Self-similar measures of bounded measure type 5 2.1

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THE ABSENCE OF SECOND-ORDER IDENTITIES

SZE-MAN NGAI AND YUANYUAN XIE

Abstract. For a self-similar measure inR^dwith overlaps but satisfies the so-calledbounded measure type conditionintroduced by Tang and the authors, we set up a framework for deriv- ing a closed formula for the L^q-spectrum of the measure forq≥0. The framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the L^q-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the L^q-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.

Contents

1. Introduction 1

2. Self-similar measures of bounded measure type 5

2.1. Finite type condition and measure type 6

2.2. Condition (B) for a class of IFSs onR² 9

3. Renewal equation and proof of Theorem 1.1 14

4. A class of finite IFSs with overlaps onR 18

5. A class of finite IFSs with overlaps onR² 27

6. Comments and questions 38

References 38

1. Introduction

Letµbe a positive finite Borel measure onR^dwhose support supp(µ) is compact. Forq ∈R, theL^q-spectrum τ(q) of µis defined as

τ(q) := lim

δ→0⁺

ln supP

iµ(B_δ(x_i))^q

lnδ ,

whereB_δ(x_i) is a disjoint family ofδ-balls with centerx_i∈supp(µ) and the supremum is taken over all such families. The function τ(q) arises in the theory of multifractal decomposition of

Date: January 23, 2017.

2010Mathematics Subject Classification. Primary: 28A80, 28A78.

Key words and phrases. Fractal,L^q-spectrum, multifractal formalism, self-similar measures, bounded measure type condition.

The authors are supported in part by the National Natural Science Foundation of China, Grant 11271122 and Construct Program of the Key Discipline in Hunan Province. The first author is also supported by the Center of Mathematical Sciences and Applications (CMSA) of Harvard University and the Hunan Province Hundred Talents Program.

1

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measures. A major goal of the theory is to compute the followingdimension spectrum:

f(α) := dimH

x∈supp(µ) : lim

δ→0⁺

lnµ(B_δ(x)) lnδ =α

,

where dim_H denotes Hausdorff dimension. The multifractal formalism, a heuristic principle first proposed by physicists (see [7, 8] and the references therein), asserts that the dimension spectrum is equal to the Legendre transform ofτ(q), i.e.,

f(α) =τ^∗(α) := inf{qα−τ(q) : q∈R}.

We are mainly interested in self-similar measures. For such measures, the multifractal formalism has been verified rigorously for those satisfying the separated open set condition [1, 3].

For self-similar measures defined by iterated function systems satisfying the weak separation condition, Lau and the first author [13] proved that if τ(q) is differentiable at q ≥ 0, then the multifractal formulism at the corresponding point holds. Feng and Lau [5] removed the differentiability condition; they also studied the validity of the multiformal formalism in the regionq <0.

The L^q-spectrum also encodes other important information of the measure. For example, τ(0) is the negative of the box dimension of µ; if τ is differentiable at q = 1, then τ⁰(1) is equal to the Hausdorff dimension ofµ(see [9, 13, 19, 23] and the references therein); for q > 1, τ(q)/(q−1) is theL^q-dimension of µ(see [24]).

The computation ofL^q-spectrum thus plays a key role in the theory of multifractal measures.

For self-similar and graph-directed self-similar measures satisfying the open set condition,τ(q) is computed by Cawley and Mauldin [1] and Edgar and Mauldin [3]. For self-similar measures with overlaps, the computation is much more difficult. Lau and the first author obtainedτ(q), q≥0, for the infinite Bernoulli convolution associated with the golden ratio [12] and a class of convolutions of Cantor measures [14]. Feng [4] computedτ(q) for infinite Bernoulli convolutions associated with a class of Pisot numbers. The graph ofτ(q) forq <0 has been studied by Lau, Wang, Feng and Olivier [4, 6, 17].

The computation of τ(q) in [12] and [14] makes use of Strichartz second-order self-similar identities. Unfortunately, very few self-similar measures satisfy these identities. Thus, closed formulas for τ(q) have been obtained for only a few classes of measures that are defined by iterated function systems onRwith the same contraction ratio. The main objective of this paper is to derive a closed formula forτ(q), q ≥ 0, for self-similar measures satisfying the so-called bounded measure type condition (Condition (B)) introduced in [22]. It is worth mentioning that recently G. Deng and the first author [2] used a infinite matrix method to obtain the differentiability of theL^q-spectrum for a class if IFSs that includes some of those studies in this paper; however, the method does not yield a closed formula forτ(q).

Throughout this paper an iterated function system (IFS) refers to a finite family of contrac- tions defined on a compact subset X of R^d. The derivation of τ(q) in this paper is based on

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the following equivalent definition, which holds forq ≥0:

τ(q) = inf

α: lim

δ→0⁺

1 δ^d+α

Z

X

µ(B_δ(x))^qdx >0

. (1.1)

(see [11, 12] and [13, Proposition 3.1]).

Letµbe a self-similar measure defined by a finite type IFS (see [10,15,21]) onR^d. In Section 2, we define the set of all level-k islands I_k (see Definition 2.6). Intuitively, each level-k island corresponds to a connected component of the level-kiterates of some fixed open set Ω; moreover, two islands I₁ and I₂ are of the same measure type (with respect to µ) if µ|_I₂ = cµ|_I₁ ◦S⁻¹ for some constantc >0 and some similitude S:I1→ I2, whereIj (j= 1,2) is the component corresponding toI_j, and µ|_F denotes the restriction of the measure µtoF ⊆R^d.

