Given a function µ satisfying (4.4) and vanishing on the
triadic squares that are not contained in [0, 1[2, we have:

dimtµ(T) 6 1 + lim inf
n→+∞

sup
|i| = |j| = n
l = 1, 2

dimνli,j
(K)

where K is the middle-thirds Cantor set.

Proof. For n ∈ N\{0} and u,v ∈ Z, let (zun,zvn) be the center of a triadic
square

[
u
3n
, u+1

3n

[
×
[
v

3n
, v+1

3n

[
from Fn.

Denote by Hn the set of intervals which are written as
]
zun,z

u+2
n

[
where

u ∈ Z. Put Un =
{
I ×J : I,J ∈Hn

}
, clearly U =

⋃
n>1
Un is a countable open

basis of R2.
To establish Theorem 4.3, considering the fact ∂T (U ∩T) ⊂ ∂U ∩T for all

U ∈ R2, where ∂T is the boundary in the Sierpiński carpet T and taking into
account (4.3), it suffices to show that for all U ∈Up we have:

dimµ(T ∩∂U) 6 inf
n>p

sup
|i| = |j| = n
l = 1, 2

dimνli,j
(K). (4.6)

Let, for U ∈Up, ∂U be the union of four cloisters. We choose C as one of
these cloisters. We first treat the case where C is a vertical cloister.

Let n > p and Ci,j be a triadic square of the n-th generation such that
Ci,j ∩ T ∩ C 6= ∅. We also choose {Ik}k, a ε-cover of K ∩ [0, 1[ by triadic
intervals, i.e. K ∩ [0, 1[⊂

⋃
k

Ik with ν
1
i,j(Ik) < ε for each k. It follows that

(see Figure 1) Ci,j ∩ T ∩C ⊂
⋃
k

Ci1,jk, where 1 = 11 . . . 1 and |1| = |k|.



the µ-topological hausdorff dimension 249

Figure 1: Sierpiński Carpet
Square U ∈U2.
Triadic square Ci,j of third generation such that Ci,j ∩∂U ∩T 6= ∅.
Triadic squares Ci1,j0 and Ci1,j2.

Moreover, µ
(
Ci1,jk

)
= ν1i,j(Ik) < ε for all k with |k| = |1|. For α > 0, we

have:
Hαµ,ε(Ci,j ∩T ∩C) 6

∑
k

(
µ(Ci1,jk)

)α
=
∑
k

(
ν1i,j(Ik)

)α
Therefore:

Hαµ,ε(Ci,j ∩T ∩C) 6 H
α
ν1i,j,ε

(K ∩ [0, 1[),

When ε goes to zero, we obtain:

Hαµ(Ci,j ∩T ∩C) 6 H
α
ν1i,j

(K ∩ [0, 1[).

Subsequently,

dimµ(Ci,j ∩T ∩C) 6 dimν1i,j (K ∩ [0, 1[) 6 dimν1i,j (K).

Clearly, taking account of convention dimµ(∅) = −1, the previous inequality
is still valid if Ci,j ∩T ∩C = ∅.



250 hela lotfi

On the other hand:

T ∩ C∩ [0, 1[2 ⊂
⋃

|i|=|j|=n

Ci,j

Thus:
T ∩ C∩ [0, 1[2 =

⋃
|i|=|j|=n

Ci,j ∩T ∩C

Then we obtain:

dimµ(T ∩C∩ [0, 1[2) = sup
|i|=|j|=n

dimµ(Ci,j ∩T ∩C) 6 sup
|i|=|j|=n

dimν1i,j
(K).

Since µ vanishes on the triadic squares that are not contained in [0, 1[2, then:

dimµ(T ∩C) = dimµ(T ∩C∩ [0, 1[2).

Thus:
dimµ(T ∩C) 6 sup

|i|=|j|=n
dimν1i,j

(K).

As a consequence:

dimµ(T ∩C) 6 inf
n>p

sup
|i|=|j|=n

dimν1i,j
(K).

If C is a horizontal cloister, we analogously obtain:

dimµ(T ∩C) 6 inf
n>p

sup
|i|=|j|=n

dimν2i,j
(K).

Finally, we have:

dimµ(T ∩∂U) 6 inf
n>p

sup
|i| = |j| = n
l = 1, 2

dimνli,j
(K).

4.3. Equality case. Let us recall that we have proved in Theorem 4.1
and Theorem 4.3 the following inequalities:

1 + sup
(

dimW1 (K), dimW2 (K)
)
6 dimtµ(T)

6 1 + lim inf
n→+∞

sup
|i| = |j| = n
l = 1, 2

dimνli,j
(K). (4.7)



the µ-topological hausdorff dimension 251

In this section, we give an example of measure µ, where the equality holds
between the upper and lower bounds of the µ-topological Hausdorff dimension
of T . For this purpose, let

(
pi,j
)
i,j∈A be a square matrix of order 3 such that

for each i,j ∈A, pi,j > 0 and
∑

06i,j62
pi,j = 1.

