Results on fractal measure of some sets
The fractal dimensions are very important characteristics of fractal sets. A problem which arises in the study of fractal sets is the determination of their dimensions. The Hausdorf dimension is dfficult to be determined, even if the box dimensions can be computed. In this article we present some re...
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irk-123456789-44732009-11-20T12:00:55Z Results on fractal measure of some sets Barbulescu, A. The fractal dimensions are very important characteristics of fractal sets. A problem which arises in the study of fractal sets is the determination of their dimensions. The Hausdorf dimension is dfficult to be determined, even if the box dimensions can be computed. In this article we present some relations between these types of measures and we estimate them for some sets. 2007 Article Results on fractal measure of some sets / A. Barbulescu // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 13-22. — Бібліогр.: 13 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4473 en Інститут математики НАН України |
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The fractal dimensions are very important characteristics of fractal sets. A problem which arises in the study of fractal sets is the determination of their dimensions. The Hausdorf dimension is dfficult to be determined, even if the box dimensions can be computed. In this article we present some relations between these types of measures
and we estimate them for some sets. |
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Results on fractal measure of some sets |
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Results on fractal measure of some sets |
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Results on fractal measure of some sets |
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Results on fractal measure of some sets |
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results on fractal measure of some sets |
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Results on fractal measure of some sets / A. Barbulescu // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 13-22. — Бібліогр.: 13 назв.— англ. |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.13-22
ALINA BARBULESCU
RESULTS ON FRACTAL MEASURE
OF SOME SETS
The fractal dimensions are very important characteristics of fractal
sets. A problem which arises in the study of fractal sets is the deter-
mination of their dimensions. The Hausdorff dimension is difficult to
be determined, even if the box dimensions can be computed. In this
article we present some relations between these types of measures
and we estimate them for some sets.
1. Introduction
The dimensions calculus is fundamental in the fractals study. The Haus-
dorff measures, the box and packing dimensions are widely used and in
many papers the relations between them are given ([5] - [7], [11]). In [1] -
[4] we gave some boundedness conditions for a class of fractal sets, in Rn.
This type of conditions is important in order to prove theorems concerning
the module and the capacities and the relations between them ([10]) or to
determine the dimensions of fractal sets ([8], [9], [12], [13]).
In what follows we shall work with the following basic notions.
Definition 1. Let Rn be the Euclidean n-dimensional metric space, E a
subset of Rn and r0 > 0.
A continuous function h(r), defined on [0, r0) , nondecreasing and such that
lim
r→0
h(r) = 0 is called a measure function.
If 0 < β < ∞ and h is a measure function, then, the Hausdorff h-measure
of E is defined by :
Hh(E) = lim
β→0
inf
{∑
i
h(|Ui|) : E ⊆
⋃
i
Ui : 0 < |Ui| < β
}
.
where |Ui| is the diameter of Ui.
2000 Mathematics Subject Classifications 28A78
Key words and phrases. Hausdorff h-measure, box dimension, boundedness
13
14 ALINA BARBULESCU
Remark. There are definitions where the cover of the set E is made with
balls. The relation between these ”spherical” Hausdorff h - measure, de-
noted by H ′
h and Hh is:
Hh(E) ≤ H ′
h(E). (1)
Definition 2. Let ϕ1, ϕ2 > 0 be functions defined in D ⊂ Rn. We say that
ϕ1 and ϕ2 are similar and we denote by: ϕ1 ∼ ϕ2, if there exists Q > 0,
satisfying:
1
Q
ϕ1(x) ≤ ϕ2(x) ≤ Qϕ1(x), ∀x ∈ D.
Definition 3. Let δ > 0 and f : D ⊂ Rn → R. f is said to be a δ - class
Lipschitz function if there exists M > 0 such that:
|f(x + α) − f(x)| ≤ M‖α‖δ, ∀x ∈ D, ∀α ∈ Rn withx + α ∈ D,
where for α ∈ Rn, α = (α1, ..., αn), ‖α‖ =
∑n
i=1 α2
i .
f is said to be a Lipschitz function if δ = 1.
Definition 4. A set E ⊂ Rn is called k - rectifiable if there are a Lipschitz
function f : Rk → Rn and a bounded subset F of Rk, such that f (F ) =E.
