# Dimensions of Fractals Generated by Bi-Lipschitz Maps

Authors: Qi-Rong Deng; Sze-Man Ngai
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

On the class of iterated function systems of bi-Lipschitz mappings that are contractions with respect to some metrics, we introduce a logarithmic distortion property, which is weaker than the well-known bounded distortion property. By assuming this property, we prove the equality of the Hausdorff and box dimensions of the attractor. We also obtain a formula for the dimension of the attractor in terms of certain modified topological pressure functions, without imposing any separation condition. As an application, we prove the equality of Hausdorff and box dimensions for certain iterated function systems consisting of affine maps and nonsmooth maps.


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## Body

## 1. Introduction

In the literature on the equality of the Hausdorff and box dimensions of the attractor of an iterated function system (IFS), it is usually assumed that the generating maps areC1 and the bounded distortion property holds (see [1–3]). For IFSs of conformal contractions, the weak separation condition is also assumed (see [3]). These three conditions are usually imposed in order to obtain a formula for the dimensions of the attractor in terms of topological pressures (see, e.g., [4, 5]). The main goal of this paper is to relax these three conditions.There are many definitions of dimension for fractal sets. As is well known, the Hausdorff and upper box dimensions may be regarded as the smallest and the greatest values of any reasonable definition of dimension. Fox example, the packing dimension introduced by Tricot Jr. [6] always lies between these two values. Motivated by this observation, McLaughlin [7] and Falconer [1] studied conditions under which the Hausdorff and box dimensions of a fractal set are equal. As an application of the so-called implicit method, Falconer [1, Examples 2 and 3] proved the equality of the Hausdorff and box dimensions for all self-similar sets and a class of graph-directed sets (called recurrent sets), without assuming any separation condition. By assuming the C1-smoothness of the maps of the IFS, thebounded distortion property (BDP), and theweak separation condition (WSC), Lau et al. [3] proved the equality of the two dimensions for self-conformal sets. Under these conditions, the authors [5] proved that the common dimension is given by the zero of some topological pressure functions. For an infinite iterated function system, by assuming the open set condition, BDP, and that the maps of the IFS are C1+ϵ smooth, Mauldin and Urbański [4] proved that the Hausdorff dimension of the limit set is given by the zero of some topological pressure function.The dimensions of self-affine sets have also been studied extensively, since the work of McMullen [8] and Falconer [9]. Our results in this paper allow us to deal with a special class of self-affine sets. A simple example in this class is the self-affine set generated by the affine maps
(1)S1(x)=A-1x,S2(x)=A-1(x+(1,1)t),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiA=[2012],
which arises in the study of connectedness of self-affine sets in [10] (see also [11] and the references therein). This IFS does not satisfy BDP. There are of course plenty of examples of IFSs that do not satisfy WSC or contain maps that are not C1. We will study such examples in Section 4. Our work is partly motivated by them.There are two main goals in this paper. First, we would like to prove the equality of the Hausdorff and box dimensions by assuming a weaker set of conditions. We weaken theC1-smoothness condition to the bi-Lipschitz condition and replace the bounded distortion property by a weaker logarithmic distortion property. Second, under these conditions, we would like to obtain a formula for the common dimension in terms of the zero of some topological pressure functions, without assuming any separation condition.As some of the mappings we consider are not necessarily contractive with respect to the Euclidean metric, but contractive with respect to some other metric, for convenience we first introduce the definition of an iterated function system of essential contractions.Definition 1.
LetX be a nonempty compact subset of Rd, equipped with the Euclidean metric, and let Si:X→X, i=1,…,N, be a finite family of mappings. If there exists a metric ϱ on X such that all the Si are contractions with respect to ϱ, then one says that {Si}i=1N areessential contractions with respect toϱ (or simplyessential contractions). In this case one calls {Si}i=1N aniterated function system (IFS) of essential contractions.Some IFSs of affine mappings are not necessarily contractions with respect to the Euclidean metric but are essential contractions (see [12]). Some of the IFSs we consider in this paper are defined by matrices that are powers of a single matrix (see Example 23). They are also essential contractions.In order to state our conditions and results, we first introduce some basic definitions and notations. LetX be a nonempty compact subset of Rd, equipped with the Euclidean metric, and let Si:X→X, i=1,…,N, be essential contractions with respect to some metric ϱ. It is well known that there exists a unique nonempty compact subset K⊆X, called theattractor, such that
(2)K=⋃i=1NSi(K)
(see [13, 14]). The set K is independent of the metric ϱ. For such an IFS, we define
(3)Σk:={1,…,N}kfork≥0,Σ*:=⋃k≥0Σk,Σ∞:={1,…,N}∞,
with Σ0:={∅}. For I=(i1,…,ik)∈Σk, we denote by |I|=k thelength of I and write SI:=Si1∘⋯∘Sik (S∅ is defined to be the identity). We also denote I=(i1,…,ik) simply by I=i1⋯ik and let I-:=i1⋯in-1 be the word obtained from I by deleting its last alphabet.Let|·| denote the Euclidean norm. Define
(4)rI≔inf⁡x≠y∈X|SI(x)-SI(y)||x-y|,RI≔sup⁡x≠y∈X|SI(x)-SI(y)||x-y|,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiI∈Σ*,r:=min⁡1≤i≤Nri,R:=max⁡1≤i≤NRi,(5)rψ:=rI,Rψ:=RI,ifψ=SIforsomeI∈Σ*.For anyI,J∈Σ*, by writing
(6)|SIJ(x)-SIJ(y)||x-y|=|SIJ(x)-SIJ(y)||SJ(x)-SJ(y)|·|SJ(x)-SJ(y)||x-y|,
we obtain the following sets of inequalities:
(7)RIJ≤RIRJ,rIJ≥rIrJ,(8)RIJ≥RIrJ,RIJ≥rIRJ,rIJ≤RIrJ,rIJ≤rIRJ.
These inequalities will be used repeatedly.Assumption 2.
Throughout this paper we assume thatr>0; equivalently, Si, i=1,…,N, are bi-Lipschitz.Remark 3.
