Dimensions for random self–conformal sets

Math. Nachr. 250, 71 – 81 (2003) / DOI 10.1002/mana.200310022

Dimensions for random self–conformal sets

Yan–Yan Liu∗1 and Jun Wu∗∗1

1 Department of Mathematics, Wuhan University, Wuhan, Hubei, 430072, P. R. China

Received 8 June 2001, revised 27 February 2002, accepted 7 March 2002Published online 27 January 2003

Key words Random self–conformal set, Hausdorff dimension, box–counting dimension

MSC (2000) Primary: 60D05, 58F03. Secondly: 28A78

A set is called regular if its Hausdorff dimension and upper box–counting dimension coincide. Inthis paper, we prove that the random self–conformal set is regular almost surely. Also we determinethe dimensions for a class of random self–conformal sets.

1 Introduction

There has been considerable interest in fractals, both in their occurrence in the sciences, and in theirmathematical theory. A wide class of fractal sets are generated by iterated function system. A self–similar set in Rd is a compact set K fulfilling the invariance

K =N⋃

i=1

SiK ,

where S1, S2, . . . , SN are contraction similarities. If S1, S2, . . . , SN are contraction conformal map-pings, we obtain self–conformal set. It is well–known (see Hutchinson [12]) that, given a family ofsuch mappings, there is a unique compact set with this property. If the open set condition (OSC) issatisfied, i. e. if there is an open set O such that Si(O) ⊂ O for all i and Si(O) ∩ Sj(O) = ∅ for alli = j, the dimensions (including Hausdorff, packing and box–counting dimension) of a self–similar setand a self–conformal set were calculated by Hutchinson [12] and Bedford [3], respectively. Moreover,Falconer [7] proved that even without OSC, they are regular sets, i. e. the Hausdorff dimension andthe upper box–counting dimension coincide. Cawley and Mauldin [5] and Patzschke [18] considered themultifractal spectrum of self–similar measures and self–conformal measures, respectively.

But, natural phenomena are complex and sometimes they are better modeled by random sets, sorandom fractals have aroused people’s great interest. In the 1980s, Falconer [6], Mauldin and Williams[17] and Graf [10] investigated random fractal sets by randomizing each step in Hutchinson’s construc-tion. Hutchinson and Ruschendorff [13] extended the contraction mapping method to prove variousexistence and uniqueness properties for the random self–similar sets. The authors [15] proved the regu-larity of a random self–similar set. Falconer [8], Arbeiter and Patzschke [2], Patzschke [20] consideredthe multifractal spectrum for the random self–similar measures and random self–conformal measures,respectively. For other references on this topics, see [1], [4], [11], [14], [21] and [22] and references therein.

In this paper, we study the regularity for a random self–conformal set. We also determine thedimensions (including Hausdorff, packing and box–counting dimension) for a random self–conformalset. We prove that even without OSC, a random self–conformal set is regular almost surely. Thusthis paper may be regarded as the generalization of the works of Bedford [3] and Falconer [7] to therandom setting. But compare to [7], the proofs in [7] cannot extend to the random case directly. In the

∗ Corresponding author: e-mail: [email protected], Phone: 0086 27 87682957, Fax: 0086 27 87640256∗∗ e-mail: [email protected], Phone: 0086 27 87642924, Fax: 0086 27 87640256

c© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0025-584X/03/25002-0071 $ 17.50+.50/0

72 Liu and Wu: Dimensions for random self–conformal sets

deterministic case, each ball centered in K with radius at most the diameter of K, contains a subset ofK which is the image of K under a map with Lipschitz constant comparable to the radius of the ball.This is an important point that underlies the argument in [7]. In the random setting although this isnot true for individual realization of K, it is true in a certain probabilistic sense. On the other hand,we adopt the approach by modifying the well known topological pressure and the Bowen’s formula togive the explicit dimensional formula for a random self–conformal set. Instead of using “singularityspectrum” to obtain the box–counting dimension in [3] and [20], we use the regularity directly, and thisexplicit dimensional formula improves Theorem 4.1 of [14].

The paper is organized as follows. Section 2 is devoted to some definitions and notations. Theregularity of a random self–conformal set is proved in Section 3 and the dimensions of a random self–conformal set are calculated in Section 4.