To compute τ(q) we divide level-kiterates of some fixed bounded open set Ω under the IFS into connected components called level-k islands, and classify them into measure types. Our main assumption is Condition (B) introduced in [22], which, loosely speaking, holds if there exists somek≥1 such that there is a uniform bound on those level-m (m > k) islands whose measure types, as well the measure types of their ancestors up to levelk+ 1, are different from that of any level-kisland. If k=:k_b is the smallest integer satisfying this condition, we call the correspondingI_k_b =:I_b thebasic set of islands (see Definition 2.15).

Assume {S_i}_i∈Λ is a finite type IFS on R^d (see [15]) with Ω being a finite type condition set and assume that Condition (B) holds withI_b being the basic set of islands. LetI:={I_1,`}_`∈Γ⊆ I_bbe a minimal subset such that the measure type of any island inI_b equals that of some island inI. Fixq≥0, define

ϕ`(δ) :=

Z

I1,`

µ(Bδ(x))^qdx and Φ^(α)_` (δ) := 1

δ^d+αϕ`(δ) for `∈Γ,

whereI_1,` :=SI_1,`(Ω). Then we can derive renewal equations for Φ^(α)_` (δ), and express them in vector form as:

f =f ∗M_α+z, whereα∈R, and

f =f^(α)(x) = [f_`^(α)(x)]`∈Γ, x∈R; f_`^(α)(x) := Φ^(α)_` (e^−x) for`∈Γ;

M_α = [µ^(α)_m`]`,m∈Γ is a finite matrix of Borel measures onR; z=z^(α)(x) = [z^(α)_` (x)]`∈Γ is a vector of error functions.

(1.2)

Let

M_α(∞) :=h

µ^(α)_m`(R)i

`,m∈Γ. (1.3)

For each`∈Γ andα∈R, define F_`(α) := X

m∈Γ

µ^(α)_m`(R) and D_`:={α∈R:F_`(α)<∞}. (1.4)

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If the error functions decay exponentially to 0 asx→ ∞, then theL^q-spectrum ofµis given by the unique α such that the spectral radius of M_α(∞) is equal to 1. The following is our main result.

Theorem 1.1. Let µ be a self-similar measure defined by a finite type IFS {S_i}i∈Λ on R^d. Assume that µ satisfies Condition (B). LetM_α(∞) andF_`(α)be defined as in (1.3)and (1.4).

(a) There exists a unique α∈R such that the spectral radius of M_α(∞) is equal to 1.

(b) If we assume, in addition, that for the unique α in (a), there exists >0 such that for alli∈Γ, z_`^(α)(x) =o(e^−x) as x→ ∞ . Then τ(q) =α for q≥0.

In Section 4, we illustrate Theorem 1.1 by the following family of IFSs on R:

S₁(x) =ρx, S₂(x) =rx+ρ(1−r), S₃(x) =rx+ 1−r, (1.5) where the contraction ratiosρ, r∈(0,1) satisfy

ρ+ 2r−ρr≤1, (1.6)

i.e., S₂(1) ≤S₃(0) (see Figure 1). This family of IFSs is first studied by Lau and Wang [16], and is used to illustrate the (general) finite type condition in [10, 15]. For a probability vector (pi)³_i=1, we define

w1(k) :=p1 k

X

j=0

p^k−j₂ p^j₃, k≥0. (1.7)

Theorem 1.2. Let µ be a self-similar measure defined by an IFS in (1.5) together with a probability vector (p_i)³_i=1, and w₁(k) be defined as in (1.7). Then for q ≥ 0, there exists a unique real number α:=α(q) satisfying

ρ^−α(1−p^q₂r^−α)(1−p^q₃r^−α)

∞

X

k=0

w1(k)^q(r^−α)^k+r^−α(p^q₂+p^q₃) = 1. (1.8) Henceτ(q) =α. Moreover, τ is differentiable on(0,∞) and

dim_H(µ) =τ⁰(1) = X³

i=2

p_ilnp_i−p₂p₃

3

X

i=2

lnp_i

∞

X

k=0

w₁(k)−Y³

i=2

(1−p_i)X^∞

k=0

w₁(k) lnw₁(k)−

3

X

i=2

p_ilnp_i

×

p2+p3−2p2p3

∞

X

k=0

w1(k) lnr−Y³

i=2

(1−pi) X^∞

k=0

w1(k) ln(ρr^k)−

3

X

i=2

pilnr −1

. Remark 1.3. Substituting q = 0in (1.8) gives ρ^−τ(0)+ 2r^−τ(0)−(ρr)^−τ(0) = 1. Hence −τ(0) is the Hausdorff dimension of the corresponding self-similar set (see [10, 15, 16]).

In Section 4, we illustrate Theorem 1.1 by the following family of IFSs on R²: S₁(x) =ρx, S₂(x) =rx+ (ρ−ρr,0),

S3(x) =rx+ (1−r,0), S4(x) =rx+ (0,1−r), (1.9)

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where the contraction ratiosρ, r∈(0,1) satisfy

ρ+ 2r−ρr≤1, (1.10)

i.e.,S2(1,0)≤S3(0,0) (see Figure 2(a)). For any probability vector (pi)⁴_i=1, define w₂(k) :=p₁

k

X

j=0

p^k−j₂ p^j₃, k≥0. (1.11)

Theorem 1.4. Let µ be a self-similar measure defined by any IFS in (1.9) together with a probability vector (p_i)⁴_i=1, and w₂(k) be defined as in (1.11). For q ≥0, there exists a unique real numberα:=α(q) satisfying

ρ^−α 1−p^q₂r^−α

1−p^q₃r^−α

∞

X

k=0

w₂(k)^q(r^−α)^k+r^−α

4

X

i=2

p^q_i = 1. (1.12) Henceτ(q) =α. Moreover, τ is differentiable on(0,∞) and

dim_H(µ) =τ⁰(1) = X³

i=2

p_ilnp_i−p₂p₃

3

X

i=2

lnp_i

∞

X

k=0

w₂(k)−Y³

i=2

(1−p_i)X^∞

k=0

w₂(k) lnw₂(k)−

4

X

i=2

p_ilnp_i

×

p₂+p₃−2p₂p₃

∞

X

k=0

w₂(k) lnr−Y³

i=2

(1−p_i)X^∞

k=0

w₂(k) ln(ρr^k)−

4

X

i=2

p_ilnr−1

. Remark 1.5. Substituting q = 0 into (1.12), we get ρ^−τ(0)+ 3r^−τ(0)−(ρr)^−τ(0) = 1. Again,

−τ(0) is the Hausdorff dimension of the corresponding self-similar set (see [15, Example 5.2]).