It is possible thatR≥1. Since all Si, 1≤i≤N, are essential contractions, RI converges uniformly to 0 as |I| tends to infinity. As a consequence, we also have r<1.For anyE⊆Rd, we let dim⁡H(E), dim⁡P(E), dim⁡B(E), Hs(E), Ld(E), |E|, and E∘ denote, respectively, the Hausdorff dimension, packing dimension, box dimension, s-dimensional Hausdorff measure, d-dimensional Lebesgue measure, Euclidean diameter, and interior of E. Given an IFS {Si}i=1N on X, a nonempty set U⊆X (not necessarily open) is said to beinvariant if ⋃i=1NSi(U)⊆U. We say that U⊆X isopen if it is open in the relative Euclidean topology of X.Fix an invariant setU⊆X and let 0<b<1. Define
(9)Ib:={I=(i1,…,in)∈Σ*:RI≤b<RI-},Ib*(U)≔{Ld(SI(U))≤bdLd(U)<Ld(SI-(U))I=(i1,…,in)∈Σ*:Ld(SI(U))≤bdLd(U)<Ld(SI-(U))},Ab:={SI:I∈Ib},Ab*(U):={SI:I∈Ib*(U)}.We make a few remarks concerning these sets of indices or mappings. First, sinceR can be greater than 1, for (i1,i2,…)∈Σ∞, it is possible that there are more than one prefix I=(i1,…,in)∈Σ* such that I∈Ib. However, in view of Remark 3, the number of such prefixes must be finite. Second, it is possible that SI=SI′ for distinct I,I′∈Σ*; we identify such SI and SI′. Last, for IFSs of contractive similitudes, Ib=Ib*(U) and so Ab=Ab*(U). In general, however, they need not be the same.Definition 4.
LetX⊂Rd be a compact subset with X∘≠∅ and let Si:X→X, i=1,…,N, be bi-Lipschitz essential contractions. We say that {Si}i=1N has thelogarithmic distortion property (LDP) if there is a constant σ>0 such that
(10)lim⁡b→0+sup⁡I∈IbbrI|ln⁡b|σ=0.Remark 5.
In the above definition, we do not assume that the maps of the IFS are differentiable. Besides this, if{Si}i=1N satisfies BDP, then there is a constant c>0 such that b/rI≤c for all b∈(0,1) and I∈Ib. Thus LDP holds. Hence LDP is an extension of BDP. Examples of IFSs satisfying LDP but not BDP will be given in Section 4.Definition 6.
LetX,{Si}i=1N,U satisfy the hypotheses of Definition 4, U⊆X be a bounded invariant set that is open in the relative topology of X with Ld(U)>0, and Φ be a finite subset of {SI:I∈Σ*}. One calls a finite subcollection {φ1,…,φk}⊆Φ apacking family for Φ with respect to U if the following conditions are satisfied: (i)
φ1(U),…,φk(U) are pairwise disjoint;(ii)
for anyφ∈Φ, φ(U) intersects at least one φj(U).
Denote the class of all packing families ofAb with respect to U by PU(b), and denote the class of all packing families of Ab*(U) by PU*(b).Example 7.
LetU=(0,3), Si(x)=(1/2)(x+i), i=0,1,2,3, and Φ={Si}i=03. Then {S0,S3}, {S1}, and {S2} are three packing families of Φ.Definition 8.
LetX, {Si}i=1N, U satisfy the hypotheses of Definition 4 and fix λ∈(0,1). Define
(11)Q_λ(s):=lim⁡_n→∞1nln⁡(inf⁡Φ∈PU(λn)∑φ∈Φ‍Rφs),s∈R,Q¯λ(s):=lim⁡¯n→∞1nln⁡(sup⁡Φ∈PU(λn)∑φ∈Φ‍Rφs),s∈R.
We call Q_λ (resp., Q¯λ) thelower (resp., upper) topological pressure function (with scaleλ). If Q_λ=Q¯λ, we denote the common function by Qλ and call it atopological pressure function (with scaleλ). Note that λ is fixed and s is the variable of the functions Q_λ(s) and Q¯λ(s).Remark 9.
The aboveQ_λ(s) and Q¯λ(s) are similar to those in [5], but they are different, since packing families are used here.The functionsQ_λ,Q¯λ, and Qλ depend on λ. However, they have a common zero (independent of λ), as is shown in the following main theorem.Theorem 10.
LetX, {Si}i=1N, U satisfy the hypotheses of Definition 6. Fix any λ∈(0,1) and any sequence of packing families {SIn,1,…,SIn,kn}∈PU(λn), where n∈N. Then (a)
Q_λ(s)=Q¯λ(s)=lim⁡n→∞(1/n)ln⁡(∑j=1knRIn,js)=(s-dim⁡HK)ln⁡λ for all s∈R;(b)
dim⁡HK=dim⁡PK=dim⁡BK=lim⁡n→∞ln⁡kn/(-nln⁡λ).For some applications, it is easier to treatLd(SI(U)) than RI and rI. Similar to Definition 8, we define
(12)Q_λ*(s)≔lim⁡_n→∞1nln⁡(inf⁡Ψ∈PU*(λn)∑φ∈Ψ‍[Ld(φ(U))]s/d),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiis∈R,Qλ*¯(s)≔lim⁡¯n→∞1nln⁡(sup⁡Ψ∈PU*(λn)∑φ∈Ψ‍[Ld(φ(U))]s/d),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiis∈R.
We have the following theorem.Theorem 11.
LetX, {Si}i=1N, U satisfy the hypotheses of Definition 6. Fix any λ∈(0,1) and any sequence of packing families {SIn,1,…,SIn,kn}∈PU*(λn), where n∈N. Then
(13)Q_λ*(s)=Qλ*¯(s)=lim⁡n→∞1nln⁡⁡(∑j=1kn‍[Ld(SIn,j(U))]s/d)=Qλ(s)=(s-dim⁡HK)ln⁡λ.A key in the proof of Theorem11 is to use the volume estimates in (15).In the following example, Theorem11 is used in computing the dimension of the attractor. Although the dimension of the self-affine set can also be computed by the method by Bárány [11], the method we use appears to be simpler (see Section 4).Example 12.