2 Notations and definitions

In this section, we recall the definitions of various dimensions and random self–conformal sets. For moredetails we refer to [9], [16] and [20].

2.1 Dimensions

Let K ⊂ Rd. For any s ≥ 0, the s–dimensional Hausdorff measure of K is given in the usual way by

Hs(K) = limδ→0

inf

∑i

|Ui|s : K ⊂⋃i

Ui, 0 < |Ui| < δ

,

where | · | denotes the diameter of a set. This leads to the definition of the Hausdorff dimension of K:

dimH K = inf s : Hs(K) < ∞ = sup s : Hs(K) > 0 .

Many other definitions of dimensions are encountered in the literature. Let Mε(K) be the smallestnumber of closed balls of radius ε that cover K, Nε(K) be the maximum number of disjoint closed ballsof radius ε with centers in K. We define the lower and upper box–counting dimensions by

dimBK = lim infε→0

log Mε(K)− log ε

,

dimBK = lim supε→0

log Mε(K)− log ε

.

In [9], it was shown that

dimBK = lim infn→∞

logN2−n(K)n log 2

= lim infn→∞

log M2−n(K)n log 2

,

dimBK = lim supn→∞

logN2−n(K)n log 2

= lim supn→∞

log M2−n(K)n log 2

.

(2.1)

Finally, we can define packing dimension of K by

dimP K = inf

sup dimBAi, K ⊂

⋃i

Ai

.

It is well known (see [9]) that

dimH K ≤ dimBK ≤ dimBK , (2.2)

dimH K ≤ dimP K ≤ dimBK . (2.3)

Math. Nachr. 250 (2003) 73

2.2 Random self–conformal sets

We use Rd to denote d–dimensional Euclidean space. Let U ⊂ Rd be a connected open set. For0 < α ≤ 1, denote by Con1+α(U) the family of conformal diffeomorphisms S : U → S(U) for whichthere exists a constant CS such that

|S′(x) − S′(y)| ≤ CS |x − y|α for all x , y ∈ U ,

where S′(x) is the differential of S at x and |S′(x)| is the operator norm of the differential.Let J ⊂ U be a compact set with J = int(J), where int(A) is the interior of A. We are given a

positive integer N ≥ 2 and a probability measure µ on(Con1+α(U)

)N , where Con1+α(U) is equippedwith the usual topology of uniform convergence on compact sets. In this paper, we always assume thefollowing:

Assumptions(I) There exists a connected open set V such that V is compact, J ⊂ V ⊂ V ⊂ U satisfying

Si

(V

) ⊂ V and Si(J) ⊂ J for all i = 1, 2, . . . , N and µ–a. s. (S1, S2, . . . , SN ).(II) There exist 0 < rmin ≤ rmax < 1 such that rmin ≤ infx∈U |S′

i(x)| ≤ supx∈U |S′i(x)| ≤ rmax for

all i = 1, 2, . . . , N and µ–a. s. (S1, S2, . . . , SN).(III) There exists C0 > 0 such that CSi ≤ C0 for all i = 1, 2, . . . , N and µ–a. s. (S1 , S2, . . . , SN).

In the sequel we often make use of symbolic dynamics. Let Σ = 1, 2, . . . , NN be the code spaceover the indices 1, 2, . . . , N, Σn = 1, 2, . . . , Nn the space of all sequences of length n, n ≥ 0, Σ0 = ∅,and Σ∗ =

⋃n Σn. For τ ∈ Σn, denote by |τ | = n the length of τ , and by τ |k the truncation of τ to the

first k entries, k ≤ n. For any τ ∈ Σ∗, σ ∈ Σ∗⋃

Σ, define τ ∗ σ = (τ1, τ2, . . . , τ|τ|, σ1, σ2, . . .). We writeτ ≺ σ if there exists a η ∈ Σ∗

⋃Σ with σ = τ ∗ η.

Define the space Ω =((Con1+α(U))N

)Σ∗ . Let be the product σ–algebra on Ω. Taking P theproduct measure with µ on each component, we get our primary probability space (Ω,, P).