We use the vector-valued renewal theorem of Lau, Wang and Chu [18] to derive the stated formulas for τ(q); the classical renewal theorem used in [12] and [14] is not sufficient, as a finite number of renewal equations arise in our derivations. New techniques are also used in estimating the error terms and in proving the differentiability of τ(q).

This paper is organized as follows. In Section 2, we briefly recall the definition of Condition (B). In Section 3 we derive renewal equations and prove Theorem 1.1. Section 4 illustrates Theorem 1.1 by the class of one-dimensional IFSs (1.5) and prove Theorem 1.2. Section 5 studies IFSs in higher dimension and prove Theorem 1.4. Finally we state some comments and open questions in Section 6.

2. Self-similar measures of bounded measure type

In this section, we recall the definition of Condition (B) and then prove that it is satisfied by the self-similar measures defined by the IFSs in (1.9).

LetXbe a compact subset ofR^dwith nonempty interior, and{S_i}_i∈Λbe an IFS of contractive similitudes on X with attractor K ⊆ R^d. To each probability vector (p_i)i∈Λ (i.e., p_i > 0 and P

i∈Λp_i = 1), letµbe the associated self-similar measure, which satisfies theself-similar identity µ=X

i∈Λ

p_iµ◦S_i⁻¹.

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Moreover, supp(µ) =K.

2.1. Finite type condition and measure type. For k≥1, define Λ^k:={(i₁, . . . , i_k) :ij ∈Λ for j = 1, . . . , k},

where we calli∈Λ^k a word of lengthk, and denote its length by|i|. Ifk= 0, we define Λ⁰:=

{∅}. Also, we let Λ^∗ :=S

k≥0Λ^k. We frequently write i:=i₁· · ·i_k instead ofi= (i₁, . . . , i_k) if no confusion is possible; in particular, we write i=:i^k₁, if i_j =i₁ for allj = 1, . . . , k. Fork≥0 and i=i₁· · ·i_k∈Λ^k,we use the standard notation

S_i:=S_i₁ ◦ · · · ◦S_i_k, r_i :=r_i₁· · ·r_i_k, p_i :=p_i₁· · ·p_i_k, with S∅:=id, r∅ =p∅ := 1,whereid is the identity map onR^d.

For two indices i,j ∈Λ^∗, we writei4j ifi is a prefix ofj ori=j, and denote byi64j if i4j does not hold. Let{M_k}^∞_k=1 be a sequence of index sets, where M_k⊆Λ^∗. Let

m_k=m_k(M_k) := min{|i|:i∈ M_k} and m_k=m_k(M_k) := max{|i|:i∈ M_k}.

We also letM₀:={∅}.

Definition 2.1. We say that {M_k}^∞_k=0 is a sequence of nested index sets if it satisfies the following conditions:

(1) both {m_k} and {m_k} are nondecreasing, and limk→∞m_k= limk→∞m_k=∞;

(2) for each k≥1, M_k is an antichain inΛ^∗;

(3) for each j∈Λ^∗ with |j|> m_k or j∈ M_k+1, there exists i∈ M_k such thati4j;

(4) for each j∈Λ^∗ with |j|< m_k or j∈ M_k−1, there exists i∈ M_k such thatj 4i;

(5) there exists a positive integer L0, independent of k, such that for all i ∈ M_k and j∈ M_k+1 with i4j, we have |j| − |i| ≤L0.

To define neighborhood types, we fix a sequence of nested index sets {M_k}^∞_k=0.

Notation 2.2. (1) For each integerk≥0, letV_k be the set oflevel-kvertices(with respect to {M_k}) defined as

V₀ :=

(id,0) and V_k:=

(Si, k) :i∈ M_k for allk≥1.

We call (id,0)the root vertex and denote it by v_root. (2) Let V :=S

k≥0V_k be the set of all vertices.

(3) For v = (S_i, k) ∈ V_k, we use the convenient notation S_v := S_i and r_v := r_i. It is possible to have v= (S_i, k) = (S_j, k) with i6=j.

(4) More generally, for any k≥0 and any subset A ⊂ V_k, we use the notation SA(Ω) := [

v∈A

Sv(Ω). (2.1)

Let Ω⊆Xbe a nonempty bounded open set which isinvariantunder{S_i}_i∈Λ, i.e.,S

i∈ΛSi(Ω)⊆ Ω. Such an Ω exists by our assumption; in particular, X^◦ is such a set.

Next, we recall the definitions of neighbors and neighborhoods.

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Definition 2.3. We say that two level-k vertices v,v⁰ ∈ V_k (allowing v = v⁰) are neighbors (with respect to Ωand {M_k}) ifS_v(Ω)∩S_v⁰(Ω)6=∅. We call the set of vertices

N_Ω(v) :={v⁰ :v⁰ ∈ V_k is a neighbor ofv}

the neighborhood of v (with respect to Ω and{M_k}).

Obviously v ∈N_Ω(v). If no confusion is possible, we omit the subscript Ω inN_Ω(v).

Let S :={S_jS_i⁻¹ :i,j∈Λ^∗}. We define an equivalence relation on the set of vertices V.