LetK be the self-affine set defined by the IFS in (1) (see Figure 1). Then dim⁡HK=dim⁡PK=dim⁡BK=1.Figure 1
The self-affine set in Example12.Remark 13.
Theorem11 makes dimension computation easier. The computation would be very complicated if we use Theorem 10 or the definitions of the Hausdorff or box dimensions.The rest of this paper is organized as follows. In Section2 we establish some basic properties of the topological pressure functions. Section 3 is devoted to the proof of the main theorems. In Section 4 we illustrate our main results by some examples.


## 2. Properties of Topological Pressures

In this section we prove some basic properties of the topological pressure functions. Let{Si}i=1N be an IFS of bi-Lipschitz essential contractions on a compact subset X⊂Rd. The following inequalities will be used repeatedly, for any E⊆X, and any I∈Σ*:
(14)rI|E|≤|SI(E)|≤RI|E|,(15)rI(Ld(E))1/d≤(Ld(SI(E)))1/d≤RI(Ld(E))1/d.We first state some basic properties of the topological pressures, without assuming LDP. The proof of the following proposition is similar to that of [5, Proposition 2.3]; we will only give an outline.Proposition 14.
LetX, {Si}i=1N, and U satisfy the hypotheses of Theorem 10 and let λ∈(0,1). Then both Q_λ(s) and Q¯λ(s) are real-valued, strictly decreasing, and continuous functions on R that tend to -∞ and ∞ as s tends to ∞ and -∞, respectively. Moreover, Q¯λ(0)≥Q_λ(0)≥0 and Q¯λ(s) is convex on R.Proof.
SinceRI→0 uniformly as |I|→∞, there is an integer k0>0 such that RI≤r for all I∈Σ* such that |I|≥k0. Let C:=max⁡{RI:|I|<k0}, n∈N, and φ=Si1⋯ik∈Φ∈PU(λn). Write k-1=lk0+m with 0≤m<k0. Then we have Ri1⋯ik-1≤Crl≤C′rk/k0 for some constant C′. Hence (7) implies
(16)rk≤rφ≤Rφ≤λn<Ri1⋯ik-1≤C′rk/k0.
It follows that
(17)nlog⁡rλ≤k<nk0log⁡rλ-k0log⁡rC′,
and thus
(18)#Φ≤Nnk0log⁡rλ-k0log⁡rC′,Rφ≥rnk0log⁡rλ-k0log⁡rC′.
Using (16)–(18) and a similar derivation as that in [5, Proposition 2.3] gives
(19)s(k0log⁡rλ)ln⁡r≤Q_λ(s)≤Q¯λ(s)≤(k0log⁡rλ)ln⁡N+sln⁡λ,ifs≥0,sln⁡λ≤Q_λ(s)≤Q¯λ(s)≤(k0log⁡rλ)ln⁡N+s(k0log⁡rλ)ln⁡r,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiifs<0.
Hence Q_λ(s) and Q¯λ(s) are real-valued, Q¯λ(0)≥Q_λ(0)≥0. Moreover, since 0<λ<1, we have lim⁡s→∞Q_λ(s)=lim⁡s→∞Q¯λ(s)=-∞ and lim⁡s→-∞Q_λ(s)=lim⁡s→-∞Q¯λ(s)=∞.
Next, for anyδ>0, by using (16)–(18), we get
(20)Q_λ(s)+δ(k0log⁡rλ)ln⁡r≤Q_λ(s+δ)≤Q_λ(s)+δln⁡λ<Q_λ(s),Q_λ(s)≤Q_λ(s-δ)≤Q_λ(s)-δ(k0log⁡rλ)ln⁡r.
Exactly the same inequalities hold for Q¯λ. Therefore, Q_λ(s) and Q¯λ(s) are strictly decreasing and continuous on R. The convexity of Q¯λ follows from Hölder’s inequality.By using the inequalities in (15), we can prove the following proposition in the same way.Proposition 15.
Under the same assumptions of Proposition14, both Q_λ*(s) and Qλ*¯(s) are real-valued, strictly decreasing, and continuous functions on R that tend to -∞ and ∞ as s tends to ∞ and -∞, respectively. Moreover, Qλ*¯(0)≥Q_λ*(0)≥0 and Qλ*¯(s) is convex on R.We now state some simple consequences of LDP.Lemma 16.
Assume the same hypotheses onX, {Si}i=1N, and U as in Theorem 10. Let 0<b0<1, RI and let rI be defined as in (4), and let σ>0 be defined as in Definition 4. The following hold.(a)
There is a constantc1>0 such that
(21)bc1|ln⁡b|σ≤rI≤RI≤b,∀I∈Ib,b∈(0,b0).(b)
There exists a constantc2>0 such that
(22)bc2|ln⁡b|σ≤rJ≤RJ≤c2|ln⁡b|σb,∀J∈Ib*(U),b∈(0,b0).Proof.
(a) By Definition4, we have
(23)0<c1:=sup⁡b∈(0,b0)sup⁡I∈IbbrI|ln⁡b|σ<∞.
Hence b/(c1|ln⁡b|σ)≤rI, and the conclusion follows.
(b) As mentioned in Remark3, the IFS is not necessarily contractive in the Euclidean metric. Nevertheless, since RJ→0 as |J|→∞, there exists some k0>0 such that RI≤r when |I|≥k0. For any J∈Σ* with |J|=n≥k0, let n=lk0+t with 0≤t<k0; that is, J can be decomposed into l parts, with l-1 of them having length k0 and one of them having length k0+t. Hence rn≤rJ≤RJ≤rl. Taking logarithm, we have
(24)nln⁡r-1≥ln⁡rJ-1≥ln⁡RJ-1≥lln⁡r-1≥(nk0-1)ln⁡r-1,|J|>k0.
