THE REGULARITY OF RANDOM GRAPH DIRECTED SELF-SIMILAR SETS
XIAO-QUN ZHANG
YAN-YAN LIU
Department of Mathematics
Wuhan University ∗
Abstract
d
A set in R is called regular if its Hausdorff dimension coincides with its upper box counting
dimension. We prove that a random graph-directed self-similar set is regular a.e..
1
INTRODUCTION
A subset of Rd is called a self-similar set if it fulfills the invariance K = ∪N
i=1 Si (K), where
S1 , · · · , SN are contractive similarities. It is well known (see Huntchison[6])that, if Si (K) ∩ Sj (K)
is almost empty(open set condition), then dimH K = dimB K = a where a is the unique solution
of the equation:
N
X
Lip(Si )a = 1.
i=1
This theory was subsequently extended to so-called graph directed self-similar sets by Mauldin
and Williams[9]. Let G = (V, E) be a directed multigraph , where V is the set of vertices and
E is the set of edges. Denote the edge from vertex i to vertex j by (i, j). Suppose {Xi }i∈V is a
sequence of subsets of Rd and S(i,j) is a contraction similarity from Xj to Xi for
S all i, j ∈ V and
(i, j) ∈ E. It is known that there is an unique list of (Ki )i∈V such that: Ki = (i,j)∈E S(i,j) (Kj ),
where Ki is a non-empty compact subset of Xi . In this case K = ∪i∈V Ki is called graph directed
self-similar set. For a ≥ 0, define a matrix by
Aa = [(LipS(i,j) )a ]i,j∈V ,
where we adopt a convention that LipS(i,j) = 0 if (i, j) ∈
/ E and 00 = 0. Let φ(a) be the spectral
radius of matrix Aa . If the graph possesses strong connection, that is, for every two vertices,
there is a directed path, and in addition for each i, {S(i,j) (Xj )}j is a non-overlapping family, then
dimH K = a, where a such that
φ(a) = 1.
It is much more difficult to determine the dimension of graph-directed self-similar sets(even for
self-similar sets) with overlap. Falconer[2] proved that graph directed self-similar sets (overlap or
not) have a nice property, the graph-directed self-similar sets are regular a.e..
The random generalization of self-similar sets was investigated by Mauldin, Willams[10], and
Graf[5] by randomizing each step in Hutchison’s construction. Subsequently, Olsen [7] extended
this theory to random graph directed self-similar sets by giving a sequence of probability measures
on the space of all similarities. For other references on this topic, see [7] and the references therein.
∗ This
reasearch was supported by the National Natural Science Foundation of China 10201027
1
Recently, Y. Y. Liu and J. Wu [8] proved that random self-similar set is regular a.s.. This result
generalized that of Falconer [2] to the random setting. In this paper, we extend the above result
to random graph-directed sets.
2
NOTATIONS AND DEFINITIONS
In this section, we recall some definitions and properties of dimensions.
Let K ⊂ Rd , for any s ≥ 0, the s-dimension Hausdorff measure of K is given in the usual way
by
X
[
Hs (K) = lim inf{
|Ui |s : K ⊂
Ui , 0 < |Ui | < δ}
δ→0
i
i
where | · | denotes the diameter of a set. This leads to the definitions of the Hausdorff dimension
of K:
dimH K = inf{s > 0, Hs (K) < ∞} = sup{s > 0 : Hs (K) > 0}
For ² > 0, let M² (K) be the maximum number of disjoint closed balls of radius ² with centers
in K, and let N² (K) be the smallest number of closed balls of radius ² covering K. We can define
the upper and lower box-counting dimensions by
dimB K
=
dimB K
=
log N² (K)
− log ²
log N² (K)
lim inf
²→0
− log ²
lim sup
²→0
In his book [3], Falconer established the following results:
dimB K
dimB K
log M² (K)
log N2−n (K)
log M2−n (K)
= lim sup
= lim sup
−
log
²
n
log
2
n log 2
n→∞
n→∞
²→0
log M² (K)
log N2−n (K)
log M2−n (K)
= lim inf
= lim inf
= lim inf
n→∞
n→∞
²→0
− log ²
n log 2
n log 2
= lim sup
(2.1)
(2.2)
Finally the packing dimension of K is by definition
dimP K = inf{sup dimB Ki , K ⊂ ∪i Ki }
i
It was also proved in[3] that
dimH K
dimH K
≤
≤
dimB K ≤ dimB K
dimP K ≤ dimB K.
