SPECTRAL ASYMPTOTICS OF ONE-DIMENSIONAL GRAPH-DIRECTED SELF-SIMILAR MEASURES WITH OVERLAPS

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SZE-MAN NGAI AND YUANYUAN XIE

Abstract. For the class of graph-directed self-similar measures onR, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of these measures. For the class of finitely ramified graph-directed self- similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [7]. The main novelty of our result is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.

Contents

1. Introduction 1

2. Graph-directed iterated function systems and measures essentially of finite type 6

2.1. Graph-directed iterated function systems 6

2.2. The essentially finite type condition for the graph-directed self-similar measure 7

3. Eigenvalue counting function 8

3.1. Eigenvalue counting function onR 8

3.2. Unitarily equivalent operators 9

4. Renewal equation and Proof of Theorem 1.1 and Theorem 1.3 9

4.1. Renewal equation 9

4.2. Proof of Theorem 1.1 and Theorem 1.3 13

5. Strongly connected GIFSs onR 14

5.1. An example of strongly connected GIFS 15

5.2. Spectral dimension ofµin Example 5.1. 18

6. GIFSs that are not strongly connected 21

6.1. An example of a GIFS that is not strongly connected 21

6.2. Spectral dimension ofµin Example 6.1. 24

References 27

1. Introduction

Let U ⊆R^d be a bounded domain with smooth boundary, ∆ be the Dirichlet Laplacian on U, {λ_n} be the eigenvalues of −∆, and N(λ,−∆) be the number of eigenvalues that do not

Date: October 19, 2018.

2010Mathematics Subject Classification. Primary: 28A80, 35P20; Secondary: 35J05.

Key words and phrases. Fractal, spectral dimension, graph-directed self-similar measure, essentially of finite type.

The authors are supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by the Hunan Province Hundred Talents Program and a Faculty Research Scholarly Pursuit Funding from Georgia Southern University.

1

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exceedλ. Weyl [25] proved the following asymptotic formula for the Dirichlet Laplacian:

N(λ,−∆) = B_d

(2π)^d|U|λ^d/2+o(λ^d/2) = 1

(4π)^d/2Γ(d/2 + 1)|U|λ^d/2+o(λ^d/2), (1.1) where |U| denotes thed-dimensional volume of U and B_d is the volume of the unit ball in R^d. We point out that in [20, Equation (1.1)], the factor (2π)^dis incorrectly typed as (4π)^d/2.

There has been considerable interest in studying spectral dimension on various domains.

McKean and Ray [16] computed the spectral dimension of the Cantor measure. Fujita [5], Naimark and Solomyak [17] studied the spectral dimension of the self-similar measures satisfying the open set condition (OSC) (see [9]). Kigami and Lapidus [12] obtained the spectral dimension of Laplacians on post-critically finite (p.c.f.) self-similar sets with a harmonic struc- ture. Croydon and Hambly [2, 6] studied the spectral dimension on the continuum random tree and random recursive affine nested fractals. For finitely ramified graph-directed self-similar set- s, Hambly and Nyberg [7] studied the spectral dimension of the associated Laplace operators.

Freiberg [4] investigated spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets. Kajino [10, 11] studied asymptotics of the partition functions associated with self-similar sets. Alonso-Ruiz and Freiberg [1] obtained the spectral dimension of Laplacians on Hanoi attractors.

We say that an iterated function system (IFS) or a graph-directed iterated function system (GIFS), as well as any associated self-similar measure or graph-directed self-similar measure, have overlaps, if (OSC) or the graph open set condition (GOSC) (see Section 2.1) fails. In this case, it is much harder to compute the spectral dimension. For a class of IFSs on R with overlaps and satisfying second-order identities (see [23]), the first author [18] computed the spectral dimension of the corresponding measures. Tang and authors [20] defined measures that are essentially of finite type (EFT), a property describing the finiteness of basic measure types, and computed the spectral dimension of the Laplacian defined by a self-similar measure satisfying (EFT). The first author and Tang [19] computed the spectral dimension of a special class of graph-directed self-similar measures with overlaps. This paper studies the eigenvalue asymptotics of the Dirichlet Laplacians defined by graph-directed self-similar measure with overlaps in much greater generality.