Hence the set {(ln⁡rJ-1)/(ln⁡RJ-1):J∈Σ*, |J|>k0} is bounded. Let c>0 be a constant such that
(25)ln⁡rJ-1ln⁡RJ-1≤c,∀J∈Σ*with|J|>k0.
LetJ∈Ib*(U). The definition of k0 shows that there is a decomposition J=J1J2 with |J2|≤k0 so that J1∈IRJ(U) (since it is possible that RIJ>RI for some I,J∈Σ*). Substituting b=RJ and I=J1 into (21) yields
(26)RJrJ1(ln⁡RJ-1)σ≤c1.
We need only prove (22) for small b>0, since for any given b1>0 the set {(b/rJ|ln⁡b|σ):J∈Ib*(U),b1≤b<b0} is finite. Without loss of generality, we can assume |J|>k0 for any J∈Ib*(U). Using (8) and the facts that J1∈IRJ(U) and |J2|≤k0, we have rk0rJ1≤rJ1rJ2≤rJ≤rJ1Rk0. As J∈Ib*(U), we have
(27)(Ld(SJ(U))Ld(U))1/d≤b<(Ld(SJ-1(U))Ld(U))1/d
and thus rJ≤b<RJ-. As RJ≥RJ-r, we get
(28)rJ≤b<r-1RJ.
From (25) and (28), we see that there exists some constant c~≥1 such that c~-1≤ln⁡RJ/ln⁡b≤c~. Combining the above estimates, we get
(29)brJ|ln⁡b|σ≤r-1RJc~σrk0rJ1|ln⁡RJ|σ≤c~σc1rk0+1=:c2<∞.
That is, b/(c2|ln⁡b|σ)≤rJ. Similarly, we can show that RJ≤c2|ln⁡b|σb. Thus, (22) holds and this completes the proof.For IFSs satisfying LDP, the definitions of the topological pressures are independent of the choice of the invariant open setU and the packing families. To see this we need the following lemma.Lemma 17.
Assume that{Si}i=1N and X satisfy the hypotheses of Theorem 10, b0∈(0,1) are fixed, and U, V are nonempty invariant open subsets of X with Ld(U)>0 and Ld(V)>0. Then there is a constant c2>0, depending only on U,V, and s, such that for any b∈(0,b0) and any two packing families {SI1,…,SIk}∈PU(b) and {SJ1,…,SJn}∈PV(b),
(30)(c2|ln⁡b|σd+σ|s|)-1∑j=1n‍RJjs≤∑i=1k‍RIis≤c2|ln⁡b|σd+σ|s|∑j=1n‍RJjs.Proof.
LetΦi:={SJl:1≤l≤n,SJl(U)∩SIi(U)≠∅}, i=1,…,k. By using the definition of rJj, the disjointness of SJ1(V),…,SJn(V), and the equality {SJ1,…,SJn}=⋃i=1kΦi, we get
(31)∑j=1n‍rJjdLd(V)≤∑j=1n‍Ld(SJj(V))≤∑i=1k‍Ld(⋃φ∈Φi‍φ(V)).
LetBγ(x0)⊂U be a ball with radius γ>0 and center x0. Then SIi(U) contains a ball BrIiγ(xi) with radius rIiγ>0 and center xi=SIi(x0). For each SJl∈Φi, SJl(U)∩SIi(U)≠∅, and both SJj(V) and SIi(V) have diameters bounded above by b|V|. Let τ=2(|U|+|V|). Then SJl(U)∪SJl(V)⊂Bτb(xi) for SJl∈Φi. Therefore, (31) implies
(32)∑j=1n‍rJjdLd(V)≤∑i=1k‍Ld(Bτb(xi))≤kbdLd(Bτ(0)).
By using the inequalityrJj≥b/(c1|ln⁡b|σ) from Lemma 16, we get
(33)n(bc1|ln⁡b|σ)dLd(V)≤kbdLd(Bτ(0)),
and so n≤ck|ln⁡b|σd with c=c1dLd(Bτ(0))(Ld(V))-1. Interchanging the roles of the two packing families and using the same argument, we get k≤cn|ln⁡b|σd. Hence
(34)c-1|ln⁡b|-σdk≤n≤ck|ln⁡b|σd.
Also, by Lemma 16, we have b≤c1|ln⁡b|σrIi and b≤c1|ln⁡b|σrJj. Hence for all s≥0,
(35)∑j=1n‍RJjs≤nbs≤ck|ln⁡b|σdbs≤cc1s|ln⁡b|σd+σs∑i=1k‍rIis≤cc1s|ln⁡b|σd+σs∑i=1k‍RIis.
By symmetry, ∑i=1k‍RIis≤cc1s|ln⁡b|σd+σs∑j=1n‍RJjs. Therefore,
(36)(cc1s|ln⁡b|σd+σs)-1∑j=1n‍RJjs≤∑i=1k‍RIis≤cc1s|ln⁡b|σd+σs∑j=1n‍RJjs,∀s≥0.
The conclusion for the case s≥0 follows by letting c2=cc1s. The proof for the case s<0 is similar; we omit the details.The following proposition follows easily from Lemma17 and its proof.Proposition 18.
LetX and {Si}i=1N satisfy the hypotheses of Theorem 10. Then for any nonempty invariant open set U⊂X with Ld(U)>0, and any sequence of packing families {SIn,j}j=1kn of Aλn, one has
(37)Q_λ(s)=lim⁡_n→∞1nln⁡∑j=1kn‍RIn,js=lim⁡_n→∞1nln⁡∑j=1kn‍rIn,js,s∈R,(38)Q¯λ(s)=lim⁡¯n→∞1nln⁡∑j=1kn‍RIn,js=lim⁡¯n→∞1nln⁡∑j=1kn‍rIn,js,s∈R.
Thus, the definitions of Q_λ,Q¯λ, and Qλ are independent of the choices of the invariant open set U⊂X and the packing families. Furthermore, in Definition 8, Rφ can be replaced by rφ.In the following, the open setU will not be mentioned unless it is necessary.In order to obtain a lower estimate for the Hausdorff dimension in Theorem10, we need themass distribution principle (Lemma 19) and Proposition 20 below.Lemma 19 (see, e.g., [13, Theorem 4.9]).