For ω ∈ Ω and τ ∈ Σ∗ write,

ω(τ ) = (Sτ∗1(ω), Sτ∗2(ω), . . . , Sτ∗N (ω)) ,

and S∅(ω) = id. By k we denote the σ–algebra generated by all Sτ with |τ | ≤ k. For any τ ∈ Σ∗,write

Sτ (ω) = Sτ|1(ω) Sτ|2(ω) . . . Sτ||τ|(ω) .

Further, let us introduce shift operators ∆τ : Σ∗⋃

Σ → Σ∗⋃

Σ by

∆τ (σ) = τ ∗ σ ,

and write

Tτω = ω(∆τ ) .

Then we have Sσ(Tτω) = Sτ∗σ(ω).Let us define a random mapping πω : Σ → J by

πω(σ) = limn→∞Sσ|n(ω)(x0) . (2.4)

This limit exists for P–a. s. ω ∈ Ω and does not depend on the choice of x0 ∈ J , (see Corollary 3.2 inSection 3). We call a random compact set K(ω) = πω(Σ) the random self–conformal set.

We call a random subset Γ ⊂ Σ∗ a Markov stopping, if(1) For each σ ∈ Σ and ω ∈ Ω, there exists a unique τ ∈ Γ(ω) with τ ≺ σ, and(2) ω ∈ Ω : τ ∈ Γ(ω) ∈ |τ| for all τ ∈ Σ∗.If Γ is a Markov stopping, let Γ be the sub–σ–algebra of generated by Sη: there is a τ ∈ Γ with

η ≺ τ .The set K(ω) fulfills the following invariance (see [2], [20]).


Lemma 2.1 For P–a. s. ω ∈ Ω, we have(1) K(ω) =

⋂∞n=0

⋃σ∈Σn

Sσ(J),

(2) K(ω) =⋃

τ∈Γ Sτ (ω)K(Tτ ω), where the K(Tτω) are i. i. d. copies of K(ω) and independentof Γ.

3 Regularity for random self–conformal sets

In this section, we prove the regularity for a random self–conformal set. We adopt the idea of [15].For brevity, for any τ ∈ Σ∗, write

cτ (ω) = infx∈V

|S′τ (ω)(x)|,

cτ (ω) = infx∈V

∣∣S′τ (ω)(x)

∣∣ ,

dτ(ω) = supx∈V

∣∣S′τ (ω)(x)

∣∣ ,

dτ(ω) = supx∈V

∣∣S′τ (ω)(x)| ,

Jτ = Sτ (J) .

The following result is due to Patzschke [20], Lemma 2.3.2 and Corollary 2.3.3.

Lemma 3.1 There exists a constant C ≥ 1 such that(1) dτ(ω) ≤ C cτ (ω) for all τ ∈ Σ∗ with probability one.(2) With probability one,

C−1dτ(ω) |x − y| ≤ ∣∣Sτ(ω)x − Sτ (ω)y∣∣ ≤ C dτ(ω) |x − y| ,

for all x, y ∈ V and τ ∈ Σ∗, and

C−1dτ(ω) |B| ≤ ∣∣Sτ (ω)B∣∣ ≤ C dτ (ω) |B| ,

if B ⊂ V . In particular,

C−1dτ(ω) |K(ω)| ≤ |Kτ (ω)| ≤ C dτ(ω) |K(ω)| ,

with probability one, where Kτ (ω) = Sτ(ω)K(ω).(3) If x ∈ V and r > 0 with B(x, r) ⊂ V ,

B(Sτ (ω)x, C−1dτ(ω)r

) ⊂ Sτ(B(x, r)) ⊂ B(Sτ (ω)x, C dτ(ω)r

).

From Lemma 3.1, it is easy to prove, (see [20])

Corollary 3.2 For any x0 ∈ J , let πω(σ) = limn→∞ Sσ|n(ω)(x0). This limit exists for P–a. s. ω ∈ Ωand does not depend on the choice of x0 ∈ J.

Now we begin to prove the main results of this paper.

Proposition 3.3 Let K(ω) be the random self–conformal set, then there exists constant a such thatP–a. s.

dimH K(ω) = a .

P r o o f. Suppose there exists b such that

0 < Pω : dimH K(ω) < b

< 1 .