If f : [0, 1] → R, we denote by Rf [t1, t2] the oscillation of f on
[t1, t2] ⊂ [0, 1] and by Γ (f) the graph of the function f .
2. Results
In ([1] − [3]), the following functions were studied:
g(x) =
⎧⎪⎪⎨
⎪⎪⎩
2x , 0 ≤ x < 1
2
−2 (x − 1) , 1
2
≤ x < 3
2
2(x − 2) , 3
2
≤ x < 2
(2)
f(x) =
∞∑
i=1
λs−2
i g(λix), (∀)x ∈ [0, 1] , (3)
where g is given in (2), s > 0 and {λi}i∈N∗ is a sequence such that
(∃) ε > 1 : λi+1 ≥ ελi > 0, (∀) i ∈ N∗. (4)
Theorem 1. ([2]) If h is a measure function, h(t)˜tp, p ≥ 2, f : [0, 1] → R
is a δ- class Lipschitz function, δ ≥ 0, then Hh(Γ(f)) < +∞. The result
RESULTS ON FRACTAL MEASURE OF SOME SETS 15
remains true if p ≥ 1 and δ > 1.
Theorem 2. If h is a measure function, h(t)˜tp, p ≥ 2, f is the function
defined in (3), with s ∈ [0 , 2) and {λi}i∈N∗ ∈ R+ is a sequence that satis-
fies (4), then Hh(Γ(f)) < +∞.
Proof. For s ∈ [1 , 2), the proof is analogous to that of theorem 2 [1]. It
remains to prove the result for s ∈ [0 , 1) .
We consider 0 < α < 1, small enough and k ∈ N∗ such that:
λ−1
k+1 ≤ α < λ−1
k . (5)
Then:
|f(x + α) − f(x)| =
∣∣∣∣∣
∞∑
i=1
λs−2
i {g(λi(x + α)) − g(λix)}
∣∣∣∣∣ ≤
≤
k∑
i=1
λs−2
i |g(λi(x + α)) − g(λix)| +
∞∑
i=k+1
λs−2
i |g(λi(x + α)) − g(λix)| .
From the definition of g it results:
|g(λi(x + α)) − g(λix)| ≤ 2.
Thus:
|f(x + α) − f(x)| ≤
k∑
i=1
λs−2
i |g(λi(x + α)) − g(λix)| + 2
∞∑
i=k+1
λs−2
i ⇒
|f(x + α) − f(x)| ≤ 2α
k∑
i=1
λs−1
i + 2
∞∑
i=k+1
λs−2
i . (6)
Using (4) we have:
εi−1λ1 < λi , s < 1 ⇒ λs−1
i < λs−1
1
(
εi−1
)s−1 ⇒
k∑
i=1
λs−1
i < λs−1
1
k∑
i=1
(
εs−1
)i−1
= λs−1
1
1 − (1
ε
)k(1−s)
1 − (1
ε
)1−s ⇒
k∑
i=1
λs−1
i < λs−1
1 · 1
1 − (1
ε
)1−s . (7)
The relations (6) and (7) give:
|f(x + α) − f(x)| ≤ α
2λs−1
1
1 − (1
ε
)1−s + 2
∞∑
i=k+1
λs−2
i . (8)
16 ALINA BARBULESCU
∞∑
i=k+1
λs−2
i ≤
∞∑
j=0
(
εjλk+1
)s−2
= λs−2
k+1
∞∑
j=0
(
εj
)s−2
= λs−2
k+1
∞∑
j=0
(
εs−2
)j
Since s − 2 < 0, ε > 1, the series
∞∑
j=0
(εs−2)
j
is convergent and
∞∑
i=k+1
λs−2
i < λs−2
k+1 ·
1
1 − (1
ε
)2−s (9)
From (8) and (9), it results:
|f(x + α) − f(x)| ≤ α
2λs−2
1
1 − (1
ε
)1−s +
2λs−2
k+1
1 − (1
ε
)2−s (10)
The relation (5) implies: λs−2
k+1 ≤ α2−s < α because α ∈ [0, 1) . Thus:
|f(x + α) − f(x)| ≤ α
(
2λs−1
1
1 − (1
ε
)1−s +
2
1 − (1
ε
)2−s
)
= αM ⇔
|f(x + α) − f(x)| < αM, (11)
where M =
2λs−1
1
1−( 1
ε)
1−s + 2
1−( 1
ε)
2−s .