LetK⊆Rn, μ a positive Borel measure on K with 0<μ(K)<∞, and s≥0. If there is a constant c>0 such that lim⁡¯r→0+(μ(Br(x))/rs)≤c for any x∈K, then dim⁡HK≥s.Recall that an IFS{Si}i=1N on X satisfies theopen set condition (OSC) if there exists a nonempty bounded invariant open (in the relative topology of X) set O⊂X, called anOSC set, such that ⋃i=1NSi(O)⊆O and Si(O)∩Sj(O)=∅ for all i≠j.The following result is similar to that of [15, Theorem 10.3] where the strong separation condition is used; we include a proof for convenience.Proposition 20.
LetK be the attractor of an IFS {Si}i=1N satisfying the hypotheses of Theorem 10. If OSC holds with an OSC set U satisfying U⊇K and ∑j=1N‍rjs>1, then dim⁡HK≥s.Proof.
Chooset>s such that ∑j=1N‍rjt=1. Let pj=rjt and let μ be the invariant probability measure associated with the weights {pi}i=1N (see [14]); that is, μ=∑j=1Npjμ∘Sj-1.
For anyx∈K and sufficiently small b>0, let
(39){I1,…,Ik}={I∈Ib:SI(U)∩Bb(x)≠∅}.
Then OSC implies that SI1(U),…,SIk(U) are disjoint and the fact that Ij∈Ib implies that |SIj(U)|≤b|U|. The definition of rI implies that Ld(SIj(U))≥rIjdLd(U) (see (15) also), and Lemma 16 implies b/(c1|ln⁡b|σ)≤rIj≤RIj≤b. Hence
(40)bdLd(B1+|U|(x))=Ld(Bb+b|U|(x))≥Ld(⋃j=1k‍SIj(U))=∑j=1k‍Ld(SIj(U))≥k(bc1|ln⁡b|σ)dLd(U).
Thus there is a constant c4>0 such that
(41)k≤c4|ln⁡b|σd.
Combining OSC and the fact that U⊇K gives
(42)μ(Si1⋯in(U))=pi1⋯pin=ri1t⋯rint≤ri1⋯int.
Using U⊇K, (41), together with the fact that rIi≤b, we get
(43)μ(Bb(x))≤∑j=1k‍rIjt≤c4bt|ln⁡b|σd.
Since t>s, lim⁡b→0+bt-s|ln⁡b|σd=0. Hence inequality (43) implies lim⁡b→0+μ(Bb(x))/bs=0 for all x∈K. The conclusion follows by using Lemma 19.


## 3. Proof of the Main Theorems

This section is devoted to the proofs of the main theorems.Proof of Theorem10.
In order to apply Proposition20, we first use Proposition 18 to require, in addition, that U⊇K. Let α and β be the zeroes of Q_λ(s) and Q¯λ(s), respectively. By Proposition 18, α and β are independent of the choice of the packing family. Proposition 14 implies that both Q_λ(0) and Q¯λ(0) are real numbers.
We first prove(44)Q_λ(s)=Q_λ(0)+sln⁡λ=(s-α)ln⁡λ,Q¯λ(s)=Q¯λ(0)+sln⁡λ=(s-β)ln⁡λ.
Substitutingb=λn and I=In,j into (21) gives
(45)λnc1(n|ln⁡λ|)σ≤rIn,j≤RIn,j≤λn,n=1,2,…,i=1,…,k.
Hence
(46)λnskn≥∑i=1knRIn,js≥(1c1(n|ln⁡λ|)σ)s‍knλns,ifs≥0,λnskn≤∑i=1knRIn,js≤(1c1(n|ln⁡λ|)σ)s‍knλns,ifs<0.
Since
(47)lim⁡n→∞1nln⁡(1c1(n|ln⁡λ|)σ)s=0,
by using Proposition 18 and the fact that Q_λ(0) and Q¯λ(0) are real numbers, we have
(48)Q_λ(s)=lim⁡_n→∞1nln⁡⁡(∑i=1kn‍RIn,js)=lim⁡_n→∞1nln⁡(λnskn)=Q_λ(0)+sln⁡λQ¯λ(s)=lim⁡¯n→∞1nln⁡⁡(∑i=1kn‍RIn,js)=lim⁡¯n→∞1nln⁡(λnskn)=Q¯λ(0)+sln⁡λ.
Equation (44) now follows from the equalities Q_λ(0)=-αln⁡λ and Q¯λ(0)=-βln⁡λ.
Next, we prove(49)β≤dim⁡HK.
Suppose, on the contrary, β>dim⁡HK. Then there exists s such that dim⁡HK<s<β. We will derive a contradiction.
By (44), Q¯λ(s)=(s-β)ln⁡λ>0. Choose a sequence of packing families {SIn,j}j=1kn of Aλn with respect to U, where n>0. Then by using (38), there exists an integer n>0 such that
(50)1nln⁡(∑j=1kn‍rIn,js)≥Q¯λ(s)2>0.
Denote the new IFS{SIn,j}j=1kn by {fj}j=1kn and let Kn be its attractor. Then this IFS satisfies OSC with U being an OSC set. Since U⊇K, by applying Proposition 20 to the new IFS {fj}j=1kn and noticing that Kn⊆K, we get dim⁡HK≥dim⁡H(Kn)≥s, a contradiction. Thus dim⁡HK≥β.
Now, we prove(51)Q_λ(s)=Q¯λ(s)=(s-dim⁡HK)ln⁡λ.
To this end we first prove α≥dim⁡HK. Let s>α. Then Q_λ(s)<0 and s>0 by the fact that Q¯λ(0)≥0 (Proposition 14). For every integer n>0, choose a sequence of packing families {SIn,j}j=1kn of Aλn with respect to U.