Math. Nachr. 250 (2003) 75

By (2) of Lemma 2.1, K(ω) =⋃N

i=1 Si(ω)K(Tiω), thus

dimH K(ω) = max1≤i≤N

dimH Si(ω)K(Tiω) = max1≤i≤N

dimH K(Tiω) ,

we have

Pω : dimH K(ω) < b

= P

ω : max

1≤i≤NdimH K(Tiω) < b

=N∏

i=1

Pω : dimH K(Tiω) < b

= (P ω : dimH K(ω) < b)N

.

and this leads a contradiction.

For any σ ∈ Σ∗ and ε > 0, let

Iσε (ω) =

τ ∈ Σ∗ :

∣∣Sτ (Tσω)(J)∣∣ ≤ ε,

∣∣S(τ||τ|−1)(Tσω)(J)∣∣ > ε

.

Define

Dσε (ω) =

A ⊂ Iσ

ε (ω) : for all ζ, η ∈ A, Sζ(Tσω)(J)⋂

Sη(Tσω)(J) = ∅

,

N(σ)

ε (ω) = maxA∈Dσ

ε (ω)A .

Let Aσε (ω) be a member of Dσ

ε (ω) which realizes N(σ)

ε (ω) and write

Aσε (ω) =

τσ,1(ω), τσ,2(ω), . . . , τ

σ,N(σ)ε (ω)

(ω)

.

For simplicity, write Iε(ω) = I∅ε (ω), Dε(ω) = D∅ε(ω), Nε(ω) = N

(∅)ε (ω), Aε(ω) = A∅

ε(ω).Lemma 3.4 There exists a constant c > 0 such that P–a. s. for any ε > 0 small enough,

Nε(ω) ≥ Nε(K(ω)) , Nε(ω) ≤ cMε(K(ω)) .

P r o o f. Let B(x1, ε), B(x2, ε), . . . , B(xNε(K(ω)), ε

)be Nε(K(ω)) disjoint closed balls of radius ε and

centers in K(ω). For any 1 ≤ i ≤ Nε(K(ω)), by Lemma 2.1 (1), there exists i ∈ Σ such that

∞⋂n=1

S(i|n)(ω)(J) = xi .

Choose n0 such that

S(i|n0)(ω)(J) ⊂ B(xi, ε) , S(i|n0−1)(ω)(J)\B(xi, ε) = ∅ .

Then we have∣∣S(i|n0−1)(ω)(J)

∣∣ > ε.Now we consider two cases:(i) If

∣∣S(i|n0)(ω)(J)∣∣ ≤ ε, choose τ (i) = (i|n0);

(ii) If∣∣S(i|n0)(ω)(J)

∣∣ > ε, then there exists m ≥ 1 such that∣∣S(i|n0+m)(ω)(J)∣∣ ≤ ε ,

∣∣S(i|n0+m−1)(ω)(J)∣∣ > ε ,

choose τ (i) = (i|n0 + m). Therefore we have

Nε(ω) ≥ Nε(K(ω)) . (3.1)


Now we prove the second inequality.Let Sτ(i) (ω)(J), i = 1, 2, . . . , Nε(ω) be Nε(ω) disjoint sets satisfying for any 1 ≤ i ≤ Nε(ω),∣∣Sτ(i) (ω)(J)

∣∣ ≤ ε ,∣∣S(τ(i)||τ(i)|−1)(ω)(J)

∣∣ > ε .

Since int(J) = ∅, there exists x ∈ J and r > 0 such that B(x, r) ⊂ J . By Lemma 3.1, Sτ(i) (ω)(J)contains a closed ball with radius C−1 · r · dτ(i)(ω). By Assumption (II) and Lemma 3.1,

C−1 · r · dτ(i)(ω) ≥ C−1 · r · c(τ(i)||τ(i)|−1)(ω) · rmin

≥ C−2 · r · rmin · d(τ(i)||τ(i)|−1)(ω)

≥ C−3 · r · rmin

∣∣S(τ(i)||τ(i) |−1)(ω)(J)∣∣

|J |>

C−3 · r · rmin

|J | · ε .

