On random fractals with infinite branching: definition, measurability, dimensions

www.imstat.org/aihp 2013, Vol. 49, No. 4, 1080–1089

DOI:10.1214/12-AIHP502

© Association des Publications de l’Institut Henri Poincaré, 2013

On random fractals with infinite branching: Definition, measurability, dimensions ¹

Artemi Berlinkov

Komendantski pr. 7-1-25, St. Petersburg, 197227, Russian Federation. E-mail:artemiberlinkov@hotmail.com Received 17 April 2011; revised 13 June 2012; accepted 15 June 2012

Abstract. We investigate the definition and measurability questions of random fractals with infinite branching, and find, under certain conditions, a formula for the upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.

Résumé. Nous étudions les questions de la définition et de la mesurabilité des fractales aléatoires avec ramification infinie. Nous trouvons sous certaines conditions une formule pour les dimensions de Minkowski supérieure et inférieure. Pour un d’ensemble aléatoire auto-similaire nous obtenons la dimension.

MSC:Primary 28A80; secondary 28A78; 60D05; 37F40

Keywords:Packing dimension; Minkowski dimension; Random fractal

1. Introduction

In this paper we study the Minkowski and packing dimensions of random fractals with infinite branching.

The almost sure Hausdorff dimension of random fractals was independently found by Mauldin and Williams in [13], and Falconer in [5]. Packing dimension and measures in case of finite branching were investigated by Berlinkov and Mauldin in [4]. It was shown that if the number of offspring is uniformly bounded, the Hausdorff, packing, lower and upper Minkowski dimensions coincide a.s.

Barnsley et al. in [1] introduced the notion ofV-variable fractals and in [2] find their Hausdorff dimension. Fraser in [7] discusses the Minkowski dimension, packing and Hausdorff measures from topological (in the Baire sense) point of view rather than probabilistic. Random fractals find interesting and important applications in other areas, e.g.

harmonic analysis [3], stochastic processes and random fields [14].

However most authors focus on the situation when the fractals are finitely branching, or, in other words, the number of offspring is bounded. In this paper we investigate the case when the number of offspring may be infinite. If it is bounded but not uniformly, the results of this paper show that all of these dimensions still coincide. If the number of offspring is unbounded, these dimensions may differ from each other, as shown in Examples1,2from Section6. As we see in these examples, the Minkowski dimensions may be non-degenerate random variables, whereas in [4] for the case of finite branching they have been shown to coincide with the a.s. constant Hausdorff dimension.

In Section2we give a precise definition of a random recursive construction and show that another definition used in [1] for random fractals coincides with it. Since in the case of infinite branching the Minkowski dimensions no longer

1Research was partially supported by NSF Grant DMS-0100078 and DFG-Graduiertenkolleg “Approximation und algorithmische Verfahren” in Jena, Germany.

have to be constant, their measurability is proven in Section4. In Section5we derive the Minkowski dimensions of random recursive constructions under some additional conditions and a formula for the packing dimensions of random self-similar sets with infinite branching.

2. On the definition of random fractals

Let n∈N∪ {∞},Δ= {1, . . . , n}ifn <∞, and Δ=Nif n= ∞. Denote byΔ^∗=_∞

j=0Δ^j the set of all finite sequences of numbers inΔ, and byΔ^Nthe set of all their infinite sequences. The result of concatenation of two finite sequencesσ andτ fromΔ^∗ is denoted byσ ∗τ. For a finite sequence σ, its length will be denoted by|σ|. For a sequenceσof length at leastk,σ|kis a sequence consisting of the firstknumbers inσ. There is a natural partial order on then-ary treeΔ^∗: σ≺τ if and only if the sequenceτ starts withσ. A setS⊂Δ^∗is called an antichain, ifσ≺τ andτ≺σfor allσ, τ∈Δ^∗.

The following construction was proposed by Mauldin and Williams in [13]. We have to modify the original defini- tion to fully take into account the case of offspring degeneration (see condition(vi)) below.

Suppose thatJ is a compact subset ofR^dsuch thatJ=Cl(Int(J )), without loss of generality its diameter equals one. The construction is a probability space (Ω, Σ, P ) with a collection of random subsets ofR^d – {J_σ(ω)|ω∈ Ω, σ∈Δ^∗}, so that the following conditions hold.

(i) J_∅(ω)=J for almost allω∈Ω.