Definition 2.4. Two verticesv ∈ V_k andv⁰∈ V_k⁰ are said to be equivalent, denoted byv ∼_σ v⁰ (or simply v ∼v⁰), if for σ := S_v⁰S_v⁻¹(∈ S) :S

u∈N(v)S_u(X) → X, the following conditions hold:

(1)

S_u⁰ :u⁰ ∈N(v⁰) =

σS_u:u∈N(v) ; in particular, σS_u is defined for allu∈N(v).

(2) foru∈N(v)andu⁰ ∈N(v⁰)such thatS_u⁰ =σS_u, and for any positive integer`≥1, an indexi∈Λ^∗ satisfies(S_uS_i, k+`)∈ V_k+`if and only if it satisfies(S_u⁰S_i, k⁰+`)∈ V_k⁰_+`. It is direct to check that ∼ is an equivalence relation. We denote the equivalence class containingv by [v] and call it the(neighborhood) type ofv (with respect to Ω and{M_k}).

We define an infinite graph G with vertex set V and directed edges defined as follows. Let v∈ V_k andu∈ V_k+1. Suppose there existsi∈ M_k,j ∈ M_k+1, and l∈Λ^∗ such that

v = (Si, k), u= (Sj, k+ 1), j = (i,l).

Then we connect a directed edge l:v → u. We call v a parent of u and u an offspring of v.

We writeG= (V,E), where E is the set of all directed edges defined above.

Definition 2.5. Let {S_i}_i∈Λ be an IFS of contractive similitudes on a compact subset X⊆R^d. We say that{S_i}_i∈Λ is offinite type (or that it satisfies thefinite type condition) if there exists a sequence of nested index sets {M_k}^∞_k=0 and a nonempty bounded invariant open set Ω⊆ X such that, with respect to Ω and {M_k}, the set of equivalence classes V/_∼:=

[v] :v ∈ V is finite. We call such anΩ a finite type condition set(or FTC set).

Definition 2.6. A subset I ⊆ V_k is called a level-k island (with respect to Ω and {M_k}) if SI(Ω)is a connected component of SV_k(Ω).

Remark 2.7. (1) For each v ∈ V_k, there exists a unique island, denoted by I(v), containing v and, moreover, N(v)⊆ I(v).

(2) If {S_i}_i∈Λ satisfies (OSC) with Ωbeing an OSC set, then I(v) ={v} for all v∈ V. Notation 2.8. (1) Let

I_k :={I :I is a level-k island} and I:= [

k≥0

I_k

be the collection of all level-k islands and the collection of all islands, respectively.

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(2) Generalizing (2.1), for anyk≥0 and any subset B ⊂I_k, we use the notation SB(Ω) := [

I∈B

SI(Ω).

Definition 2.9. We say that two islands I ∈I_k and I⁰ ∈I_k⁰ are equivalent, and denote it by I ≈_σ I⁰ (or simply, I ≈ I⁰), if there exists some σ∈S such that {S_v⁰ :v⁰ ∈ I⁰}={σS_v :v ∈ I} and, moreover,v ∼_σ v⁰ for any v∈ I andv⁰ ∈ I⁰ satisfyingS_v⁰ =σS_v.

Notation 2.10. (1) We denote the equivalence class of I by[I]and we call[I]the(island) type of I.

(2) ForI ∈I_k, I⁰ ∈I_k+1, I is said to be aparent of I⁰ andI⁰ an offspringof I if for any v∈ I⁰, I contains some parent of v. For anyk≥0 and I ∈I_k, let

O(I) :={J :J is an offspring of I} (2.2) be the collection of all offspring ofI.

Definition 2.11. Let µ be a self-similar measure defined by an IFS {S_i}_i∈Λ of finite type with Ω being an FTC set. Two equivalent vertices v ∈ V_k andv⁰ ∈ V_k⁰ are µ-equivalent, denoted by v ∼_µ,σ,wv⁰ (or simply v ∼_µv⁰) if for σ=S_v⁰ ◦S_v⁻¹, there exists a number w >0 such that

µ|_S

N(v0)(Ω)=w·µ|_S

N(v)(Ω)◦σ⁻¹.

As∼is an equivalence relation, so is∼_µ. Denote theµ-equivalence class ofvby [v]_µand call it the (neighborhood) measure typeof v (with respect to Ω, {M_k}and µ). Intuitively,v ∼_µv⁰ means that the measures µ|_S

N(v)(Ω) and µ|_S

N(v0)(Ω) have the same structure. The following proposition shows that µ-equivalent vertices generate the same number of offspring of each neighborhood measure type. The proof can be found in [22].

Proposition 2.12. For two equivalent vertices v∈ V_k andv⁰ ∈ V_k⁰, let{u_i}i∈Λ₁ and{u⁰_i}_i∈Λ⁰

1

be the offspring of v and v⁰ in G, respectively. If [v]_µ = [v⁰]_µ, then, counting multiplicity, {[u_i]_µ:i∈Λ₁}={[u⁰_i]_µ:i∈Λ⁰₁}.

Definition 2.13. Let µ be a self-similar measure defined by a finite type IFS {S_i}_i∈Λ on R^d with Ωbeing an FTC set. Two islandsI ∈I_k and I⁰ ∈I_k⁰ are said to beµ-equivalent, denoted I ≈_µ,σ,wI⁰ (or simply I ≈_µI⁰), if I ≈_σ I⁰ and there exists some w >0 such that

µ|_S

I0(Ω)=w·µ|_S_I_(Ω)◦σ⁻¹. (2.3) We remark that (2.3) holds if and only if v ∼_µ,σ,w v⁰ for any v ∈ I and v⁰ ∈ I⁰ satisfying S_v⁰ =σS_v. We note that ≈_µ is an equivalence relation. We denote theµ-equivalence class ofI by [I]_µ, and call [I]_µthe(island) measure typeofI(with respect to Ω,{M_k}andµ). From the definition of≈_µ, we obtain an analog of Proposition 2.12 concerning≈_µ. That is,µ-equivalent islands generate the same number of offspring of each island measure type.