From (11) it results that f is a Lipschitz function. Since the hypothesis
of the theorem 1 is satisfied, then Hh(Γ(f)) < +∞.
Theorem 3. Let h be a measure function, such that
h(t)˜P (t)eT (t), t ≥ 0, (12)
where P and T are polynomials:
P (t) = a1t + a2t
2 + ... + apt
p, p ≥ 1,
T (t) = b0 + b1t + ... + amtm,
with the property
P ′ (t) + P (t) · T (t) > 0, t ≥ 0. (13)
If f : [0, 1] → R is a δ - class Lipschitz function, δ ≥ 1, then Hh(Γ(f)) <
+∞.
The result remains true if p ≥ 2, a1 = 0 and δ ∈ [0, 1] .
Proof. The condition (13) means that P (t)eT (t), t ≥ 0 is itself a measure
function.
The first part of the proof follows that of [5] .
RESULTS ON FRACTAL MEASURE OF SOME SETS 17
We suppose that the Lipschitz constant is M = 1.
To any x corresponds an interval (x− k, x + k) such that, for any x + α
of this interval:
|f(x + α) − f(x)| ≤ |α|δ .
Since [0, 1] is a compact set, there exists a finite set of overlapping in-
tervals covering (0, 1):
(0, k0), (x1 − k1, x1 + k1), ..., (xn−1 − kn−1, xn−1 + kn−1), (1 − kn, 1).
If ci are arbitrary points, satisfying:
c1 ∈ (0, x1), ci ∈ (xi−1, xi), i = 2, ..., n − 1, cn ∈ (xn−1, 1)
ci ∈ (xi−1 − ki−1, xi−1 + ki−1)
⋂
(xi − ki, xi + ki), i = 2, ..., n − 1.
we have: 0 < c1 < x1 < c2 < x2 < ... < xn−1 < cn < 1·
The oscillation of f(x) in the interval (ci−1, ci) is less than 2 (ci − ci−1)
δ
and thus the part of the curve corresponding to the interval (ci−1, ci) can
be enclosed in a rectangle of height 2 (ci − ci−1)
δ and of base ci − ci−1,
and consequently in
[
2 (ci − ci−1)
δ−1
]
+ 1 squares of side ci − ci−1 or in the
number of circles of radius ci−ci−1√
2
circumscribed about each of these squares.
The integer part of x was denoted by [x].
Given an arbitrary r ∈ (0, 1
2
)
it can always be assumed that: ci−ci−1 < r,
i = 2, 3, ..., n.
Denote by Cr the set of all the above circles and consider
∑
Cr
h(2r) =
∑
Cr
{
h(2r)
eT (2r) · P (2r)
· eT (2r) · P (2r)
}
.
From (12) it results that:
(∃)Q > 0 :
h(2r)
eT (2r) · P (2r)
≤ Q.
Then, ∑
Cr
h(2r) ≤ Q
∑
Cr
{
eT (2r) · P (2r)
}⇔
∑
Cr
h(2r) ≤ Q
∑
Cr
P (2r) · e m
k=0 bk·(2r)k
.
r ∈
(
0,
1
2
)
⇒
∑
Cr
h(2r) ≤ Q · e m
k=0|bk| ·
∑
Cr
P (2r)·
We have to estimate
∑
Cr
P (2r).
18 ALINA BARBULESCU
The sum of the terms corresponding to the interval (ci−1, ci), i = 2, ..., n
is:
Si =
{[
2 (ci − ci−1)
δ−1
]
+ 1
}
·
p∑
k=1
ak
{
(ci − ci−1)
√
2
}k
,
where [x] is the integer part of x.