(ii) For all σ ∈Δ^∗ the maps ω→J_σ(ω) are measurable with respect to Σ and the topology generated by the Hausdorff metric on the space of compact subsets.

(iii) For allσ∈Δ^∗andω∈Ω, the setsJ_σ, if non-empty, are geometrically similar toJ².

(iv) For almost everyω∈Ω and allσ∈Δ^∗,i∈Δ,J_σ∗iis a proper subset ofJ_σ providedJ_σ =∅.

(v) The construction satisfies the randomopen set condition: ifσandτ are two distinct sequences of the same length, then Int(Jσ)∩Int(Jτ)=∅a.s. and, finally.

(vi) The random vectorsTσ=(T_σ∗1, T_σ∗2, . . .),σ∈Δ^∗, are conditionally i.i.d. given thatJ_σ(ω)=∅, whereT_σ∗i(ω) equals the ratio of the diameter ofJ_σ∗i(ω)to the diameter ofJ_σ(ω).

The object of study is the random set K(ω)=

∞ k=1

σ∈Δ^k

Jσ(ω).

In general in condition(iii)other classes of functions instead of similarities may be used, e.g. conformal or affine mappings.

The meaning of condition(vi)is the following. Given thatJ_σ is non-empty, we ask that the random vectors of reduction ratiosTσ =(T_σ∗1, T_σ∗2, . . .), have the same conditional distribution and be conditionally independent, i.e.

for any finite antichainS⊂Δ^∗and any collection of Borel setsB_s⊂ [0,1]^Δ,s∈S, P (Ts∈B_s∀s∈S|J_s=∅∀s∈S)=

s∈S

P (Ts∈B_s|J_s=∅)

andTσ has the same distribution asT_∅, providedJ_σ=∅, i.e. for anyσ∈Δ^∗and any Borel setB⊂R^Δ, P (Tσ ∈B|Jσ=∅)=P (T_∅∈B).

Following [13] the above is called a random recursive construction.

The second term commonly used is “random fractals” (see, e.g. [1]), where condition(vi)is replaced by existence of an i.i.d. sequence of random vectors of reduction ratios. We note that the following holds:

2The setsA, B⊂R^dare geometrically similar, if there existS:R^d→R^dandr >0 such that for allx, y∈R^ddist(S(x), S(y))=rdist(x, y)and S(A)=B, suchSis called a similarity map.

Proposition 1. Random fractals and random recursive constructions are the same class of sets.

Proof. That every random recursive construction is a random fractal is obvious because we can set the distributions ofTσ givenJ=∅the same asT_∅.

Suppose that we have a random fractal. Then the random vectorTσ is independent of vectorsTτ withτ≺σ and, in particular, of the eventJ_σ =∅, therefore the second equality for the random vectors being conditionally i.i.d. holds.

In the first equality the right-hand side equals

s∈S

P (Ts∈B_s)

becauseSis an antichain andTs do not depend on events{J_s=∅},s∈S, while the left hand side equals the same

expression for the same reason.

Another definition in [13] for random stochastically geometrically self-similar sets made no reference to indepen- dence in the construction but a similar kind of conditional independence condition is needed to find the dimension of the limit set. We call such setsrandom self-similar sets, and for them not only the reduction ratios but also the maps (see Section5) that map parent to its offspring are conditionally i.i.d.

3. Preliminaries

If the average number of offspring does not exceed one, thenK(ω)is almost surely an empty set or a point, and we exclude that case from further consideration. Mauldin and Williams in [13] have found the Hausdorff dimension of almost every non-empty setK(ω),

α=inf

βE _n

i=1

T_i^β

≤1

.

In casen <∞,αis the solution of equation E

_n

i=1

T_i^α

=1.

The definitions and properties of Hausdorff and packing measures and dimensions, as well as definitions of upper and lower Minkowski dimensions, can be found in the book of Mattila [9]. We denote the Hausdorff, packing, lower and upper Minkowski dimension by dimH, dimP, dim_Band dimBrespectively.

For anyK⊂R^d denote byN_r(K)the smallest number of closed balls with radiir, needed to coverK. Then the upper Minkowski dimension,

dimBK=lim

r→0−N_r(K)/logr and the lower Minkowski dimension,

dim_BK=lim

r→0

−N_r(K)/logr.