(2.3)
(2.4)
A more detailed introduction and proof of above properties can be found in [3].
3
RANDOM GRAPH DIRECTED SELF-SIMILAR SETS
Let (V, E) be a finite directed graph defined as above. For u, v ∈ V , let Euv denote the set of edges
from u to v and Eu = ∪v∈V Euv . A path in the graph is a finite string e1 · · · en of edges such that
the terminal vertex of the edge ei is the initial vertex of the next edge ei+1 and an infinite path
in the graph is an infinite string e1 e2 · · · of edges such that e1 · · · en is a path for all n ∈ N. For a
2
path or an infinite path α, let ι(α) denote the initial vertex of α. If α is a path , then denote the
terminal vertex of α by τ (α). Moreover, for u, v ∈ V, n ∈ N, we write:
n
Euv
= {e : e = e1 · · · en is a path such that ι(e) = u, τ (e) = v}
N
Eu = {e
infinite path such that
S : e = en1 e2 · · · isn an S
S ι(e) =n u}
∗
n
n
Euv
=S n∈N Euv
Eu =
E
E
=
uv
u∈V
S v∈V S
S Eu
Eu∗ = n∈N Eun
E ∗ = u∈V Eu∗ {∅}
EN = u∈V EuN
If α = α1 · · · αn ∈ E n , then |α| = n is the length and write a|k = α1 · · · αk for k ≤ n. Similarly,
if σ ∈ E N , denote σ|k = σ1 · · · σk for all k ∈ N. For α ∈ E ∗ , σ ∈ E N ∪ E ∗ , if τ (α) = ι(σ), then
define α ∗ σ = α1 · · · α|α| σ1 · · · .We write α ≺ σ if there exists β ∈ E ∗ ∪ E N with σ = α ∗ β.
n
(V, E) is strongly connected if for all u, v ∈ V , there exists n ∈ N such that Euv
6= ∅. Finally,
n
N
except special declaration, if α ∈ E , σ ∈ E , we denote α, σ by α1 · · · αn and σ1 σ2 · · · .
For each u ∈ V , let Xu ⊂ Rd be a compact set with int(Xu ) = Xu and H(Xu ) be the family of
all non-empty compact subsets of Xu equipped with Hausdorff metric du . Let νu be a probability
measure on the space Πe∈Eu Sime where Sime is the contraction similarity space from Xτ (e) to
Xι(e) , equipped with the usual topology of uniform convergence on compact sets. We assume in
the sequel that the following assumptions are fulfilled:
Assumptions:
(i) (V, E) is strongly connected.
(ii) for all u ∈ V, #Eu ≥ 2, that is , there are at least two edges initiating from every vertex.
(iii) there exist 0 < r̂ < 1, such that for all u ∈ V , (Se )e∈Eu ∈ Πe∈Eu Sime , and Lip(Se ) ≥ r̂,
νu − a.e..
For fixed u ∈ V , we denote:
Q
Sime
Γu =
e∈EQ
u
Ωu = Γu × α∈Eu∗ Γτ (α)
Q
Ω =
u∈V Ωu
for α = α1 α2 · · · αn ∈ Eu∗ , define projection maps on Ω:
πu
πα
Sα
: Ω → Γu
: Ω → Γτ (α)
: Ω → Simαn
by the requirements: for each ω = (ωu )u∈V , ωu = (πu (ω), (πα (ω))α∈Eu∗ ), πu (ω) = (Se (ω))e∈Eu ,
πα (ωu ) = (Sαe (ω))e∈Eτ (α) and S∅ (ω) = id.