For anyφ∈Aλn, there is at least one j such that φ(U)∩SIn,j(U)≠∅. Choose xn,j∈SIn,j(U). Since (14) implies |φ(U)|, |SIn,j(U)|≤λn|U|, φ(U) is contained in the ball B2λn|U|(xn,j) with radius 2λn|U|. Since K⊂U, we have
(52)K=⋃φ∈Aλn‍φ(K)⊂⋃φ∈Aλn‍φ(U)⊂⋃j=1kn‍B2λn|U|(xn,j).
Hence {B2λn|U|(xn,j)}j=1kn is a δn:=4λn|U|-cover of K. By (48),
(53)lim⁡_n→∞1nln⁡[kn(4λn|U|)s]=Q_λ(0)+sln⁡λ=Q_λ(s)<0.
Hence, kn(4λn|U|)s<1 for infinitely many integers n. Therefore,
(54)Hs(K)=lim⁡n→∞Hδns(K)≤lim⁡_n→∞kn(4λn|U|)s<∞,
and thus s≥dim⁡HK. Since s>α is arbitrary, we conclude that α≥dim⁡HK.
Sinceα≤β, by combining this with (49), we get α=β=dim⁡HK. Equation (51) now follows by substituting this into (44).
Finally, we provedim⁡HK=dim⁡BK. Let U be as above. For any n>0 and λ∈(0,1), choose a packing family {SIn,j}j=1kn of Aλn with respect to U. Let
(55)Bn≔{∏i=1d‍[(mi-1)λn,miλn):mi∈Z,K∩∏i=1d‍[(mi-1)λn,miλn)≠∅}
and let Nn:=#Bn, the cardinality of Bn. According to (52), we define
(56)Bn,j={∏i=1d‍[(mi-1)λn,miλn)∈Bn:mi∈Z,B2λn|U|(xn,j)∩∏i=1d‍[(mi-1)λn,miλn)≠∅}.
Then (52) implies Bn=⋃j=1knBn,j. If B2λn|U|(xn,j)∩∏i=1d‍[(mi-1)λn,miλn)≠∅, then ∏i=1d‍[(mi-1)λn,miλn) is contained in B2λn|U|+dλn(xn,j). Since {∏i=1d‍[(mi-1)λn,miλn)} are disjoint, we have
(57)#Bn,j≤Ld(B2λn|U|+dλn(xn,j))Ld(∏i=1d‍[(mi-1)λn,miλn))=Ld(B2|U|+d(0)).
Therefore,
(58)Nn≤∑j=1kn‍#Bn,j≤knLd(B2|U|+d(0)).
Hence
(59)dim⁡¯BK=lim⁡¯n→∞ln⁡Nn-ln⁡λn≤lim⁡¯n→∞ln⁡kn-nln⁡λ=Q¯λ(0)-ln⁡λ=β=dim⁡HK.
It follows immediately that dim⁡¯BK=dim⁡_BK=β=dim⁡HK. Since dim⁡HK≤dim⁡PK≤dim⁡¯BK, the proof is complete.A similar argument shows the following corollary.Corollary 21.
Assume the same hypotheses of Lemma17. Also, for any given λ∈(0,1), L∈N and {SIn,j}j=1mn⊂Aλn for n∈N, assume that the following conditions hold.(1)
For anyφ∈Aλn, there is at least one j∈{1,…,mn} such that φ(U) intersects SIn,j(U).(2)
For anySIn,l, there are at most L maps SIn,j such that SIn,l(U)∩SIn,j(U)≠∅.
Then Theorem10 holds by replacing the packing families with {SIn,j}j=1mn, n∈N.Remark 22.
For IFSs consisting ofC1 conformal contractions and satisfying BDP and WSC (see [3]), Theorem 1.1 of [5] gives a method for computing dim⁡HK by solving the equation P(s)=0. We remark that, in computing the function P(s), the sum in the definition of P(s) is over distinct maps, and thus in numerical computations the following two types of mistakes may occur:(a)
SI≠SJ, but numerical approximations show SI=SJ;(b)
SI=SJ, but numerical approximations show SI≠SJ.
In view of Corollary21 and the definition of packing families, the formula dim⁡HK=lim⁡n→∞(ln⁡kn/nln⁡λ) is numerically much more stable.Proof of Theorem11.
In view of (15), we have
(60)∑j=1kn‍rIn,js≤∑j=1kn‍[Ld(SIn,j(U))]s/d[Ld(U)]s/d≤∑j=1kn‍RIn,js≤sup⁡Ψ∈PU*(λn)∑ψ∈Ψ‍Rψs,s≥0,∑j=1kn‍rIn,js≥∑j=1kn‍[Ld(SIn,j(U))]s/d[Ld(U)]s/d≥∑j=1kn‍RIn,js≥inf⁡Ψ∈PU*(λn)∑ψ∈Ψ‍Rψs,s<0.
Thus, by using (37) and (38) we need only prove
(61)Qλ(s)=lim⁡n→∞1nln⁡⁡(sup⁡Ψ∈PU*(λn)∑ψ∈Ψ‍Rψs)=lim⁡n→∞1nln⁡(inf⁡Ψ∈PU*(λn)∑ψ∈Ψ‍Rψs),s∈R.
For anyb∈(0,λ] and any two packing families {SI1,…,SIk}∈PU(b) and {SJ1,…,SJm}∈PU*(b). Similar to the proof of Lemma 17, let Ψi={SJj:SJj(U)∩SIi(U)≠∅} and Φj={SIi:SJj(U)∩SIi(U)≠∅}.
Using LDP, we have(62)(RJjc1c22|ln⁡b|3σ)dLd(U)≤(rJjc1|ln⁡b|σ)dLd(U)≤(1c1|ln⁡b|σ)dLd(SJj(U))≤(bc1|ln⁡b|σ)dLd(U)≤rIidLd(U)≤Ld(SIi(U))≤RIidLd(U),
where the first and fourth inequalities follow from (22) and (21), respectively, the second, fifth, and last ones follow from (15), and the third one follows from the definition of Ib*(U). We assume, without loss of generality, that c2≥c1. It follows that
(63)RJj(c2|ln⁡b|)3σ≤RIi∀1≤i≤k,1≤j≤m.