Denote byMthe closure of a setM. Obviously, ifMis bounded, then dimBM=dimBMand dim_BM=dim_BM(see, e.g., [6], Proposition 3.4). One can use the maximal number of disjoint balls of radiirwith centers inK(which will be denoted byP_r(K)) instead of the minimal number of balls needed to cover setKin the definition of Minkowski dimensions because of the following relation ([6], (3.9) and (3.10)):

N2r(K)≤Pr(K)≤Nr/2(K).

The packing dimension can be defined using upper Minkowski dimension:

dimPK=inf

sup dimBFi|K⊂

i

Fi

.

4. Measurability of Minkowski dimensions

The measurability questions of dimension functions in deterministic case have been studied by Mattila and Mauldin in [10]. We start by exploring these questions for random fractals. In case of finite branching there is an obvious topology with respect to which the functionsω→dimBK(ω)andω→dim_BK(ω)are measurable – the topology generated on the space of compact subsets ofJby the Hausdorff metric. However, it is unknown to the author, with repect to which topology these maps would be measurable in the case of infinite branching. Therefore we circumvent this problem as follows.

Denote byK(J )the space of compact subsets ofJ equipped with the Hausdorff metric d_H(L₁, L₂)=max

sup

x∈L₁

dist(x, L2),sup

y∈L₂

dist(L1, y)

.

Lemma 2. Suppose thatL_i∈K(J ),i∈N.Then

k→+∞lim k

i=1

L_i=

+∞

i=1

L_i in the Hausdorff metric.

Proof. Suppose that

n→+∞lim d_{H n}

i=1

L_i,

+∞

i=1

L_i

>0.

Since n

i=1L_i ⊂_+∞

i=1L_i, there exists an ε >0 such that for every n∈ N there exists p_n ∈ _+∞

i=1L_i with dist(pn,_n

i=1L_i)≥ε. Without loss of generality we can assume that p_n converges to somep∈_+∞

i=1L_i. Then dist(p,_+∞

i=1Li)≥ε/2 which is a contradiction.

Corollary 3. The mapω→

|τ|=n,Jτ∩K=∅J_τ(ω)is measurable.

Corollary 4. Ifτ_i,i∈N,is an enumeration of{τ∈Δⁿ|J_τ∩K=∅},then

klim→∞P_{r k}

i=1

J_τi

=P_r

|τ|=n J_τ∩K=∅

J_τ

.

Proof. The statement follows from the fact that the functionP_r:K(J )→R is lower semicontinuous (see, [10],

remark after Lemma 3.1).

Lemma 5. In the Hausdorff metric, limn→∞

|τ|=n,J_τ∩K=∅Jτ(ω)=K(ω)for a.e.ω∈Ω.

Proof. According to [13], (1.14), limn→∞sup_τ∈Δnl_τ=0 for a.e.ω∈Ω. Consider such anω. Suppose that

nlim→∞dH

|τ|=n Jτ∩K=∅

Jτ(ω), K(ω)

>0,

then there exists anε >0 such that for everyn∈Nthere existsp_n∈

|τ|=n,Jτ∩K=∅J_τ(ω)with dist(pn, K(w))≥ε.

Choosen0∈Nsuch that for allτ∈Δ^∗of length at leastn0the following holds:

l_τ(ω) < ε/4.

Without loss of generalityp_nconverges to somep∈J. Thus dist(p, K(w))≥ε. Next choosen₁∈N,n₁≥n₀such that for alln≥n₁

dist(pn, p) < ε/4.

Since a 3ε/4 neighborhood ofp_ncontains a point ofK(ω), we get a contradiction.

Corollary 6. limn→+∞N_r(

|τ|=n,Jτ∩K=∅J_τ(ω))=N_r(K(ω))for a.e.ω.The equality holds if either set is replaced with its closure.

Proof. This follows from the facts that the functionNr:K(J )→Ris upper semicontinuous (see, e.g., [10], proof of

Lemma 3.1) andN_r(A)=N_r(A).

From the statements above follows:

Theorem 7. The mapsω→dimBK(ω)andω→dim_BK(ω)are measurable.

Proof. Since the maps ω→K(ω), ω→Nr

K(ω)

and ω→Nr

K(ω)

are measurable, the measurability of the lower and upper Minkowski dimensions ofK(ω)follows from their defini-

tion.