Definition 2.14. Let µ be a self-similar measure defined by a finite type IFS. Let B ⊆I_k for k ≥ 0 and B_µ := {[I]_µ : I ∈ B}. We call I a level-2 nonbasic island with respect to B if

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I ∈O(J) for some J ∈B and [I]_µ∈ B/ _µ. Inductively, for`≥3, we call I a level-` nonbasic island with respect to B if I is an offspring of some level-(`−1) nonbasic island with respect toB and [I]_µ∈ B/ _µ.

We remark that, by definition, for any ` ≥2, I is a level-` nonbasic island with respect to Bif and only if there exists a finite sequence of {I_k}^`_k=1 such that I₁ ∈B, I_` =I, [I_i]µ∈ B/ _µ, and I_i is an offspring of I_i−1 for all i= 2, . . . , `. In particular, I_i is a level-i nonbasic island with respect toB for alli= 2, . . . , `.

Definition 2.15. Let µ be a self-similar measure defined by a finite type IFS on R^d. We say thatµsatisfiesCondition (B)if there exists somek≥1such that the number of level-`nonbasic islands with respect to I_k is uniformly bounded for all ` > k. If k=: k_b is the minimum non- negative integer satisfying this condition, then we call the corresponding I_k_b =:I_b the basic set of islands.

The following two classes of examples for Condition (B) are proved in [22].

Example 2.16. Let µ be a self-similar measure defined by an IFS {S_i}_i∈Λ in R^d satisfying (OSC). Thenµ satisfies Condition (B).

Let {S_i}³_i=1 be defined as in (1.5) and µbe the self-similar measure associated with a probability vector (p_i)³_i=1. Letw₁(k), k≥0, be defined as in (1.7). We remark that for k≥0,

p₁p^k+1₃ +p₂w₁(k) =p₁p^k+1₂ +p₃w₁(k) =w₁(k+ 1) and w₁(k+ 1)≤w₁(k)≤p₁. (2.4) Example 2.17. Let µ be the self-similar measure defined by any of the IFSs {S_i}³_i=1 in (1.5) together with a probability vector(pi)³_i=1. Then µ satisfies Condition (B).

0 1

S1

S2 S3

Figure 1. The first iteration of {S_i}³_i=1 defined in (1.5). The figure is drawn with ρ= 1/3 and r= 2/7.

2.2. Condition (B) for a class of IFSs on R². In this subsection, we prove that any self- similar measure defined by an IFS in (1.9) satisfies Condition (B).

Let {S_i}⁴_i=1 be defined as in (1.9) and µbe the self-similar measure associated with a probability vector (p_i)⁴_i=1. Letw₂(k), k≥0, be defined as in (1.11). We remark that for k≥0,

p1p^k+1₃ +p2w2(k) =p1p^k+1₂ +p3w2(k) =w2(k+ 1), w2(k+ 1)≤w2(k)≤p1. (2.5)

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Throughout this subsection we let X = [0,1]×[0,1],

Ω =X^◦ and W_n:={2ⁿ⁻ⁱ13ⁱ :i= 0,1, . . . , n}, n≥1. (2.6) To simplify notation we let

γk:= 1−r^k, k≥0. (2.7)

Define

I_1,1:={(S₁,1),(S2,1)}, I_1,2 :={(S₃,1)}, I_1,3 :={(S₄,1)} (2.8) (see Figure 2(a)) and

I1,1 :=SI_1,1(Ω) =S1(Ω)∪S2(Ω) = (0, ργ1)×(0, ρ)∪(ργ1, ργ1+r)×(0, r), I_1,2 :=SI_1,2(Ω) =S₃(Ω) = (γ₁,1)×(0, r),

I_1,3 :=SI1,3(Ω) =S₄(Ω) = (0, r)×(γ₁,1),

(2.9)

whereI_1,i, i= 1,2,3, are defined in (2.8).

Example 2.18. Let µ be a self-similar measure defined by an IFS {S_i}⁴_i=1 in (1.9) together with a probability vector(p_i)⁴_i=1. LetΩ andW_n be as in (2.6). Then µsatisfies Condition (B).

To prove Example 2.18, we first summarize without proof some elementary properties.

Proposition 2.19. Let {S_i}⁴_i=1 be as in (1.9) and {I_1,i}³_i=1 be as in (2.8). The following relations hold:

(a) S13=S21. Moreover, for any i,j∈Wn, S_i =S_j. (b) I₁ ={I_1,1,I_1,2,I_1,3}.

Proposition 2.20. Assume the hypotheses of Example 2.18 and {I_1,i}³_i=1 defined as in (2.9).

Then (a)–(c) below hold, and (d)–(f ) hold for allk≥0:

(a)

S3(I1,1) =(γ1,(1 +ρr)γ1)×(0, ρr)∪((1 +ρr)γ1,(1 +ρr)γ1+r²)×(0, r²), S₄(I_1,1) =(0, ρrγ₁)×(γ₁, γ₁+ρr)∪(ρrγ₁, ρrγ₁+r²)×(γ₁, γ₁+r²).

(b)

S₁(I_1,2) = (ργ₁, ρ)×(0, ρr), S₃(I_1,2) = (γ₂,1)×(0, r²),

S4(I1,2) = (rγ1, r)×(γ1, γ1+r²).

(c) S3(I1,3) = (γ1, γ1+r²)×(rγ1, r) and S4(I1,3) = (0, r²)×(γ2,1).