Let B(x1 , ε), B(x2, ε), . . . , B(xMε(K(ω)), ε

)be Mε(K(ω)) closed balls of radius ε such that⋃Mε(K(ω))

i=1 B(xi, ε) ⊃ K(ω). For any 1 ≤ i ≤ Mε(K(ω)), define

Bi(ω) =

j : 1 ≤ j ≤ Nε(ω), Sτ(j) (ω)(J)⋂

B(xi, ε) = ∅

.

Then for any j ∈ Bi(ω), Sτ(j) (ω)(J) ⊂ B(xi, 2ε). By volume estimating, we have

λd(B(xi, 2ε)) ≥ Bi(ω) · λd

(B

(0,

C−3 · r · rmin

|J | · ε))

,

where λd is the d–dimensional Lebesgue measure. That is

Bi(ω) ≤ (2ε)d(C−3·r·rmin

|J|)d · εd

=2d ·C3d · |J |drd · (rmin)d

:= c .

Thus,

Mε(K(ω)) ≥ 1c

Nε(ω) . (3.2)

From Lemma 3.4 and (2.1), we get the following corollary immediately.

Corollary 3.5 For P–a. s. ω ∈ Ω,

dimBK(ω) = lim supn→∞

logN2−n(ω)n log 2

.

Now let’s prove the main result of this section.

Theorem 3.6 Let K(ω) be a random self–conformal set. Then there exists a constant a such thatP–a. s.,

dimH K(ω) = dimBK(ω) = a .

P r o o f. By Proposition 3.3, there exists a constant a such that dimH K(ω) = a, P–a. s.For any n ≥ 1, choose ε = 2−n, and define a random set Kn(ω) ⊂ K(ω) as following.

Math. Nachr. 250 (2003) 77

Let

Fn,0(ω) = J ,

Fn,1(ω) =⋃

σ∈Aε(ω)

Sσ(ω)(J) ,

Fn,2(ω) =⋃

σ∈Aε(ω)

⋃τ∈Aσ

ε (ω)

Sσ∗τ (ω)(J) ,

. . .

Fn,k(ω) =⋃

σ1∈Aε(ω)

⋃σ2∈A

σ1ε (ω)

· · ·⋃

σk∈Aσ1∗σ2∗···∗σk−1ε (ω)

Sσ1∗σ2∗···∗σk(ω)(J) ,

. . .

Let

Kn(ω) =∞⋂

k=0

Fn,k(ω) .

By the construction of K(ω), it is clear that Kn(ω) ⊂ K(ω), P–a. s. On the other hand, by Lemma3.1 (2), we have for any σ1 ∈ Aε(ω), σ2 ∈ Aσ1

ε (ω), . . . , σk ∈ Aσ1∗σ2∗···∗σk−1ε (ω),

C−1 cσ1∗σ2∗···∗σk(ω) |x − y| ≤ ∣∣ Sσ1∗σ2∗···∗σk (ω)x − Sσ1∗σ2∗···∗σk(ω)y∣∣

≤ Cdσ1∗σ2∗···∗σk(ω) |x − y| .Thus for each n ≥ 1, Kn(ω) is a random net fractal studied in [14], Section 3, Theorem 3.3, (see remark3.7 below). By Theorem 3.3 of [14], we have for P–a. s. ω,

dimH Kn(ω) ≥ b ,

where b satisfies the following equation

E

N2−n (ω)∑

i=1

(C−1cτ∅,i(ω)(ω)

)b

= 1 .

Therefore

E

N2−n (ω)∑

i=1

(C−1cτ∅,i(ω)(ω)

)a

≤ 1 .

By Lemma 3.1 (1) (2), we have for any τ ∈ Σ∗,

S(τ||τ|−1)(J) ≤ C2 c(τ||τ|−1) |J | , and c(τ||τ|−1)(ω) ≤ dτ(ω)r−1min ,

thus

S(τ||τ|−1)(J) ≤ C3cτ (ω)r−1min |J | ,

therefore, by the definition of τ∅,i(ω), we have

C−1cτ∅,i(ω)(ω)) ≥ rmin ·C−4

|J | · 2−n .

Thus

E N2−n(ω) ≤ C4a · |J |a(rmin)a

· 2na .