5. Dimensions of random fractals

In this section we derive several expressions for Minkowski and packing dimensions of random self-similar fractals with infinite branching.

Lemma 8. Suppose that t >dimHK a.s., 0< p=E[

i∈ΔT_i^t]<1 and q∈N. If Γ is an arbitrary (random) antichain such that|τ| ≥q for allτ ∈Γ a.s.,thenE[

τ∈Γl_τ^t] ≤₁^p₋^q_p. Proof. Indeed,E[

τ∈Γl_τ^t] ≤_+∞

k=qE[

|τ|=kl_τ^t] ≤_+∞

k=qp^k=₁^p₋^q_p.

We will also need the following 2 conditions:

(vii) the construction is pointwise finite, i.e. each element ofJ belongs a.s. to at most finitely many setsJi,i∈Δ (see [11]) and

(viii) J possesses theneighborhood boundedness property(see [8]): there exists ann₀∈Nsuch that for everyε >

diam(J ), ifJ1, . . . , J_kare non-overlapping sets which are all similar toJ with diam(Ji)≥ε >dist(J, Ji);i= 1, . . . , k, thenk≤n0.

As we will see, knowledge of similarity maps is essential to find the Minkowski dimension. For τ ∈ Δ^∗, let K_τ(ω)=

η∈Δ^N,η||τ|=τ

_∞

i=1J_η|i(ω)⊂J_τ(ω)∩K(ω). Fix a point a ∈R^d with dist(a, J )≥1. Denote by S_σ^τ:R^d→R^d a random similarity map such thatS_σ^τ(Jτ)=Jτ∗σ. IfJτ =∅orJτ∗σ =∅, then we letS_σ^τ(R^d)=a.

For a finite word σ ∈Δ^∗, let l_σ =diam(Jσ). From [13] we know that limk→∞sup_|τ|=kl_τ =0 a.s. For x∈J_τ and n∈N, consider the random n-orbit of x within J_τ, Oτ(x, n)=

|σ|=n,Jτ∗σ∩K=∅S_σ^τ(x). For I ⊂Δ^∗, let Oτ(x, I )=

σ∈I,Jτ∗σ∩K=∅S_σ^τ(x). In caseτ =∅, Oτ(x, I ) is denoted by O(x, I ), Oτ(x, n)by O(x, n), and S_σ^τ bySσ.

Lemma 9. For allω∈Ω,n∈N,and any two collections of pointsX= {x_k}^∞_k=1,Y= {y_k}^∞_k=1⊂

|σ|=nJ_σ such that for allσ∈Δⁿcard(Y∩Jσ)=card(X∩Jσ)=1or0, dimBX=dimBY anddim_BX=dim_BY.

Proof. Without loss of generality we assume thatn=1 since for everyn >1 the collection of sets{Jτ}such that|τ| is divisible bynforms a random recursive construction. First we note that there exists anM >0 such that

∀r >0,∀z∈R^d card

i∈Δ|B(z, r)∩J_i(ω)=∅andl_i(ω)≥r/2

≤M.

Fix ω∈Ω,z∈R^d,r >0. ObviouslyB(z, r) can be covered by 12^d balls of radius r/6. LetB1 be one of them and place insideB₁a set similar toJ. By the neighborhood boundedness property withε=r/2, we obtain card{i∈ Δ|B₁∩J_i=∅andl_i≥r/2} ≤n₀. Therefore it suffices to takeM=12^dn₀.

Finally take 0< r≤2, letI_r(ω)=

li(ω)<r/2J_i(ω)andI_r(ω)=

li(ω)≥r/2J_i(ω). ThenN_r(Y∩I_r)≤N_r/2(X∩ I_r). Clearly, for any collection of pointsZ= {z_k}^∞_k=1, such that card(Z∩J_i)=0 or 1 for alli, we haveN_r(Z∩I_r)≤ card(I_r). On the other handN_r(Z∩I_r)≥card(I_r)/M. Hence,

Nr(Y )≤Nr/2(X∩Ir)+Nr

Y ∩I_r

≤Nr/2(X)+MNr

X∩I_r

≤(1+M)Nr/2(X).

The result follows.

Remark. From the proof of Lemma 9, we see that if for some x ∈J, D >0 and 0≤u≤d for all 0< r ≤2, N_r(O(x,1))≤Dr^−u,then for ally∈J,N_r(O(y,1))≤2^d(12^dn₀+1)Dr^−u.