(d)

S₂k1(I_1,1) =(ργ_k, ργ_k+ρ²r^kγ₁)×(0, ρ²r^k)∪

(ργ_k+ρ²r^kγ₁, ργ_k+ρ²r^kγ₁+ρr^k+1)×(0, ρr^k+1), S₂k1(I1,3) =(ργ_k, ργ_k+ρr^k+1)×(ρr^kγ1, ρr^k).

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(e)

S₂k(I1,1) =(ργ_k, ργ_k+1)×(0, ρr^k)∪(ργ_k+1, ργ_k+1+r^k+1)×(0, r^k+1), S₂^k(I1,2) =(r^kγ1+ργk, r^k+ργk)×(0, r^k+1),

S₂k(I_1,3) =(ργ_k, ργ_k+r^k+1)×(r^kγ₁, r^k).

(f) S₂k1(Ω) = (ργ_k, ρ)×(0, ρr^k) and S₂k(Ω) = (ργ_k, ργ_k+r^k)×(0, r^k).

Proof. (a)–(c) follow from (2.9), and (d)–(f) can be proved directly by induction; we omit the

details.

Lemma 2.21. Assume the hypotheses of Proposition 2.20. Then µ(S1(Ω)∩S₂k(Ω)) =µ[³

i=1

S1(I1,i)∩S₂k(Ω)

=µ(S₂k1(Ω)) for k≥1. (2.10) Proof. First, we prove the first equality in (2.10). Since µ(S₁(Ω)) = µ(∪³_i=1S₁(I_1,i)), we have µ(S₁(Ω)∩A) =µ((∪³_i=1S₁(I_1,i))∩A) =µ ∪³_i=1S₁(I_1,i)∩S₂k(Ω)

for any A⊆Ω.

Next, we show that

3

[

i=1

S₁(I_1,i)∩S₂k(Ω) =S₂k1(Ω) for all k≥1. (2.11) By Proposition 2.20(b,d,f), we have

S₁(I_1,1) = (0, ρ²γ₁)×(0, ρ²)∪(ρ²γ₁, ρ²γ₁+ρr)×(0, ρr), S1(I1,2) = (ργ1, ρ)×(0, ρr), S1(I1,3) = (0, ρr)×(ργ1, ρ),

and S₂(Ω) = (ργ₁, ργ₁ +r)×(0, r). It follows from (1.10) that ρr+ρ²γ₁ ≤ ργ₁ and hence S₁(I_1,1)∩S₂(Ω) =∅. Sinceρ < r+ργ₁, we haveS₁(I_1,2)∩S₂(Ω) = (ργ₁, ρ)×(0, ρr) =S₂₁(Ω), where in the last equality we use Proposition 2.20(f). Sincer < γ₁, we haveρr < ργ₁, and thus S₁(I_1,3)∩S₂(Ω) =∅. HenceS3

i=1S₁(I_1,i)∩S₂(Ω) =S₂₁(Ω). Assume that the stated inequality holds fork=m, i.e.,S₃

i=1S₁(I_1,i)∩S₂^m(Ω) =S₂^m₁(Ω). ThenS₁(I_1,2)∩S₂^m(Ω) =S₂^m₁(Ω) and S1(I1,i)∩S2^m(Ω) =∅fori= 1,3. Fork=m+ 1, sinceS1(I1,i)∩S₂^m+1(Ω)⊆S1(I1,i)∩S2^m(Ω), we haveS1(I1,i)∩S₂^m+1(Ω) =∅ fori= 1,3. By (2.9) and Proposition 2.19(a), we have

S₁(I_1,2)∩S₂m+1(Ω)

= S₁₃(Ω)∩S₂^m+1(Ω) =S₂₁(Ω)∩S₂^m+1(Ω)

= S2(S1(Ω)∩S2^m(Ω)) =S2((∪³_i=1S1(I1,i))∩S2^m(Ω))

= S2

[³

i=1

S1(I1,i)∩S2^m(Ω)

=S2(S2^m1(Ω)) =S₂^m+1₁(Ω).

This proves (2.11). Hence the second inequality in (2.10) holds.

Part (a) of the following lemma explains the meaning of the factor w₂(k).

Lemma 2.22. Assume the hypotheses of Proposition (2.20) and let w2(k) be defined as in (1.11). Then

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(a) for k≥0 and i= 1,3, µ|_S

2k1(I1,i)=w₂(k)µ◦S₂⁻¹_k₁; (b) for k≥1, µ|_S

2k(I1,1)=w2(k−1)µ◦S₂⁻¹k−11+p^k₂µ◦S₂⁻¹k ; (c) for k≥1 and i= 2,3, µ|_S

2k(I1,i)=p^k₂µ◦S₂⁻¹k ; (d) for i= 1,2,3 andj = 3,4, µ|_S_j_(I_1,i₎ =pj◦µ|_I_1,i.

Proof. We only prove (a) for i= 1 as an example. By Proposition 2.20(d), we have S₁(I_1,1) = (0, ρ²γ₁)×(0, ρ²)∪(ρ²γ₁, ρr+ρ²γ₁)×(0, ρr).