Let Ω ⊆ R^d be a bounded open set, and µ be a positive finite Borel measure on R^d with supp(µ) ⊆Ω and µ(Ω)>0. We assume that the Poincar´e inequality (PI) for µ holds: There exists a constant C >0 such that

Z

Ω

|u|²dµ≤C Z

Ω

|∇u|²dx, for all u∈C_c^∞(Ω). (1.2) (see, e.g., [8, 15, 17]) (PI) implies that each equivalence classu ∈H₀¹(Ω) contains a unique (in the L²(Ω, µ) sense) memberu that belongs toL²(Ω, µ) and satisfies both conditions below:

(1) there exists a sequence {u_n} in C_c^∞(Ω) such that un → u in H₀¹(Ω) and un → u in L²(Ω, µ);

(2) u satisfies (1.2).

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We call u the L²(Ω, µ)-representative of u. Define a mapping I : H₀¹(Ω) → L²(Ω, µ) by I(u) = u. It is easy to see that I is a bounded linear operator, but not necessarily injective.

Consider the subspaceN ofH₀¹(Ω) defined as N :=n

u∈H₀¹(Ω) :kI(u)k_L2(Ω,µ) = 0o .

It follows from the continuity of I that N is a closed subspace of H₀¹(Ω). Let N^⊥ be the orthogonal complement of N in H₀¹(Ω). Then I : N^⊥ → L²(Ω, µ) is injective. With a slight abuse of notation, we will denoteu by u.

Consider a nonnegative bilinear form E(·,·) in L²(Ω, µ) given by E(u, v) :=

Z

Ω

∇u· ∇v dx (1.3)

with domain domE =N^⊥. (PI) implies that (E,domE) is a closed quadratic form onL²(Ω, µ).

Hence these exists a nonnegative self-adjoint operator on L²(Ω, µ), which we denote by −∆_µ and call the (Dirichlet) Laplacian with respect to µ, such that domE = dom(−∆_µ)^1/2 and E(u, v) =h(−∆_µ)^1/2u,(−∆_µ)^1/2vi_L2(Ω,µ)for allu, v∈domE. Letu∈domE. Thenu∈dom∆_µ holds if and only if there existsf ∈L²(Ω, µ) such thatE(u, v) =hf, vi_L2(Ω,µ)for all v∈domE, where−∆_µu=f. We remark that if d= 1, then (PI) holds for any such µ, and thus −∆_µ is well defined.

We assume thatL²(Ω, µ) is infinite dimensional. It is known (see, e.g., [8]) that there exists an orthonormal basis{ϕ_n}^∞_n=1ofL²(Ω, µ) consisting of the eigenfunctions of−∆_µ. The eigenvalues λn=λn(−∆_µ) satisfy 0< λ1 ≤λ2 ≤ · · · and lim

n→∞λn=∞.

Let N(λ,−∆_µ) be the number of eigenvalues of −∆_µ (counting multiplicity) which do not exceedλ, i.e.,

N(λ,−∆_µ) := #{n:λ_n≤λ}, (1.4)

where #A denotes the cardinality of a set A. Define the lower and upper spectral dimensions of−∆_µ(or µ), respectively, as

d_s(−∆_µ) := lim

λ→∞

2 lnN(λ,−∆_µ)

lnλ and ds(−∆_µ) := lim

λ→∞

2 lnN(λ,−∆_µ) lnλ .

Ifd_s(−∆_µ) =d_s(−∆_µ), the common value, denoted d_s(−∆_µ) (or d_s(µ)), is called the spectral dimension of −∆_µ (or µ); it measures the asymptotic growth rate of the eigenvalue counting function as well as the magnitude of then-th eigenvalue.