Forτ ∈Δ^∗, letγ_τ =dimBOτ(x,1)for somex∈J_τ and letγ=sup_τ∈Δ∗γ_τ. By Lemma9,γ_τ does not depend on the choice ofx. Similarly we defineγ

τ =dim_BOτ(x,1)andγ=sup_τ∈Δ∗γ

τ. For the rest of the paper, suppose additionally that

(ix) there existsA >0 such that for allτ∈Δ^∗,x∈J_τ,t >0 and 0< r≤2 we haveN_r(Oτ(x,1))1_{γ_τ<t}≤Ar^−tl_τ^t. Lemma 10. For anyx∈J,

max

dimHK,sup

n dimBO(x, n)

=max{dimHK, γ} and

max

dimHK,sup

n

dim_BO(x, n)

=max{dimHK, γ} a.s.

Proof. Fix ω∈ Ω. Since for any τ ∈Δ^∗, Oτ(S_τ(x),1)⊂O(x,|τ| +1), we have γ_τ =dimBOτ(S_τ(x),1)≤ dimBO(x,|τ| +1)≤sup_ndimBO(x, n), andγ≤sup_ndimBO(x, n).

In the opposite direction we prove by induction onnthat ifP (max{dimHK, γ}< t ) >0 for somet >0, then there exists a random variableB_n>0 such thatE[B_n]<+∞andN_r(O(x, n))1_{γ <t}≤B_nr^−t a.s. for all 0< r≤1. When n=1, we letB₁=A. Suppose that for alln≤kand for all 0< r≤1, there existsB_n>0 withE[B_n]<+∞such that Nr(O(x, n))1_{γ <t}≤Bnr^−ta.s. To prove the statement forn=k+1, fixr >0 and setIr(ω)= {τ∈Δ^k|lτ(ω) < r/2}.

Then N_r

O(x, Ir×Δ)

≤N_r/2

O(x, Ir)

≤N_r/2

O(x, k) . For a fixedτ∈Δ^k,

N_r Oτ

S_τ(x),1

1τ /∈Ir1_{γ <t}≤Al^t_τr^−t. Therefore

N_r

O(x, k+1)

1_{γ <t}≤N_r/2

O(x, k) 1_{γ <t}

+

|τ|=k

Nr

Oτ

Sτ(x),1

1τ /∈I_r1_{γ <t}≤2^tBkr^−t+Ar^−t

|τ|=k

l_τ^t.

SetB_k+1=2^tB_k+A

|τ|=kl_τ^t. If we fixn, then by Markov’s inequality for everyε >0

∞ i=0

P

B_n2^it>2^i(t+ε)

≤

∞ i=0

E[B_n]2^−iε<∞

and therefore by Borel–Cantelli lemma for a.e.ω∈Ω B_n2^it>2^i(t+ε)only finitely many times, hence for a.e.ω∈Ω N₂₋i(O(x, n))1_{γ <t}>2^i(t+ε)only finitely many times. Therefore

ilim→∞

logN₂₋i(O(x, n)) ilog 2 < t+ε

for almost everyω such that max{dimHK(ω), γ (ω)}< t for everyε >0. Thus for almost every such ωwe have dimBO(x, n)≤t. The same argument holds for the lower Minkowski dimension.

From the proof of the last lemma and the fact that there cannot be more than 10^d offspring in the construction of diameter at least 1/5 follows

Corollary 11. Suppose thatq∈N,construction satisfies property(ix),for somet >0P (max{dimHK, γ}< t ) >0 and let

Γ_τ,q=

η∈Δ^|τ|+q: l_η< l_τ/5

∪

η∈Δ^∗: |η|>|τ| +q, l_η< l_τ/5, lη||η|−1≥l_τ/5 . Then there exists a random variableB_q withE[B_q]<+∞such that

N_r

O(x, Γτ,q)

1_{γ <t}≤B_ql_τ^tr^−t. Proof. Let

Γ_0,τ,q=

σ∈Δ^∗: |σ| ≥ |τ| +q, l_σ≥l_τ/5,∃τ ∈Γ_τ,q: τ|_|τ|−1=σ . Then

Nr

O(x, Γτ,q)

1_{γ <t}≤Nr

Oτ

Sτ(x), q

1_{γ <t}+

σ∈Γ0,τ,q

Nr

Oσ

Sσ(x),1 1_{γ <t}

≤Bql_τ^tr^−t+Al_τ^tr^−tcard

σ ∈Δ^∗|lσ≥1/5 ,

whereB_q and the estimate on the first term come from the proof of Lemma10, and the second term is bounded according to condition(ix).