Note thatS2(Ω) = (ργ1, ργ1+r)×(0, r). Moreover,ργ1−(ρ²γ1+ρr) =ρ(1−2r−ρ+ρr)≥0, we haveS₁(I_1,1)⊆S₁(Ω)\S₂(Ω). Henceµ(A) =p₁µ◦S₁⁻¹(A) for anyA⊆S₁(I_1,1). Assume that the stated equality holds fork=m, i.e.,µ|_S

2m1(I1,1)=w₂(m)µ◦S₂⁻¹m1. Fork=m+1, by Proposition 2.19(a), we have S₂m+11(I_1,1) = S₁₃m+1(I_1,1). Then S₁⁻¹(A) ⊆ S₃m+1(I_1,1) and S₂⁻¹(A) ⊆ S₂^m₁(I_1,1) for any A⊆S₂^m+1₁(I_1,1). It follows thatµ(S₁⁻¹(A)) =p^m+1₃ µ◦S₃⁻¹m+1(S₁⁻¹(A)) and µ(S₂⁻¹(A)) =w₂(m)µ◦S₂⁻¹m1(S⁻¹₂ (A)). Thus

µ(A) = p1µ◦S₁⁻¹(A) +p2µ◦S₂⁻¹(A)

= p₁p^m+1₃ µ◦S₃⁻¹m+1(S₁⁻¹(A)) +p₂w₂(m)µ◦S₂⁻¹m1(S₂⁻¹(A))

= p₁p^m+1₃ µ◦S₁₃⁻¹m+1(A) +p₂w₂(m)µ◦S₂⁻¹m+11(A)

= (p1p^m+1₃ +p2w2(m))µ◦S⁻¹₂m+11(A)

= w2(m+ 1)µ◦S₂⁻¹m+11(A).

The last equality follows from (2.5). This proves part (a) fori= 1. For the proof of part (c) in

the casei= 3, we use Lemma 2.21.

Proof of Example 2.18. By (2.6), we have Ω = (0,1)×(0,1). For each k ≥ 0, let M_k = {1,2,3,4}^k. We show thatµsatisfies Condition (B) withI_b :=I₁ being the basic set of islands.

LetI_1,i be defined as in (2.8). Thus I_b ={I_1,1,I_1,2,I_1,3}. Let I_b,µ:= {[I_1,1]µ,[I_1,2]µ,[I_1,3]µ}.

It suffices to show that I_k,1,3 :={(S₂k−11, k),(S₂k, k)} is the only level-k nonbasic island with respect to Ib for any k≥ 2 (see Figure 2(b)). For i= 2,3, since [I(v_root)]µ = [I_1,i]µ, none of theI ∈ O(I_1,i) is a nonbasic island with respect to Ib (see Figure 4). Upon iterating the IFS once,I_1,1 generates the following five islands:

I_2,1,1:={(S₁₁,2),(S12,2)}, I_2,1,2 :={(S₁₄,2)},

I_2,1,3:={(S₂₁,2),(S₂₂,2)}, I_2,1,4 :={(S₂₃,2)}, I_2,1,5 :={(S₂₄,2)}

(see Figure 3). Lemma 2.22 implies that I_2,1,i ∈ [I_1,i]_µ for i = 1,2,4,5, and [I_2,1,3]_µ ∈/ I_b,µ. Thus I_2,1,3 is the only level-2 nonbasic island with respect to I_b. Assume that I_k,1,3 :=

{(S₂k−11, k),(S₂k, k)} is the only level-k nonbasic island with respect to I_b. Similarly, I_k,1,3 generates five islands, namely,

I_k+1,1,1:={(S₂k−111, k+ 1),(S₂k−112, k+ 1)}, I_k+1,1,2 :={(S₂k−114, k+ 1)}, I_k+1,1,3:={(S₂k1, k+ 1),(S₂k+1, k+ 1)}, I_k+1,1,4 :={(S₂k3, k+ 1)}, I_k+1,1,5:={(S₂k4, k+ 1)}.

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Lemma 2.22 again implies thatI_k+1,1,i ∈ [I_1,i]_µ for i= 1,2,4,5, and [I_k+1,1,3]_µ ∈/ I_b,µ. Thus, I_k+1,1,3 is the only level-(k+ 1) nonbasic island with respect toI_b, completing the proof.

I_1,1 ^-I_1,2

I_1,3

(a) level-1 islands{I_1,`}

{I_2,1,i}⁵_i=1 ^-{I_2,2,i}³_i=1 {I_2,3,i}³_i=1

(b) level-2 islands {I_2,`,i}

Figure 2. (a) First level iterations containing {I_1,`}³_`=1. (b) Second level iterations containing {I_2,1,i}⁵_i=1 and {I_2,`,i}³_i=1 for `= 2,3. The figures are drawn with ρ= 1/4 and r= 7/20.

(a) I_1,1

I_2,1,1 I_2,1,2

6 I_2,1,3

- I_2,1,4 I_2,1,5

(b) {I_2,1,i}⁵_i=1 Figure 3. I_1,1 and its offspring {I_2,1,i}⁵_i=1.

(a) I_2,`

I_2,`,1 ^-I_2,`,2

I_2,`,3

(b){I_2,`,i}³_i=1 Figure 4. I_2,` and its offspring{I_2,`,i}³_i=1 for`= 2,3.

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3. Renewal equation and proof of Theorem 1.1

Let {S_i}_i∈Λ be a finite type IFS on a compact subset X ⊆R^d with FTC set Ω⊆X and let µbe the self-similar measure defined by{S_i}_i∈Λtogether with a probability vector (p_i)i∈Λ. To computeτ(q) for q ≥0, we will use the equivalent definition in (1.1). We show in this section that for the class of self-similar measures under consideration and for each q ≥0, there exists α:=α(q) such that

0< lim

δ→0⁺

1 δ^d+α

Z

X

µ(B_δ(x))^qdx <∞. (3.1)

In the rest of this section, we assume that µ satisfies Condition (B). Let I_b := I_k_b be the basic set of islands and I_b,µ :=

[I]_µ : I ∈ I_b . We choose a subset I := {I_1,`}_`∈Γ ⊆I_b such that for anyI ∈I_b, there exists a unique `∈Γ satisfying I ∈[I_1,`]µ. Define

ϕ`(δ) :=

Z

I_1,`

µ(Bδ(x))^qdx, Φ^(α)_` (δ) := 1

δ^d+αϕ`(δ) for`∈Γ, (3.2) whereI_1,`:=SI_1,`(Ω).