Q
By =u we denote the product Borel σ−algebra on Ωu ; Pu = νu × α∈Eu∗ ντ (α) denotes the
Q
Q
product probability measure. Let = = u∈V =u be the σ−algebra on Ω and P = u∈V Pu the
product measure, we get our primary probability space (Ω, =, P ). For more detailed investigation
of the measures see [7, 5, 11]. For convenience, we shall write:
S̄α =
rα =
r̄α =
Sα|1 ◦ Sα|2 ◦ · · · ◦ Sα||α|
Lip(Sα )
LipS̄α = rα|1 rα|2 · · · rα|n
for any α ∈ E n . Let Ω0 = {ω ∈ Ω| lim r̄σ|n = 0, for any σ ∈ E N }, for fixed (K̄u )u∈V ∈
n→∞
Q
H(X
),
we
define
a
random
mapping
Ku : Ω → H(Xu ) :
u
u∈V
T
½ S
N
σ∈Eu
n∈N Sσ|n (ω)(Xτ (σ|n) ) ω ∈ Ω0
Ku (ω) =
K¯u
ω∈
/ Ω0
3
It have been proved (see [11]) that P (Ω0 ) = 1 and Ku (ω) is a Borel mapping for P- a. s. ω.
(Ku (ω))u∈V is called random graph directed self-similar sets.
For any α ∈ E ∗ , α 6= ∅, define the shift operators ∆α : Ω → Ω by:
∆α (ω) = ω 0
½
0
where ω = (ωu )u∈V , ω =
(ωu0 )u∈V
,
ωu0
Sαβ (ω), for each β ∈ Eτ∗(α) .
=
(πa (ω), (παβ (ω))β∈Eu∗ ))
ωu
u = τ (α),
, then Sβ (∆α (ω)) =
u 6= τ (α)
Remark 1. For any α ∈ Eu∗ , β ∈ Eτ∗(α) , Sβ (∆α (ω)) is an i.i.d. copy of Sβ (ω).
The set Ku (ω) fulfills the following invariance(see [7, 1]).
Theorem 1. For P- a. s. ω , we have
T
S
(a)Ku (ω) = n∈N α∈Eun S̄α (ω)(Xτ (α) ).
S
(b)Ku (ω) = e∈Eu Se (ω)Kτ (e) (∆e (ω)) where (Kτ (e) (∆e (ω)))e∈Eu are independent and every
Kτ (e) (∆e (ω)) has the same distribution with Kτ (e) (ω).
4
MAIN RESULT AND ITS PROOF
In this section, we state and prove the main theorem.
Lemma 1. For all u, v ∈ V and any c ∈ R,
P {ω : dimH Ku (ω) < c} = P {ω : dimH Kv (ω) < c}
Proof : First we show the inequality
P {ω : dimH Ku (ω) < c} ≤ P {ω : dimH Kv (ω) < c}.
By the strong connection of the graph, there S
exists a path α ∈ E ∗ such that ι(α) = u, τ (α) = v.
It is clear by (b) of Theorem 1 that Ku (ω) = e∈Eu Se (ω)Kτ (e) (∆e (ω)), so
dimH Ku (ω) = max dimH Se (ω)Kτ (e) (∆e ω) = max dimH Kτ (e) (∆e ω)
e∈Eu
e∈Eu
leading to
P {ω : dimH Ku (ω) < c}
=
P {ω : max dimH Kτ (e) (∆e ω) < c}
≤
≤
≤
P {ω : dimH Kτ (α1 ) (∆α1 ω) < c} = P {ω : dimH Kτ (α1 ) (ω) < c}
···
P {ω : dimH Kτ (αn ) (ω) < c} = P {ω : dimH Kv (ω) < c}.
e∈Eu
The inverse inequality can be shown in the same way, we omit the details.