Note that if 0< p=E[

i∈ΔT_i^t]<1, then E

|τ|=q

l_τ^t (1/5)^t

=5^tE

|τ|=q

l^t_τ

=5^tp^q≥E card

τ|τ∈Δ^q, l_τ≥1/5 .

Hence E

card

σ∈Δ^∗|lσ ≥1/5

=

+∞

k=1

E card

σ∈Δ^k|lσ≥1/5

≤ 5^t 1−p

and we can putB_q =Bq+Acard{σ∈Δ^∗|lσ≥1/5}.

Lemma 12. For everyt∈Rsuch thatP (max{dimHK, γ}< t ) >0, dimBK≤tfor a.e.ωsuch thatγ (ω) < t.

Proof. Suppose thatP (max{dimHK, γ}< t ) >0. Letp∈(0,1)be defined by equalityp=E[

i∈Δl_i^t]. We will prove by induction onnthat there existsB >0 such that for eachn, for everyτ ∈Δ^∗there exists a random variable B_τ,n, independent of theσ-algebra generated by the mapsω→l_τ|i(ω), 1≤i≤ |τ|, withE[B_τ,n] ≤Bsuch that

N_r(K_τ)1_{γ <t}≤B_τ,nr^−tl^t_τ for a.e.ωsuch that 1/n≤r/ l_τ(w)≤1.

Choose q∈N such thatp^q<1/2. Then putB=max{2^d,4^t+1E[B_q]}, whereB_q is the random variable from Corollary11. The induction base obviously holds forn=1,2.

Suppose the statement is true forn₀∈N, and 1/(n0+1)≤r <1/n0. We can assume thatK_τ=∅. Let Cτ,1(ω)=

σ∈Γτ,qlσ ≤ l_τ 2n0+2

, Cτ,2(ω)=

σ∈Γτ,qlσ> l_τ 2n0+2

,

where Γ_τ,q=

σ∈Δ^q+|τ|: l_σ < l_τ/5

∪

σ∈Δ^∗: |σ|> q+ |τ|, l_σ< l_τ/5, lσ||σ|−1≥l_τ/5 . Since

K_τ=

σ∈Cτ,1

K_σ

∪

σ∈Cτ,2

K_σ

,

we have

N_r(K_τ)≤N_1/(n0+1)

σ∈C_τ,1

K_σ

+

σ∈C_τ,2

N_r(K_σ).

We note that N_1/(n0+1)(

σ∈Cτ,1K_σ)≤N_1/(2n0+2)(O(x, Γτ,q)) because if B(y_j,_2n¹

0+2)is a collection of balls of radius _2n¹

0+2 covering O(x, Γτ,q), then the balls B(y_j,_n¹

0+1)cover

σ∈Cτ,1K_σ, since diam(Jσ) <_2n¹

0+2 for all σ∈Cτ,1. Therefore by Corollary11

Nr(Kτ)1_{γ <t}≤B_ql^t_τ2^t(n0+1)^t+

σ∈Γτ,q

Nr(Kσ)1_{l_σ∈C_τ,2}1_{γ <t} a.s.

The following chain of inequalities ensures applicability of the induction hypothesis to estimate the terms in the last sum:

r

l_σ >5r≥ 5 n₀+1> 1

n₀, therefore

N_r(K_σ)1_{lσ∈Cτ,2}1_{γ <t}≤B_σ,nr^−tl_σ(ω)^t a.s.

Sincer≤2/(n0+1), N_r(K_τ)1_{γ <t}≤r^−t

4^tl^t_τB_q +

σ∈Γτ,q

B_σ,n0l_σ^t

=r^−tl_τ^t

4^tB_q +

σ∈Γτ,q

B_σ,n0l_σ^t/ l_τ^t

a.s.

Note that E

4^tB_q +

σ∈Γτ,q

B_σ,nl^t_σ/ l_τ^t

≤4^tE B_q

+Bp^q< B/4+B/2< B.

Applying the same argument as in Lemma10we come to the desired conclusion.