Proposition 3.1. Let q ≥0. If 0< lim

δ→0⁺Φ^(α)_` (δ)<∞ for all `∈Γ, (3.3) thenτ(q) =α.

Proof. To findτ(q), it suffices to look for αsuch that (3.1) holds. Since SI_1,`(Ω) is a connected component of Ω, we have

δ→0lim⁺ 1 δ^d+α

Z

X

µ(B_δ(x))^qdx ≤ X

`∈Γ δ→0lim⁺

1 δ^d+α

Z

I1,`

µ(B_δ(x))^qdx

= X

`∈Γ

lim

δ→0⁺Φ^(α)_` (δ). (3.4)

Combining (3.3) and (3.4) yields (3.1), and hence the proposition follows.

For I ∈ I, let SI(Ω) and O(I) be defined as in (2.1) and (2.2), respectively. We denote the contraction ratio of a contractive similitude σ by r_σ. In view of Proposition 3.1, to find α satisfying (3.1), it suffices to study Φ^(α)_` (δ) for `∈Γ.

Step 1. Derivation of a functional equation forΦ^(α)_` (δ) for `∈Γ. For`∈Γ, define I2,`:={I ∈O(I_1,`) : [I]_µ∈Ib,µ} and I⁰_2,`:={I ∈O(I_1,`) : [I]_µ∈/ Ib,µ}.

ThusO(I_1,`) =I2,`∪I⁰_2,`. Fork≥3, ifI⁰_k−1,`6=∅, we define I_k,`:=n

I ∈ [

J ∈I⁰_k−1,`

O(J) : [I]_µ∈I_b,µo

and I⁰_k,`:=n

I ∈ [

J ∈I⁰_k−1,`

O(J) : [I]_µ∈/I_b,µo .

We remark that for any k≥2,S

`∈ΓI⁰_k,` is the set of all level-knonbasic islands with respect toI.

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Without loss of generality, we assume that Γ can be partitioned into two sub-collections, Γ∗

and Γ⁰_∗, defined as follows. For`∈Γ, we say`∈Γ∗ if there exists some κ_` ≥2, depending on

`, such that κ_` is the smallest number satisfyingI⁰_κ

`,` =∅; otherwise, `∈ Γ⁰_∗. Define κ_` := ∞ for`∈Γ⁰_∗.

Condition (B) implies that P

`∈Γ#I⁰_k,`is uniformly bounded for anyk≥2. Fix`∈Γ. Then for any 2≤k≤κ_`, we use two finite disjoint subsetsG_k,`, G⁰_k,`⊆Zto label the elements ofI_k,`

and I⁰_k,`; more precisely,

I_k,`={I_k,`,i:i∈G_k,`} and I⁰_k,`={I_k,`,i:i∈G⁰_k,`}.

Condition (B) implies that 0 ≤ #G⁰_k,` ≤ M, where M > 0 is a constant. We remark that G⁰_κ

`,`=∅. Define

I_k,`,i:=SI_k,`,i(Ω) for 2≤k≤κ_` and i∈G_k,`∪G⁰_k,`. Then for all`∈Γ∗ we have

ϕ_`(δ) =

κ`

X

j=2

X

i∈G_j,`

Z

Ij,`,i

µ(B_δ(x))^qdx, (3.5)

while for all`∈Γ⁰_∗ andn≥2, ϕ_`(δ) =

n

X

j=2

X

i∈G_j,`

Z

Ij,`,i

µ(B_δ(x))^qdx+ X

`∈G⁰_n,`

Z

In,`,i

µ(B_δ(x))^qdx. (3.6)

For `∈Γ,2≤k≤κ`, i∈Gk,` and δ >0, let Iek,`,i(δ) be the largest subset ofIk,`,i satisfying B_δ(x)⊆I_k,`,i for anyx∈Ie_k,`,i(δ). We denote Ib_k,`,i(δ) :=I_k,`,i\eI_k,`,i(δ). So for`∈Γ∗ (3.5) can be written as

ϕ`(δ) =

κ_`

X

j=2

X

i∈G_j,`

Z

Ie_j,`,i(δ)

µ(Bδ(x))^qdx+

κ_`

X

j=2

X

i∈G_j,`

Z

Ib_j,`,i(δ)

µ(Bδ(x))^qdx, while for `∈Γ⁰_∗ and n≥2, (3.6) can be expressed as

ϕ_`(δ) =

n

X

j=2

X

i∈Gj,`

Z

Iej,`,i(δ)

µ(B_δ(x))^qdx+

n

X

j=2

X

i∈Gj,`

Z

Ibj,`,i(δ)

µ(B_δ(x))^qdx

+ X

i∈G⁰_n,`

Z

In,`,i

µ(B_δ(x))^qdx.

For ` ∈ Γ,2 ≤ k ≤ κ_` and i ∈ G_k,`, there exist unique σ(k, `, i) ∈ S, w(k, `, i) > 0 and c(k, `, i)∈Γ such thatI_1,c(k,`,i)≈µ,σ(k,`,i),w(k,`,i)I_k,`,i. By Definition 2.13, we have

µ|_S_I

k,`,i(Ω)=w(k, `, i)◦µ|_S_I

1,c(k,`,i)(Ω)◦σ(k, `, i)⁻¹.

ForIe_k,`,i(δ)⊆I_k,`,i, letIe_1,c(k,`,i)(δ/r_σ(k,`,i)) be the largest subset ofI_1,c(k,`,i)satisfyingB_δ/r_σ(k,`,i)(x)⊆ I_1,c(k,`,i) for any x∈Ie_1,c(k,`,i)(δ/r_σ(k,`,i)). Thus

µ|Iek,`,i(δ)=w(k, `, i)◦µ|

Ie1,c(k,`,i)(δ/rσ(k,`,i))◦σ(k, `, i)⁻¹.