Lemma 2. There exists a constant a such that for P- a. s. ω and all u ∈ V ,
dimH Ku (ω) = a
4
Proof : Suppose that there exists u ∈ V, c > 0 such that
0 < P {ω : dimH Ku (ω) < c} < 1.
then by Theorem 1,
P {ω : dimH Ku (ω) < c}
= P {ω : max dimH Kτ (e) (∆e ω) < c}
e∈Eu
Y
=
P {ω : dimH Kτ (e) (∆e ω) < c}
e∈Eu
=
Y
P {ω : dimH Kτ (e) (ω) < c}
e∈Eu
=
Y
P {ω : dimH Ku (ω) < c}.
e∈Eu
The last equality is followed by Lemma 1 which leads to a contradiction with #Eu ≥ 2, this proves
Lemma 2.
For any ² > 0, u ∈ V , we define
I²,u (ω) = {α ∈ Eu∗ : |S̄α (ω)(Xτ (α) )| ≤ ², |S̄α||α|−1 (ω)(X
τ (α||α|−1) )| > ²}
T
D²,u (ω) = {A ⊂ I²,u (ω) : ∀ζ, η ∈ A, S̄ζ (ω)(Xτ (ζ) ) S̄η (ω)(Xτ (η) ) = ∅}
N̄²,u (ω) = maxA∈D²,u (ω) #A.
Let A²,u (ω) is the member of D²,u (ω) which realizes the maximum N̄²,u (ω). Write A²,u (ω) for:
A²,u (ω) = {α1 , α2 , · · · , αN̄²,u (ω) }
For fixed σ ∈ Eu∗ , τ (σ) = v, we shall also need the following definitions:
σ
(ω) = {α ∈ Ev∗ : |S̄α (∆σ (ω))(Xτ (α) )| ≤ ², |S̄α||α|−1 (∆
I²,u
T σ (ω))(Xτ (α||α|−1) )| > ²}
σ
σ
D²,u (ω) = {A ⊂ I²,u
(ω) : ∀ζ, η ∈ A, S̄ζ (∆σ (ω))(Xτ (ζ) ) S̄η (∆σ (ω))(Xτ (η) ) = ∅}
σ
σ (ω) #A.
N̄²,u
(ω) = maxA∈D²,u
σ
Denote the member which realizes the maximum N̄²,u
(ω) by Aσ²,u (ω).
Lemma 3. If the assumptions are satisfied, then there exists a constant C > 0 such that for P- a.
s. ω and any ² > 0,
N̄²,u (ω) ≥ M² (Ku (ω)) and N̄²,u (ω) ≤ CN² (Ku (ω)).
Proof : For ² > 0, let B(x1 , ²), · · · , B(xM² (Ku (ω)) , ²) be M² (Ku (ω)) disjoint close balls with centers
in Ku (ω) and radius ². For each 1 ≤ i ≤ M² (Ku (ω)), there exists i ∈ EuN satisfying
lim S̄i|n (ω)(Xτ (i|n) ) = xi
n→∞
Choose n0 such that S̄i|n0 (Xτ (i|n0 ) ) ⊂ B(xi , ²), S̄i|n0 −1 (Xτ (i|n0 −1) ) \ B(xi , ²) 6= ∅, then
|S̄i|n0 −1 (Xτ (i|n0 −1) )| > ².
5
We consider two cases:
(1) If |S̄i|n0 (Xτ (i|n0 ))| ≤ ², choose αi = i|n0 ;
(2) If |S̄i|n0 (Xτ (i|n0 ))| > ², there exists l ≥ 1 such that
|S̄i|n0 +l (Xτ (i|n0 +l) )| ≤ ², |S̄i|n0 +l−1 (Xτ (i|n0 +l−1) )| > ²,
then choose αi = i|n0 + l.
It is clear that {αi } ∈ D²,u (ω), which implies
N̄²,u (ω) ≥ M² (Ku (ω))
.