Theorem 13. If there existsA >0such that for allx∈J_τ,t >0and0< r≤2we have N_r

Oτ(x,1)

1_{γ_τ<t}< Ar^−tl_τ^t,

thendimBK=max{dimHK, γ}a.s.providedK=∅.Similarly,if N_r

Oτ(x,1) 1_{γ

τ<t}< Ar^−tl_τ^t,

thendimBK=max{dimHK, γ}a.s.on{K=∅}.

Proof. Fixn∈Nand consider a collection of pointsX= {x_i}^∞_i=1⊂K such that for allσ ∈Δⁿ,J_σ ∩K=∅⇒ card(X∩J_σ)=1 and J_σ ∩K =∅⇒card(X∩J_σ)=0. By Lemma 9, dimBX=dimBO(x, n), and therefore dimBK≥max{dimHK,sup_n∈NdimBO(x, n)}. By Lemma12,P (dimBK >max{dimHK, γ})=0.

Corollary 14. If the number of offspring is finite almost surely,thendimHK=dimPK=dim_BK=dimBKa.s.

Theorem 15. Suppose that we have a random self-similar set and there existsA >0such that Nr

O(x,1)

< Ar^−γ

a.s.for all 0< r≤2.ThendimPK=dimBK=max{dimHK,ess sup dimBO(x,1)}anddim_BK=max{dimHK, ess sup dim_BO(x,1)}a.s.on{K=∅}.

Proof. Since for a random self-similar set γ_τ, τ ∈Δ^∗ are conditionally i.i.d., we obtain that if K(ω)=∅, then γ =ess sup dimBO(x,1)a.s. To see this, let z=ess sup dimBO(x,1), then ess supγτ ≤z for allτ ∈Δ^∗ andγ = sup_τγ_τ≤za.s. Ifz=0 orγ_∅=za.s., we are done. Otherwise consider 0< y < zsuch that

0< P

dimBO(x,1)≤y

=b <1.

For allτ∈Δ^∗,b=P (γ_τ ≤y|J_τ =∅). Now we prove that for everyε∈(0,1) P

{∀τ γτ≤y} ∩ {K=∅}

≤εP (K=∅).

Findm∈Nsuch thatb^m< εP (K=∅)/2. From [13] it is known that ifSkdenotes the number of non-empty offspring on levelk, then for almost everyω∈ {K=∅}, limk→∞S_k= ∞, and for almost everyω∈ {K=∅}, limk→∞S_k=0.

Therefore we can findΩ₀⊂ {K=∅},k₀∈N, and perhaps a biggermsuch that P

{K=∅} \Ω₀

< εP (K=∅)/2 and ∀ω∈Ω₀, S_k0(w)≥m.

Next we enumerate somehow all indices ofΔ^k0 and fix this enumeration, then denote allm-element subsets ofΔ^k0 byF_i,i∈N. Forω∈Ω₀denote the event, that the firstmnon-empty setsJ_σ(ω), σ∈Δ^k0, concide withF_i, byΩ_i. ThenΩi form a partition ofΩ0and

P

{γ≤y} ∩ {K=∅}

=P

{γ≤y} ∩Ω0

+P

{K=∅} \Ω0

≤

i

P

{γ≤y} ∩Ω_i

+εP (K=∅)/2=

i

P (γ≤y|Ω_i)P (Ω_i)+εP (K=∅)/2

≤

i

b^mP (Ωi)+εP (K=∅)/2≤εP (K=∅).

Examination of the proofs of Lemmas 10, 12 and Theorem 13 shows that for every τ ∈ Δ^∗, dimBK_τ = max{dimHK,ess sup O(x,1)} provided K_τ =∅. Now using Baire’s category theorem we see that for t <

max{dimHK,ess sup dimBO(x,1)},P^t(K)= ∞. The result follows.

What is the packing dimension of infinitely branching random fractals in general is unknown.

6. Examples

As we see for a random self-similar set the packing dimension is almost surely constant even with infinite branching.

In the following example we see that if we drop the condition that the similarity maps are conditionally independent, packing dimension is no longer a constant.