We consider the Bernoulli measure supported on [0, 1[2 and defined by:

µ(Ci,j) =
n∏
k=1

pik,jk

where i = i1i2 . . . in and j = j1j2 . . .jn. Choose δ and β as two positive real
numbers such that:

pδ1,0 + p
δ
1,2 = 1 and p

β
0,1 + p

β
2,1 = 1.

Theorem 4.4. (Equality case) If matrix (pi,j) satisfies

p0,1 6 min(p0,0,p0,2), p2,1 6 min(p2,0,p2,2),

p1,0 6 min(p0,0,p2,0), p1,2 6 min(p0,2,p2,2),
(4.8)

then
dimtµ(T) = 1 + sup(β,δ).

Remark. A class of matrices satisfying (4.8) is:

A =




1−4a
5

a 1−4a
5

a 1−4a
5

a
1−4a

5
a 1−4a

5


 ,

where 0 < a 6 1
9
. Note that this class of matrices contains Lebesgue measure

(case when a = 1
9
.)

Proof. The proof of Theorem 4.4 is split into three steps:

Step 1. We begin by proving that for all i,j ∈A∗ such that |i| = |j|, we have:

dimν1i,j
(K) = δ and dimν2i,j

(K) = β.

Let K̃ be the middle-thirds Cantor set deprived of extremities of triadic
intervals. Therefore, we obtain:

dimν1i,j
(K) = dimν1i,j

(K̃).



252 hela lotfi

Let i = i1i2 . . . iq, j = j1j2 . . .jq ∈ A∗. It is observed that if Ik1k2...kn
is a triadic interval crossing K̃, then for all l ∈ {1, 2, . . .n}, we have
kl ∈{0, 2} and (

ν1i,j(Ik1k2...kn)
)δ

= ξδ pδ1,k1 p
δ
1,k2

. . .pδ1,kn,

where ξ =
q∏

k=1

pik,jk > 0. Thus, function (ν
1
i,j)

δ behaves as a measure.

Then for all disjoint covering {Is}s of K̃ by triadic intervals we have:∑
s

(
ν1i,j(Is)

)δ
= ξδ

Whence, dimν1i,j
(K) = δ. Similarly, we prove that dimν2i,j

(K) = β.

Step 2. Now we verify that dimW1 (K) = α and dimW2 (K) = ρ.

Put for all k ∈{0, 2}:

lk = min
06j62

pk,j and tk = min
06j62

pj,k.

Choose α and ρ as two positive real numbers such that:

lα0 + l
α
2 = 1 and t

ρ
0 + t

ρ
2 = 1.

By step 1, we have:

dimW1 (K) = dimW1 (K̃).

It is remarkable that if Ii1i2...in is a triadic interval that crosses K̃, then
for all k ∈{1, 2, . . .n}, we have ik ∈{0, 2}. Therefore:

(W1(Ii1i2...in))
α

= lαi1 l
α
i2
. . . lαin.

Hence, Wα1 behaves as a measure. Consequently, if {Is}s is a covering
of K̃ by disjoint triadic intervals, we have:∑

s

(W1(Is))
α

= 1.

It results that HαW1 (K̃) = 1. Thus, dimW1 (K̃) = α, and then:

dimW1 (K) = α.

Analogously, we prove that dimW2 (K) = ρ.



the µ-topological hausdorff dimension 253

Step 3. Finally, from conditions (4.8), we obtain:

l0 = min
06j62

p0,j = p0,1

l2 = min
06j62

p2,j = p2,1
and

t0 = min
06j62

pj,0 = p1,0

t2 = min
06j62

pj,2 = p1,2

By hypothesis, we have:

lα0 + l
α
2 = 1

p
β
0,1 + p

β
2,1 = 1

and
t
ρ
0 + t

ρ
2 = 1

pδ1,0 + p
δ
1,2 = 1

Then:

α = β and ρ = δ.

Thus, based on (4.7), we obtain:

1 + sup(α,ρ) 6 dimtµ(T) 6 1 + sup(δ,β).

Hence, the result yields.

Corollary 4.5.

dimtH(T) =
ln 6

ln 3
.

Proof. Matrix (pi,j)i,j∈A, where pi,j =
1
9

for all i,j ∈ A satisfies (4.8).
Then by Theorem 4.4, and seeing that α = ρ = ln 2

2 ln 3
, we have:

dimtµ(T) = 1 +
ln 2

2 ln 3
.

Let us recall that triadic squares allow the calculation of the Hausdorff di-
mension (see [16]).

It is noticeable that if we choose µ(C) = 1
2
|C|2, where C is a triadic square

in [0, 1[2, we have for all A ⊂ [0, 1[2:

dimH(A) = 2 dimµ(A).

As a result:

dimtH(T) = 2 dimtµ(T) − 1.

This achieves the proof of this corollary.



254 hela lotfi

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	Introduction
	Topological Hausdorff dimension
	-Topological Hausdorff dimension
	Calculating -topological Hausdorff dimension of Sierpinski carpet
	 Lower bound of -topological Hausdorff dimension of Sierpinski carpet T.
	 Upper bound of -topological Hausdorff dimension of Sierpinski carpet T.
	Equality case.