By using (22) we see that ∪{SJj(U):SJj∈Ψi} is contained in a ball with center in SIi(U) and radius (c1+c2)|ln⁡b|σb|U|. Hence it follows from (22) again that
(64)#ΨiLd(U)(bc2|ln⁡b|σ)d≤∑SJj∈Ψi‍Ld(SJj(U))≤((c1+c2)|ln⁡b|σb|U|)dLd(B1(0)).
Therefore, there is a constant c3>0 such that
(65)m≤∑i=1k‍#Ψi≤c3|ln⁡b|2dσk.
By interchanging the roles of the two packing families, it can be proved in the same way that there exist constantsc4>0 and c5>0 such that
(66)RIi(c4|ln⁡b|)3σ≤RJj,for1≤i≤k,1≤j≤m,k≤c5|ln⁡b|2dσm.
Now, for any two sequencesΦn={SI1,…,SIk}∈PU(λn) and Ψn={SJ1,…,SJm}∈PU*(λn), by combining (63)–(66), we have
(67)∑φ∈ΦnRφs‍≤kmax⁡{RIis:1≤i≤k}≤(c5|ln⁡λn|2dσ)m(c4|ln⁡λn|)3σsmin⁡⁡{RJjs:1≤j≤m}≤(c5|ln⁡λn|2dσ)(c4|ln⁡λn|)3σs∑ψ∈ΨnRψs‍≤(c5|ln⁡λn|2dσ)(c4|ln⁡λn|)3σs×(c3|ln⁡λn|2dσ)(c2|ln⁡λn|)3σs∑φ∈ΦnRϕs‍,s≥0.
Similarly,
(68)∑φ∈ΦnRφs≤kmax⁡{RIis:1≤i≤k}≤(c5|ln⁡λn|2dσ)(c2|ln⁡λn|)3σsmmin⁡⁡{RJjs:1≤j≤m}≤(c5|ln⁡λn|2dσ)(c2|ln⁡λn|)3σs∑ψ∈ΨnRψs‍≤(c5|ln⁡λn|2dσ)(c2|ln⁡λn|)3σs(c3|ln⁡λn|2dσ)(c4|ln⁡λn|)3σs∑φ∈ΦnRφs,‍s<0.
It now follows from these inequalities and Theorem 10 that (61) holds. The proof is complete.


## 4. Examples

In this section we illustrate the applications of our results by some examples.Example 23.
LetA be a d×d real matrix, A={Ak:k=0,1,2,…}, and let Sj(x)=ρjAj(x+dj), j=1,…,N, be an IFS with Aj∈A, and 0<|ρj|<1. Assume that all eigenvalues of A have moduli 1. Then (a)
LDP is satisfied and thus the conclusions of Theorems10 and 11 hold;(b)
BDP holds if and only if there is a real invertible matrixB and a real orthogonal matrix Q such that A=BQB-1. In this case, the attractor is similar to a self-similar set generated by the IFS with A replaced by Q;(c)
if OSC holds, thendim⁡HK is the unique solution of the equation
(69)∑j=1N‍|ρj|s=1.Proof.
Letting0<γ<min⁡{ρj-1:1≤j≤N} and using the following norm |·|′ in [12]:
(70)|x|′:=∑k=0∞‍γk|A-kx|,
we see that {Si}i=1N is an IFS of essential contractions.
For the matrixA, by the Jordan decomposition theorem, there is an invertible complex matrix P such that
(71)P-1AP=diag⁡(J1,…,Jl),
where each Jj is a Jordan block with all diagonal entries being the same and equal to 1 in modulus.
(a) We need only show that the IFS satisfies LDP. Let(72)Aj=Akj,k¯=max⁡{kj},ρ¯=max⁡{ρj}.
Since all eigenvalues of A are 1 in modulus, using (71), it is not difficult (see, e.g., [5]) to prove that
(73)c-1k-d≤∥Ak∥≤ckd,c-1k-d≤∥A-k∥≤ckd,iiiiiiiiiiiiiiiiiik=1,2,…
for some constant c>0.
For anyI=i1⋯in∈Ib, let Si1⋯in(x)=ρi1⋯inAk(x+α) for some α∈Rd with k=ki1+⋯+kin. Then the RI and rI defined in (4) become
(74)ri1⋯in=|ρi1⋯in|∥A-k∥-1,Ri1⋯in=|ρi1⋯in|∥Ak∥.
By using these, (73), and the inequality k≤nk-, we get
(75)Ri1⋯inri1⋯in≤c2k2d≤c2n2dk¯2d.
By the definition of Ab, we have |ρI-|·∥Ak-kn∥>b, and hence (73) implies
(76)c(nk¯)d(ρ¯)n-1≥ckd|ρI-|≥c(k-kin)d|ρI-|≥|ρI-||Ak-k0|>b
and thus
(77)n<ln⁡(bρ¯/ck¯d)-dln⁡nln⁡ρ¯.
Therefore, (75) and (77) imply
(78)Ri1⋯inri1⋯in|ln⁡b|3d≤c2k2d|ln⁡b|3d≤c2n3dk¯2d|ln⁡b|3dnd≤(ln⁡⁡((bρ¯)/ck¯d)-dln⁡n)3d|ln⁡b|3dnd·c2k¯2dln⁡⁡(ρ¯)3d.
Since n→∞ as b→0+,
(79)lim⁡b→0+|ln⁡b|i(ln⁡n)j|ln⁡b|3dnd=0,∀0≤i≤3d,0≤j≤3d.
Hence (78) implies
(80)lim⁡b→0+sup⁡I∈IbRIrI|ln⁡b|3d=0;
that is, LDP holds. Part (a) follows.
(b) The sufficiency is obvious. For the necessity, assume BDP holds, and letAj=Akj. Let 1n:=(1,…,1)∈Σn. Then S1n(x)=ρ1nAnk1(x+αn) for some αn. Since all eigenvalues of A have moduli 1, it follows that Ld(S1n(V))=ρ1nLd(V). Hence, by (15), r1n≤ρ1n≤R1n=ρ1n∥Ank1∥. Thus BDP implies sup⁡n>0∥Ank1∥<∞. For any l, choosing n so that nk1≤l<k1(n+1), we have ∥Al∥≤∥Ank1∥·∥Al-nk1∥. Hence
(81)sup⁡n>0∥An∥<∞.