Example 1 (Random fractal for which the zero–one law does not hold). Let J = [0,1] and take p(ω), ω∈Ω with respect to the uniform distribution on [1,2]. We build a random recursive construction so that on level 1, the right endpoints of the offspring are the points 1/n^p, n∈N, and the length of the nth subinterval is V_n= (1/16ⁿ)inf1≤p≤2{1/n^p−1/(n+1)^p}.On all other levels, the offspring are formed from a scaled copy of [0,1] and its disjoint subintervals of lengthV_nwith right endpoints at1/n^p,n∈N.Obviously,_∞

n=1V_n^1/4<∞,and hence for eachω∈Ω,we havedimHK≤1/4.On the other hand we can use the results from[12]to determine that for each ω∈Ω, dimPK(ω)=dimBK(ω)= _p(ω)¹₊₁.So,the reduction ratios are constant,but random placement of subintervals gives non-trivial variation of the packing dimension.

Example 2 (Random recursive construction for whichdimBKis a non-degenerate random variable anddimHK <

dimPK <ess inf dimBKa.s.). Note that forp >0, dimB{1/n^p, n∈N} =1/(p+1).LetJ= [0,1]and takepwith respect to the uniform distribution on[1,2].We build a random recursive construction so that on level1,the right endpoints of offspring are the points1/n^p,n∈N.On all other levels,the offspring are formed from a scaled copy of [0,1]and its disjoint subintervals with right endpoints at1/n⁴,n∈N.Let(V₁, V₂, . . .)be a fixed vector of reduction ratios so thatV_n=(1/1024)ⁿinf1≤p≤4{1/ i^p−1/(i+1)^p}.Then_∞

n=1V_n^1/8<1,K(ω)=∅, dimHK≤1/8and dimBK=max{dimHK,1/(p+1)} =1/(p+1),wherepis chosen according to the uniform distribution on[1,2].

Hence, ess inf dimBK=1/3.By Theorem15, dimPK=1/5.

Acknowledgement

Sufficiency of neighborhood boundedness property, condition (viii), for the proof of Lemma 9, instead of the “cone condition” from reference [11] as in [12], Proposition 2.9, became known to the author during conversation with R. D. Mauldin.

References

[1] M. F. Barnsley, J. E. Hutchinson and Ö. Stenflo.V-variable fractals: Fractals with partial self similarity.Adv. Math.218(2008) 2051–2088.

MR2431670

[2] M. F. Barnsley, J. E. Hutchinson and Ö. Stenflo.V-variable fractals: dimension results.Forum Math.24(2012) 445–470.

[3] D. Beliaev and S. Smirnov. Random conformal snowflakes.Ann. of Math. (2)172(2010) 597–615.MR2680427

[4] A. Berlinkov and R. D. Mauldin. Packing measure and dimension of random fractals.J. Theoret. Probab.15(2002) 695–713.MR1922443 [5] K. J. Falconer. Random Fractals.Math. Proc. Cambrige Philos. Soc.100(1986) 559–582.MR0857731

[6] K. J. Falconer.Fractal Geometry: Mathematical Foundations and Applications, 2nd edition. Wiley, Chichester, UK, 2003.MR2118797 [7] J. M. Fraser. Dimensions and measure for typical random fractals. Preprint, 2011. Available at arXiv:1112.4541.

[8] S. Graf, R. D. Mauldin and S. C. Williams. The exact Hausdorff dimension in random recursive constructions.Mem. Amer. Math. Soc.381 (1988).MR0920961

[9] P. Mattila.Geometry of Sets and Measures in Euclidean Spaces.Cambridge Univ. Press, Cambridge, UK, 1995.MR1333890

[10] P. Mattila and R. D. Mauldin. Measure and dimension functions: measurability and densities.Math. Proc. Cambridge Philos. Soc.121(1997) 81–100.MR1418362

[11] R. D. Mauldin and M. Urbanski. Dimensions and measures in iterated function systems.Proc. London Math. Soc. (3)73(1996) 105–154.

MR1387085

[12] R. D. Mauldin and M. Urbanski. Conformal iterated function systems with applications to the geometry of continued fractions.Trans. Amer.

Math. Soc.351(1999) 4995–5025.MR1487636

[13] R. D. Mauldin and S. C. Williams. Random recursive constructions: asymtotic geometric and topological properties.Trans. Amer. Math. Soc.

295(1986) 325–346.MR0831202

[14] N.-R. Shieh and Y. Xiao. Hausdorff and packing dimensions of the images of random fields.Bernoulli16(2010) 926–952.MR2759163