Next we give the details of the proof of the second inequality. Let A²,u (ω) = {α1 , α2 , · · · , αN̄²,u (ω) },
then {S̄αi (ω)(Xτ (αi ) )}i are N̄²,u (ω) disjoint sets such that:
|S̄αi (ω)(Xτ (αi ) )| ≤ ²,
|S̄αi ||αi |−1 (ω)(Xτ (αi ||αi |−1) )| > ²
For all v ∈ V , since intXv 6= ∅, then there exists a constant c1 > 0 such that every Xv contains a
ball of radius c1 . Furthermore S̄αi (ω)(Xτ (αi ) ) contains a ball of radius c1 · r̄αi By assumption (iii),
we find
c1 · r̄αi ≥ c1 · r̂ · r̄αi ||αi |−1
≥ |X i r̂ i | · c1 · r̄αi ||αi |−1 · |Xτ (αi ||αi |−1) |
≥
≥
r̂
M
r̂
M
τ (α ||α |−1)
· c1 · |S̄αi ||αi |−1 (ω)(Xτ (αi ||αi |−1) )|
· c1 · ²
where M = maxu∈V |Xu |.
Let B(x1 , ²), B(x2 , ²), · · · , B(xN² (Ku (ω)) , ²) be closed balls with radius ² such that Ku (ω) ⊂
SN² (Ku (ω))
B(xj , ²). Let
j=1
Bj (ω) = {i : 1 ≤ i ≤ N̄²,u (ω), S̄αi (ω)(Xτ (αi ) ) ∩ B(xj , ²) 6= ∅} for each 1 ≤ j ≤ N² (Ku (ω)),
then S̄αi (ω)(Xτ (αi ) ) ⊂ B(xj , 2²), for each i ∈ Bj (ω).
By volume estimating,
λd (B(xj , 2²)) ≥ #Bj (ω) · λd (B(0,
r̂
· c1 · ²))
M
where λd is the d− dimensional Lebesgue measure. Thus
#Bj (ω) ≤
(2²)d
2d · M d
=
:= C
r̂ d
r̂d · cd1
(M
) · cd1 · ²d
which leads to N² (Ku (ω)) ≥ C1 N̄²,u (ω).
From Lemma 3 and (2.1), we can immediately get the following:
Corollary 1. If the assumptions are satisfied, then for P- a. s. ω
dimB Ku (ω) = lim sup
n→∞
log N̄2−n ,u (ω)
.
n log 2
Now, we shall prove the above defined random graph directed self-similar sets are regular, more
precisely the following holds.
6
Theorem 2. If the assumptions are satisfied, then there exists a constant a ≥ 0such that for all
u ∈ v and P- a. s. ω
dimH Ku (ω) = dimB Ku (ω) = a
Proof : Lemma 2 shows that there exists a constant a such that P- a. s. ω , and all u ∈ V ,
dimH Ku (ω) = a.
n
Let p0 = max(u,v) min{n : Eu,v
6= ∅}. By the strong connection of the direct graph, for fixed
i
u ∈ V, α ∈ A²,u (ω), we can choose ζ i = e1 · · · en such that ι(e1 ) = τ (αi ), τ (en ) = u, n ≤ p0 . Write
p0 +1
β i = αi ∗ ζ i , then S̄β i (ω)(Xu ) ⊂ S̄αi (ω)(Xτ (αi ) ), r̄β i ≥ r̂ M ² and S̄β i (ω) ∩ S̄β j (ω) = ∅, for any
1 ≤ i < j ≤ N̄²,u (ω). Let B²,u (ω) = {β 1 , β 2 · · · , β N̄²,u (ω) }. Similar arguments show that for fixed
σ
σ ∈ Eu∗ , we can find B²,u
(ω) from Aσ²,u (ω).