Si ≤
{
2 (ci − ci−1)
δ−1 + 1
}
·
p∑
k=1
{
ak · (ci − ci−1)
k · 2k/2
}
⇒
Si ≤ 2
p
2 · max
k∈1, p
|ak| ·
p∑
k=1
{
2 (ci − ci−1)
k+δ−1 + (ci − ci−1)
k
}
ci − ci−1 < 1, k + δ − 1 ≥ 1 ⇒ (ci − ci−1)
k+δ−1 ≤ ci − ci−1 ⇒
Si ≤ 3 · 2 p
2 · p · max
k∈1, p
|ak| (ci − ci−1) ⇒
∑
Cr
P (2r) ≤ 3 · 2 p
2 · p · max
k∈1, p
|ak|
n∑
i=2
(ci − ci−1) ≤ 3 · 2 p
2 · p · max
k∈1, p
|ak| ⇒
∑
Cr
h(2r) ≤ 3 · 2 p
2 · p · Q · max
k∈1, p
|ak| · e m
k=0|bk|.
Then H ′
h(Γ(f)) < +∞ ⇒ H ′
h(Γ(f)) < +∞.
If M �= 1, then∑
Cr
h(2r) ≤ 3 · 2 p
2 · p · Q · M · max
k∈1, p
|ak| · e m
k=0|bk | ⇒ Hh(Γ(f)) < +∞.
If p ≥ 2 and δ > 0, then k ≥ 2,(ci − ci−1)
k+δ−1 < ci − ci−1 and the proof is
the same as above.
Theorem 4. If Γ(f) is the graph of the function defined in (3), s ∈
[0, 2), {λi}i∈N∗ ∈ R+ is a sequence of numbers, that satisfies (4) and h is a
measure function satisfying (12), then Hh(Γ(f)) < +∞.
Proof. The proof is analogous to that of the Theorem 3.
The following lemma will be used:
Lemma 1. ([6]) Let f ∈ C [0, 1] , 0 < β < 1 and m be the least integer
greater than or equal to 1/β. If Nβ(Γ(f)) is the number of the squares of
the β− mesh that intersects Γ (f) , then
β−1
m−1∑
j=0
Rf [jβ, (j + 1)β] ≤ Nβ(Γ(f)) ≤ 2m + β−1
m−1∑
j=0
Rf [jβ, (j + 1)β] .
RESULTS ON FRACTAL MEASURE OF SOME SETS 19
Let us consider δ > 0 and a δ - class Lipschitz function, f : [0, 1] → R.
Then:
|f(x) − f(y)| ≤ M |x − y|δ, (∀)x, y ∈ [0, 1].
Rf [jβ, (j + 1)β] = sup
jβ≤t, u≤(j+1)β
|f (t) − f (u)| ⇒
Rf [jβ, (j + 1)β] ≤ M sup
jβ≤t, u≤(j+1)β
|t − u|δ ⇒
Rf [jβ, (j + 1)β] ≤ Mβδ.
Let N ′
β(Γ(f)) be the number of β - mesh squares that cover the set Γ(f).
Denoting by [x], the integer part of x ∈ R and using lemma 1, it can be
deduced that:
N ′
β(Γ(f)) ≤ 2
[
1
β
]
+ β−1
[
1
β
]
Mβδ ⇒
N ′
β(Γ(f)) < Mβδ−2 +
2
β
. (14)
But,
Nβ
√
n(Γ(f)) ≤ N ′
β(Γ(f)) ≤ 2nNβ(Γ(f)), (15)
where Nβ(Γ(f)) is the smallest number of discs of diameters at most β that
cover Γ(f).
The relations (14) and (15) give for n = 2:
Nβ(Γ(f)) ≤ N ′
β√
2
(Γ(f)) < M
(
β√
2
)δ−2
+
2
√
2
β
⇒
Nβ(Γ(f)) ≤ N ′
β√
2
(Γ(f)) <
3
β
+ M ′βδ−2, (16)
with M ′ = M√
2
δ−2 .
Theorem 5. If f : [0, 1] → R is a δ-class Lipschitz function, δ > 0 and
h is a measure function such that h(t) ∼ tp, p > 2, then Hh(Γ(f)) = 0. The
assertion remains true if p ≥ 1 and δ > 1.
Proof. By hypotheses, Γ(f) is a compact set. Therefore, if β > 0, for every
cover of Γ(f) with open discs Ui, i ∈ N∗, with diameters di ≤ β, there is a
finite number of discs, nβ, that covers Γ(f).