For any δ > 0,∞∑

n=1

P

ω : N2−n(ω) ≥ C4a · |J |a(rmin)a

· 2n(a+δ)

≤

∞∑n=1

E N2−n(ω) · (rmin)a

C4a · |J |a · 2n(a+δ)

≤∞∑

n=1

2−nδ

< ∞ .

By the Borel–Cantelli Lemma, we have

P

ω : N2−n(ω) ≥ C4a · |J |a(rmin)a

· 2n(a+δ) i. o.

= 0 . (3.3)

By Corollary 3.5 and (3.3), we have for P–a. s. ω,

dimBK(ω) ≤ lim supn→∞

log C4a·|J|a(rmin)a · 2n(a+δ)

n log 2= a + δ .

Since δ is arbitrary, we have for P–a. s. ω ∈ Ω

dimH K(ω) = dimBK(ω) = a ,

and we finish the proof of Theorem 3.6.Remark 3.7 For random self–conformal set, by Lemma 3.1 (2), (3), we have there exists r > 0

such that for every σ ∈ Σ∗ and for almost all ω, there exists a ball Bσ ⊂ Jσ such that |Bσ| ≥ r |Iσ|. Ifthe model discussed in Subsection 3.3 in [14] has this property, then Theorem 3.3 in [14] is definitelytrue by following the proof of Theorem 3.1 of that paper, (see also [17]). It should be mentionedthat the last sentence in the proof of Theorem 3.1 in [14] seems to be not correct. Without theproperty mentioned above, it seems we cannot use Kinney–Pitcher–Billingsley theorem directly. Withthe property mentioned above, we can follow the method in [17] p. 328 to get the desired result.

By (2.2), (2.3) and Theorem 3.6, we have the following corollary immediately.Corollary 3.8 Let K(ω) be a random self–conformal set. Then there exists a constant a, such that

for P–a. s. ω ∈ Ω,

dimH K(ω) = dimP K(ω) = dimBK(ω) = dimBK(ω) = a .

4 Dimensions for random self–conformal sets

In this section, besides the Assumptions (I), (II), (III), we further assume(IV) For µ–a. s. (S1, S2, . . . , SN), if 1 ≤ i = j ≤ N , then Si(int(J))

⋂Sj(int(J)) = ∅.

Assumption (IV) is known as the open set condition (OSC).From the construction of the random self–conformal set K(ω), we know

(cσ∗1(ω), cσ∗2(ω), . . . , cσ∗N (ω)), σ ∈ Σ∗are independent and identically distributed. Also

(dσ∗1(ω), dσ∗2(ω), . . . , dσ∗N(ω)), σ ∈ Σ∗are independent and identically distributed. By Lemma 3.1 (2), we have with probability one,

C−1cτ(ω) |x − y| ≤ ∣∣Sτ (ω)x − Sτ(ω)y∣∣ ≤ Cdτ(ω) |x − y| ,

for all x, y ∈ V and τ ∈ Σ∗. Using Theorem 3.3 of [14], (see Remark 3.7 above), we have

Math. Nachr. 250 (2003) 79

Lemma 4.1 Suppose the Assumptions (I), (II), (III) and (IV) be satisfied, K(ω) be the randomself–conformal set. Let s1, t1 be unique solutions of the expectation equations

EN∑

i=1

(C−1ci(ω)

)s1 = 1 and EN∑

i=1

(Cdi(ω))t1 = 1 .

Then for P–a. s. ω ∈ Ω,

s1 ≤ dimH K(ω) ≤ t1 .

For any k ≥ 2, let Σ(k)∗ = σ ∈ Σ∗, k divides |σ|. For any ω ∈ Ω and τ ∈ Σ(k)

∗ \ ∅, let

Sτ,k(ω) = Sτ||τ|−k+1(ω) Sτ||τ|−k+2(ω) . . . Sτ||τ|(ω) ,

cτ,k(ω) = infx∈V

∣∣S′τ,k(x)

∣∣ ,dτ,k(ω) = sup

x∈V

∣∣S′τ,k(x)

∣∣ .

It is easy to see

K(ω) =∞⋂

n=1

⋃σ∈Σnk

(Sσ|1 Sσ|2 . . . Sσ|k

) . . . (Sσ|nk−k+1 Sσ|nk−k+2 . . . Sσ|nk

)(J) .