Using (71), we have P-1AnP=diag⁡(J1n,…,Jln). Hence (81) implies
(82)sup⁡n>0∥Jjn∥<∞,j=1,…,l.
As the operator norm of a matrix is larger than or equal to the maximum of the absolute values of its entries, (82) implies that the entries of Jjn are uniformly bounded for all n>0 and j=1,…,l. Hence, similar to the proof of [16, Lemma 2.3], each Jj is of order 1×1, that is, a number with modulus 1. Thus l=d and each column of P is an eigenvector of A.
Since the eigenvalues ofA must be ±1 or pairs of complex conjugates with moduli 1, by rearranging the columns of P, we may assume, without loss of generality,
(83)A=Pdiag⁡[λ1In1,λ-1In1,…,λsIns,λ-sIns,Iu,-Iv]P-1,
where λ-j is the complex conjugate of λj, all λj are distinct (when s>1), and 2(∑j=1s‍nj)+u+v=d.
DecomposeP as P=[P1,Q1,…,Ps,Qs,U,V] according to (83). Then the columns of Pj, Qj, U, V consist of a basis for the eigenspace of λj, λ-j, 1, -1, respectively.
By rechoosing a basis for each eigenspace if necessary, we may assume further thatP is of the form P=[P1,P¯1,…,Ps,P¯s,P~1,P~-1], where P¯j is the conjugate of Pj (j=1,…,s), the columns of P1,P¯1,…,Ps,P¯s,P~1,P~-1 consist of bases for the eigenspaces of the eigenvalues λ1,λ-1,…,λs,λ-s,1,-1, respectively, and P~1, P~-1 are real matrices.
Let{αj,k+iβj,k}k=1nj be the columns of Pj. Then {αj,k-iβj,k}k=1nj are the columns of P¯j. Since P is invertible, column operations show that so is
(84)B=[α1,1,β1,1,…,α1,n1,β1,n1,…,αs,1,βs,1,…,αs,ns,βs,ns,U,V].
Moreover, as each column of Pj is an eigenvector of A, by letting λj=aj+ibj, we have
(85)aj2+bj2=1,Aαj,k=ajαj,k-bjβj,k,Aβj,k=bjαj,k+ajβj,k.
Therefore,
(86)A[αj,k,βj,k]=[αj,k,βj,k][ajbj-bjaj],j=1,…,s,k=1,…,nj,
with [ajbj-bjaj] being real orthogonal. Hence AB=BQ for a real orthogonal matrix Q, and our conclusion follows.
(c) SinceAλn*(U)={Si1⋯ik:ρi1⋯ik≤λn<ρi1⋯ik-1} is the unique packing family of itself for every n>0, a simple calculation shows that the solution s of (69) satisfies
(87)∑φ∈Aλn*(U)‍[Ld(φ(U))]s/d=[Ld(U)]s/d.
Hence Qλ(s)=0 and so the conclusion follows by using Theorem 11.Example12, proved below, is an illustration of Example 23(c).Proof of Example12.
The IFS satisfies LDP by the conclusion of Example23(a). By a result in [10], OSC holds. Note that ρ1=ρ2=1/2. The conclusion dim⁡HK=dim⁡PK=dim⁡BK=1 follows from Example 23(c).The following IFS consists of a nondifferentiable map. It satisfies LDP but not the natural extension of WSC.Example 24.
LetSi:[0,1]→[0,1], i=1,…,N, be an IFS of contractions defined as follows:
(88)S1(x)={ρ0(x+a1),x≤12,ρ0(12+a1)+ρ1(x-12),x>12,Si(x)=ρi(x+ai),i=2,…,N.
Assume ρ0ρ1>0, 0<|ρi|≤1/2 for i=0,1,…,N, and, for each i, either Si((0,1))⊆(0,(1/2)) or Si((0,1))⊆(1/2,1). Then the IFS {Si}i=1N satisfies LDP.Proof.
It follows from the assumptions that0<ri≤Ri≤1/2 for 0≤i≤N. Let
(89)c=max⁡{|ρ0|,|ρ1|}min⁡{|ρ0|,|ρ1|}.
We need only prove RI≤crI for all I∈Σ*. We use induction on the length of I. It is easy to see that this is true when |I|=1. Assume it is true for the case |I|≤n. Consider the case |I|=n+1. Let I=i1⋯in+1.
Ifi1>1, then SI(x)=ρi1(Si2⋯in+1(x)+ai1) for all x∈[0,1]. Hence RI=|ρi1|Ri2⋯in+1 and rI=|ρi1|ri2⋯in+1, and so RI≤crI by induction hypothesis.
Now assumei1=1. If Si2[0,1]⊆[0,1/2], then SI(x)=ρ0(Si2⋯in+1(x)+a1) for all x∈[0,1]. Hence RI=|ρ0|Ri2⋯in+1 and rI=|ρ0|ri2⋯in+1. If i1=1 and Si2[0,1]⊆[1/2,1], then SI(x)=ρ0(1/2+a1)+ρ1(Si2⋯in+1(x)-1/2). Hence RI=|ρ1|Ri2⋯in+1 and rI=|ρ1|ri2⋯in+1. In both cases it follows by induction hypothesis that RI≤crI. Thus LDP holds.The following is an explicit IFS from Example24 (see Figure 2):
(90)S1(x)={2xπ2,x≤12,xπ2+12π2,x>12,S2(x)=xπ+124,S3(x)=xπ+112,S4(x)=xπ+18,S5(x)=xπ+16,S6(x)=xπ+23.
The IFS in (90) does not satisfy the natural extension of WSC to the IFSs we consider in this paper, since the sub-IFS {S2,S3,S4,S5} does not.(a) Graphs of the maps in the IFS in (90). (b) First two levels of iterations of the interval [0,1] under the IFS.
(a)
(b)



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