H ′
h(Γ(f)) = lim
β→0
inf
{∑
i
h(|Ui|) : E ⊆
⋃
i
Ui : 0 < |Ui| ≤ β
}
=
20 ALINA BARBULESCU
= lim
β→0
inf
{ nβ∑
i=1
h(|Ui|)
}
≤ lim
β→0
inf {h(β)nβ} ,
since h is nondecreasing. Then
H ′
h(Γ(f)) ≤ lim
β→0
{h(β)Nβ(Γ(f))} ,
where Nβ(Γ(f)) is the smallest number of open discs of diameters at most
β that cover Γ(f).
Denoting by N ′
β(Γ(f)) the number of β - mesh squares that cover Γ(f)
and using the relations (1) and (16), the previous inequality becomes:
Hh(Γ(f)) ≤ H ′
h(Γ(f)) ≤ lim
β→0
{
h(β)N ′
β√
2
(Γ(f))
}
⇒
Hh(Γ(f)) ≤ lim
β→0
{
h(β)(3β−1 + M ′βδ−2)
}⇒
Hh(Γ(f)) ≤ lim
β→0
{
h(β)
βp
(3βp−1 + M ′βp+δ−2)
}
.
Since h(t) ∼ tp, p > 2, there is Q > 0 such that:
1
Q
tp ≤ h(t) ≤ Qtp,
and then
Hh(Γ(f)) ≤ Qlim
β→0
(3βp−1 + M ′βp+δ−2) = 0,
because p − 1 > 0 and p + δ − 2 > 0.
So, Hh(Γ(f)) = 0.
If p ≥ 1 and δ > 1 the proof is the same because p + δ − 2 > 0.
Remark. Theorem 5 gives a better result as the theorem 6 [2], where it
was proved in the same hypotheses, that H ′
h(Γ(f)) < ∞.
Lemma 2. ([4]) If E ⊂ Rm, F ⊂ Rn, f : E → F is a surjective Lipschitz
function, with the Lipschitz constant M and h is a measure function, then:
Hh(F ) ≤ Hh(M · E).
Theorem 6. If E ⊂ Rn is a k-rectifiable set and h is a measure function
such that h(t) ∼ tp, p > 2, then Hh(E) = 0.
Proof. If E ⊂ Rn is k - rectifiable, there exists a bounded set G ⊂ Rk and
f : Rk → Rn such that:
‖f(x) − f(y)‖ < M‖x − y‖, (∀)x, y ∈ Rk
RESULTS ON FRACTAL MEASURE OF SOME SETS 21
and f(G) = E.
The restriction of f at G is a surjection and, by Lemma 2,
Hh(f(G)) ≤ Hh(M · G) ⇔ Hh(E) ≤ Hh(M · G).
M ·G is bounded, so there is a disc B(z0, r), (r > 0) such that M ·G ⊂
B(z0, r). Therefore,
Hh(E) ≤ H ′
h(E) ≤ H ′
h(B(z0, r)) =
= lim
β→0
inf
{∑
i
h(|Ui|) : B(z0, r) ⊆
⋃
i
Ui : 0 < |Ui| ≤ β
}
=
= lim
β→0
inf {nβ · h(|Ui|)} ,
where nβ is the number of the open discs Ui with the diameters |Ui| ≤ β,
that covers B(z0, r).
nβ ≥ π · r2
π·|Ui|2
4
= 4 · r2
|Ui|2 ⇒
Hh(E) ≤ H ′
h(B(z0, r)) ≤ lim
β→0
inf {nβ · h(β)} =
= lim
β→0
h(β) inf nβ = lim
β→0
h(β) · 4r2
β2
= 4r2 lim
β→0
h(β)
β2
=
= 4r2 lim
β→0
{
h(β)
βp
· βp−2
}
≤ 4Qr2 lim
β→0
βp−2 = 0,
where it was used that h(t) ∼ tp, p > 2. So, Hh(E) = 0.
Remark. If in Theorem 6, p = 2, it results that Hh(E) ≤ 4Qr2, so the
Hausdorff h-measure of the k - rectifiable set E is finite.
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Department of Mathematics, Ovidius University of Constanta, 900527
Constanta, Romania.
E-mail address: abarbulescu@univ-ovidius.ro
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