From the construction of the random self–conformal set K(ω), we know

(cσ∗τ,k(ω))τ∈Σk

, σ ∈ Σ(k)∗

are independent and identically distributed. Also

(dσ∗τ,k(ω))τ∈Σk, σ ∈ Σ(k)

∗

are independent and identically distributed. Let sk, tk be unique solutions of the expectation equations

E∑

τ∈Σk

(C−1cτ,k(ω)

)sk = 1 and E∑

τ∈Σk

(Cdτ,k(ω))tk = 1 .

If we look the composition of k times iterations as 1 time iteration, by Lemma 3.1 (2) and Lemma 4.1,we have for P–a. s. ω ∈ Ω,

sk ≤ dimH K(ω) ≤ tk . (4.1)

Now we prove the main results in this section.

Theorem 4.2 Suppose the Assumptions (I), (II), (III) and (IV) be satisfied, K(ω) be the randomself–conformal set. For any k ≥ 1, let sk, tk be unique solutions of the expectation equations

E∑

τ∈Σk

cτ,k(ω)sk = 1 and E∑

τ∈Σk

dτ,k(ω)tk = 1 .


dimH K(ω) = limk→∞

sk = limk→∞

tk .


P r o o f. From (4.1), we have

lim supk→∞

sk ≤ dimH K(ω) ≤ lim infk→∞

tk .

Note that sk ≥ sk and tk ≤ tk for any k ≥ 1, it is enough to prove

limk→∞

(tk − sk) = 0 . (4.2)

Note that for any τ ∈ Σk, cτ,k(ω) = cτ(ω) and dτ,k(ω) = dτ (ω), we have

E∑

τ∈Σk

(C−1cτ(ω)

)sk = 1 and E∑

τ∈Σk

(Cdτ(ω)

)tk = 1 .

Since sk ≤ dimH K(ω) ≤ d and C ≥ 1, by Lemma 3.1, we have

1 ≤ E∑

τ∈Σk

(Cdτ(ω)

)sk ≤ E∑

τ∈Σk

C2sk · cτ (ω)sk ≤ C3d . (4.3)

Suppose

E∑

τ∈Σk

(Cdτ(ω))sk = D , where 1 ≤ D ≤ C3d .

If tk − sk > log C3d

−k log rmax−log C , write δk = log C3d

−k log rmax−log C , we have

E∑

τ∈Σk

(Cdτ(ω)

)sk> E

∑τ∈Σk

(Cdτ(ω)

)tk · dτ (ω)−δk ·C−δk

≥ E∑

τ∈Σk

(Cdτ(ω)

)tkC−δk · r−kδk

max

= C3d ,

which contradicts that D ≤ C3d. So limk→∞(tk − sk) = 0.

Theorem 4.3 Suppose the Assumptions (I), (II), (III) and (IV) be satisfied, K(ω) be the randomself–conformal set. Let αk be unique solution of the expectation equations

E∑

τ∈Σk

|Jτ |αk = 1 .



αk .

P r o o f. By Lemma 3.1,

C−2cτ(ω) |J | ≤ |Jτ | ≤ C2cτ (ω) |J | ,thus

C−2d min(1, |J |d) ≤ E

∑τ∈Σk

|Jτ |sk ≤ C2d max(1, |J |d) .

In the same way as the proof of Theorem 4.2, we have

limk→∞

(αk − sk) = 0 .

Thus


αk .

Combine Theorem 3.6, Theorem 4.2 and Theorem 4.3, we can get the following corollary immediately.

Math. Nachr. 250 (2003) 81

Corollary 4.4 Suppose the Assumptions (I), (II), (III) and (IV) be satisfied, K(ω) be the randomself–conformal set. For any k ≥ 1, let sk, tk and αk be defined as above. Then for P–a. s. ω ∈ Ω,

dimH K(ω) = dimP K(ω) = dimBK(ω) = dimBK(ω) = limk→∞

tk = limk→∞

sk = limk→∞

αk .

Acknowledgements The authors thank the referee for his (her) valuable suggestions. The research was

supported by the Special Funds for Major State Basic Research Projects and NSFC 10201027.

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Dimensions for random self–conformal sets