(EFT) is introduced in [20]. Let µ = Pq

i=1µ_i be the graph-directed self-similar measure defined by the GIFS G = (V, E) on R^d, where V = {1, . . . , q} and E = {e_i : i ∈ V}. We say thatµ satisfies (EFT) (see Definition 2.1) if there exist a family of bounded open subsets {Ω_i}^q_i=1 with Ωi ⊆R^d, supp(µi)⊆Ωi, andµ(Ωi)>0, and a finite familyB:={B_1,`:`∈Γ}of measure disjoint cells,B_1,`⊆Ωi` for some i_` ∈V, such that for any `∈Γ, there is a family of µ-partitions {P_k,`}_k≥1 of B1,` satisfying the following conditions: (1)P1,` ={B_1,`}, and there exists someB ∈ P¹_2,` such that B 6= B1,`; (2) for any k ≥ 2, P¹_k+1,` contains all cells in P¹_k,`

that are µ-equivalent to some cell inB; (3) the sum of theµ-measures of those cells B ∈Pk,`

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that are notµ-equivalent to any cell in Btends to 0 as k→∞. In this case, we call{Ω_i}^q_i=1 an EFT-family, B a basic family of cells, and (B,P) := ({B_1,`},{P_k,`}_k≥1)`∈Γ a basic pair. We say that (B,P) isregular if each cell B ∈S

k≥1,`∈ΓP_k,` is connected, and for any`∈Γ, there exist some similitude τ_`, some Ω_j_`, and some constant w(`) > 0 such that τ_`(Ω_j_`) ⊆ B_1,` and µ≥w(`)µ◦τ_`⁻¹ onτ_`(Ω_j_`).

Let µ=Pq

i=1µ_i be the graph-directed self-similar measure defined by the GIFSG= (V, E) on R. Assume that µ satisfies (EFT) with {Ω_i}^q_i=1 being an EFT-family and assume that there exists a regular basic pair (B,P) := ({B_1,`},{P_k,`}_k≥1)`∈Γ. Then we can derive renewal equations for the eigenvalue counting functions, and express them in vector form as:

f =f ∗Mα+z, whereα∈R, and

f =f^(α)(t) = [f_`^(α)(t)]`∈Γ, t∈R;

Mα = [µ^(α)_`0`]_`,`⁰∈Γ is a finite matrix of Borel measures onR; z=z^(α)(x) = [z^(α)_` (x)]`∈Γ is a vector of error functions.

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Let

M_α(∞) :=h

µ^(α)_`0`(R)i

`,`⁰∈Γ. (1.6)

For each`∈Γ andα≥0, define F_`(α) :=X

`⁰∈Γ

µ^(α)_`0`(R), D_`:={α≥0 :F_`(α)<∞}, αe_`:= infD_`. (1.7) If the error functions decay exponentially to 0 ast→ ∞, then ds(µ) is given by the uniqueα such that the spectral radius ofMα(∞) is equal to 1.

Theorem 1.1. Let µ=Pq

i=1µi be a graph-directed self-similar measure defined by a strongly connected GIFSG= (V, E) on R. Assume that µ satisfies (EFT) with {Ω_i}^q_i=1 being an EFT- family and there exists a regular basic pair. Let Ω = Sq

i=1Ω_i, ∆_µ be the Dirichlet Laplacian defined by µ, and let M_α(∞), F_`(α) and α˜_` be defined as in (1.6) and (1.7). Assume that for each`∈Γ, lim

α→˜α⁺_`

F_`(α)>1.

(a) There exists a unique α >0 such that the spectral radius of Mα(∞) equals 1.

(b) If we assume, in addition, that for the uniqueαin (a), there existsσ >0such that for all

`∈Γ, z_`^(α)(t) =o(e^−σt) ast→ ∞, thends(µ) = 2α; moreover, ifMα(∞) is irreducible, then there exist positive constants C₁ and C₂ such thatC₁λ^α ≤N(λ,−∆_µ)≤C₂λ^α for all sufficiently large λ.

In section 5, we illustrate Theorem 1.1 by the strongly connected GIFS G = (V, E) with V ={1,2} and E ={e_i : 1 ≤i≤5}, wheree₁, e₃ ∈ E^1,1, e₂ ∈E^1,2, e₄ ∈E^2,2, e₅ ∈E^2,1. The six similitudes are defined by

S_e₁(x) =ρx, S_e₂(x) =rx+ρ(1−r), S_e₃(x) =rx+ (1−r),

Se4(x) =rx+ (1−r), Se5(x) =ρx, (1.8)

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where

ρ+ 2r−ρr≤1, (1.9)

i.e.,S_e₂(1)≤S_e₃(0) (see Figure 1).