For any n ∈ N, choose ² = 2−n and we define a sequence of random sets :
Fn,0 (ω) = Xu S
Fn,1 (ω) =
S̄β (ω)(Xu )
β∈B²,u (ω)
S
S
Fn,2 (ω) =
S̄β1 ∗β2 (ω)(Xu )
β1
β1 ∈B²,u (ω) β2 ∈B²,u
(ω)
······
Fn,k (ω) =
S
S
β1 ∈B²,u (ω)
β1
β2 ∈B²,u
(ω)
S
···
β1 ∗β2 ∗···∗βk−1
βk ∈B²,u
(ω)
S̄β1 ∗β2 ∗···∗βk (ω)(Xu )
······
Let
Kn,u (ω) =
∞
\
Fn,k (ω).
k=1
then Kn,u (ω) ⊂ Ku (ω) by the construction of Kn,u (ω) and Theorem 1. On the other hand,
{Fn,k (ω)}k is a random recursive construction which was studied by Mauldin and Willams[10]. By
Theorem 1. 3 and Theorem 3. 6 of [10], we have
dimH Kn,u (ω) = c, for P-a. s.ω,
where c satisfies the following equation:
N̄2−n ,u (ω)
E
X
r̄βc i = 1.
i=1
notice that c ≤ a, we have
N̄2−n ,u (ω)
E
X
r̄βai ≤ 1
i=1
combine with the fact that r̄β i ≥
r̂
p0 +1
M
· 2−n , we find
E N̄2−n ,u (ω) ≤ r̂−(p0 +1)a 2na M a
then for any δ > 0,
∞
X
n=1
P {ω : N̄2−n ,u (ω) ≥ 2n(a+δ) } ≤
∞
∞
X
X
E N̄2−n ,u (ω)
a −(p0 +1)a
≤
M
r̂
·
2−nδ < ∞.
n(a+δ)
2
n=1
n=1
7
by Borel-Cantelli lemma, we find that
P {ω : N̄2−n ,u (ω) ≥ 2n(a+δ) i.o.} = 0
(4.1)
which, with corollary 1, gives that for P- a. s. ω ,
dimB Ku (ω) = lim sup
n→∞
log N̄2−n ,u (ω)
log(2n(a+δ) )
≤ lim sup
= a + δ.
n log 2
n log 2
n→∞
Since δ is arbitrary, we have for P- a. s. ω
dimH Ku (ω) = dimB Ku (ω) = a
which finishes the proof of Theorem 2.
(2.3) (2.4) and Theorem 2 lead to the following corollary immediately:
Corollary 2. Let the Assumptions be satisfied. Then there exists a constant a, such that for Pa. s. ω ,
dimH Ku (ω) = dimP Ku (ω) = dimB Ku (ω) = dimB Ku (ω) = a.
References
[1]
M. Arbeiter and N.Patzschke, Random seld-similar multifractals, Math. Nachr. 181(1996)
5-42.
[2]
K. J. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc.
106(1989) 543-554.
[3]
K. J. Falconer, Fractal Geometry:Mathemaical Foundations and Applications, John Wiley and
Sons Ltd. Chichester, 1990.
[4]
K. J. Falconer, Techniques in Fractal Geometry, John Wiley and Sons Ltd. Chichester, 1997.
[5]
S. Graf, Statistically self-similar fratals, Probab. Th. Rel. Fields 74(1987) 358-392.
[6]
J. E. Hutschinson, Fractal and self-similarity, Indiana Univ. Math. J. 30(1981) 713-747.
[7]
Olsen L., Random Geometrically Graph Directed Multifractals, Pitman Reserch Notes in math
serties.
[8]
Y. Y. Liu and J. Wu, A dimension result for random self-similar sets, Proc. Amer. Math. Soc.
130(2002) 2125-2131.
[9]
R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Tran.
Amer. Math. Soc. 309(1988), 811-829.
[10] R. D. Mauldin and S. C. Williams, Random recursive constructions:asymptotic geometric and
topological properties,Tran. Amer. Math Soc. 295(1986) 325-346.
[11] Y. Tsujii, Markov-self-similar sets, Hiroshima Math. J. 21(1991) 491-519.
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