Corollary 1.2. Let µ = µ₁+µ₂ be a graph-directed self-similar measure defined by a GIFS G = (V, E) in (1.8) together with a probability matrix (p_e)e∈E. Then there exists a unique positive real numberα satisfying

1−(p_e₄r)^α

(1−(p_e₁ρ)^α)(1−(p_e₃r)^α)− (p_e₁_e₃ +p_e₂_e₅)ρrα

− p_e₂_e₄_e₅ρr²α

= 0. (1.10) Moreover,d_s(µ) = 2α.

Theorem 1.3. Letµ=Pq

i=1µ_ibe a graph-directed self-similar measure onRdefined by a GIFS G= (V, E) that is not strongly connected. Assume thatGhasγ strongly connected components.

Form = 1, . . . , γ, let 2α_m be the spectral dimension of the graph-directed self-similar measure corresponding to the m^th strongly connected component, and let

SCm:={i∈V :iis contained in the m^th strongly connected component}. (1.11) Assume that µ satisfies (EFT) with {Ω_i}^q_i=1 being an EFT-family and there exists a regular basic pair. LetΩ =Sq

i=1Ωi and∆µ be the Dirichlet Laplacian defined byµ. LetMα(∞), F_`(α) andα˜` be defined as in (1.6)and (1.7). Assume that for m= 1, . . . , γ, each i∈SCm, and each

`∈Γi, lim

αm→˜α⁺_`

F`(αm)>1.

(a) There exists a unique α= max{α_m:m= 1, . . . , γ}>0 such that the spectral radius of Mα(∞) equals 1.

(b) If we assume, in addition, that for the unique α in (a), there exists σ > 0 such that for all ` ∈ Γ, z_`^(α)(t) = o(e^−σt) as t → ∞, then we have ds(µ) = 2α; moreover, if Mα(∞) is irreducible, then there exist positive constants C1 and C2 such that C1λ^α ≤ N(λ,−∆_µ)≤C₂λ^α for all sufficiently large λ.

In section 6, we illustrate Theorem 1.3 by a GIFSG= (V, E) that is not strongly connected, withV ={1,2} andE ={e_i : 1≤i≤5}, where e1, e2, e3∈E^1,1, e4∈E^2,2, e5 ∈E^2,1. The five similitudes are defined by

Se1(x) =ρx, Se2(x) =rx+ρ(1−r), Se3(x) =rx+ (1−r),

Se4(x) =rx+ (1−r), Se5(x) =ρx, (1.12)

whereρ+ 2r−ρr ≤1, i.e., S_e₂(1)≤ S_e₃(0) (see Figure 5). For a probability matrix (p_e)e∈E, we define

w(k) :=pe1

k

X

j=0

p^j_e₂p^k−j_e₃ , k≥0, (1.13) Corollary 1.4. Let µ=µ1+µ2 be a graph-directed self-similar measure defined by the GIFS G= (V, E) in (1.12) together with a probability matrix (pe)e∈E, and let w(k) be defined as in

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(1.13). Then there exists a unique positive real number α satisfying (p_e₂r)^α+ (p_e₃r)^α+ 1−(p_e₂r)^α

1−(p_e₃r)^α

∞

X

k=0

(w(k)ρr^k)^α = 1. (1.14) Moreover,d_s(µ) = 2α.

This paper is organized as follows. In Section 2, we give a modified version of the definition of (EFT). In Section 3, we introduce some properties of the eigenvalue counting function. In Section 4, we derive renewal equations and prove Theorem 1.1 and Theorem 1.3. Section 5 illustrates Theorem 1.1 by a class of one-dimensional strongly connected GIFSs defined as in (1.8); we also prove Corollary 1.2. In Section 6, we study the one-dimensional GIFSs in (1.12), which are not strongly connected, and prove Corollary 1.4.

2. Graph-directed iterated function systems and measures essentially of finite type

2.1. Graph-directed iterated function systems. Agraph-directed iterated function system (GIFS) of contractive similitudes is an ordered pairG= (V, E) described as follows (see [14]).

V is a set of vertices labeled by{1, . . . , q} and E is a set of directed edges with each begining and ending at a vertex. It is possible for an edge to begin and end at the same vertex and we allow more than one edge between two vertices. To each edge e∈E, there corresponds a contractive similitudeSe(x) :R^d→R^d defined as

Se(x) =ρeRex+be,

whereρe∈(0,1) is the contraction ratio,Reis an orthogonal transformation, andbe∈R^d. Let E^i,j denote the set of all edges that begin at vertexiand end at vertexj. We calle=e1. . . ek

a path with length k, if the terminal vertex of each edgeei (1 ≤i≤ k−1) equals the initial vertex of the edgee_i+1. It is well known that there exists a unique family of nonempty compact setsK₁, . . . , K_q satisfying

Ki=

q

[

j=1

[

e∈E^i,j

Se(Kj), i= 1, . . . , q. (2.15) Define

K:=

q

[

i=1

Ki. (2.16)

We callK the graph self-similar set associated with G = (V, E). Assume that for each edge e∈E, there corresponds a transition probabilityp_e>0, and the weights of all edges leaving a given vertexisum to 1, namely,

X

j∈V

X

e∈E^i,j

p_e= 1. (2.17)

Then for eachi∈V, there exists a unique Borel probability measuresµi such that µi=

q

X

j=1

X

e∈E^i,j

peµj◦S_e⁻¹. (2.18)

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We note that supp(µ_i) =K_i for alli∈V. Finally, letµ:=Pq

i=1µ_i and call it agraph-directed self-similar measure. We say that G = (V, E) satisfies the graph open set condition (GOSC) (see [24]) if there exists a family {O_i}^q_i=1 ⊆R^d of nonempty bounded open sets such that for alli= 1, . . . , q,

[

e∈E^i,j

S_e(O_j)⊆O_i and S_e(O_j)∩S_e⁰(O_j) =∅ for all distinct e, e⁰ ∈E^i,j.

It is obvious that Ki ⊆ Oi, i.e., supp(µi) ⊆ Oi. A GIFS, as well as any associated graph- directed self-similar measure, are said to be have overlaps if (GOSC) fails. Let {Ω_i}^q_i=1 be a family of nonempty bounded open subsets of R^d. We say that {Ω_i}^q_i=1 is invariant under the GIFS G = (V, E) if S

e∈E^i,jS_e(Ω_j) ⊆ Ω_i for i = 1, . . . , q. We say G is connected if for each pair of vertices i, j∈V, there is a (non-directed) path between them. G is said to bestrongly connected if for each pair of vertices i, j ∈V, there is a directed path from ito j. A strongly connected component of G is a maximal subgraph H of G such that H is strongly connected.

Strongly connected components are pairwise disjoint and do not necessarily cover G. A single vertex may be a strongly connected component if it loops to itself. In this paper, we assume that each graph has at least one strongly connected component.

2.2. The essentially finite type condition for the graph-directed self-similar measure.

Let Ω⊆R^dbe a bounded open subset andµbe a positive finite Borel measure with supp(µ)⊆Ω and µ(Ω)>0. We call a µ-measurable subset U of Ω is a cell (in Ω) ifµ(U)>0. Clearly, Ω itself is a cell.

We say that two cellsU andV areµ-equivalent, denoted byU '_µ,τ,w V (or simplyU '_µV), if there exist some similitudeτ :U →V and some constant w >0 such thatτ(U) =V and

µ|_V =wµ|_U◦τ⁻¹. (2.19)

It is easy to check that'_µ is an equivalence relation.

LetU ⊆Ω be a cell. Two cellsV, W inU aremeasure disjoint with respect toµifµ(V∩W) = 0. We call a finite familyPof measure disjoint cells a µ-partition ofU ifV ⊆U for allV ∈P, and µ(U) = P

V∈P

µ(V). A sequence of µ-partitions {P_k}_k≥1 is refining if for anyV ∈ P_k and any W ∈ Pk+1, either W ⊆ V or they are measure disjoint, i.e., each member of Pk+1 is a subset of some member ofPk.

Let B := {B_1,`}_`∈Γ be a finite family of measure disjoint cells in Ω, and for each `∈Γ, let {P_k,`}k≥1 be a family of refiningµ-partitions of B_1,` withP_1,`:={B_1,`}. We divide eachP_k,`, k≥ 2, into two (possibly empty) subcollections, P¹_k,` and P²_k,`, with respect to B, defined as follows:

P¹_k,`:=

B ∈P_k,`:B'_µB_1,i for somei∈Γ , P²_k,`:=P_k,`\P¹_k,`=

B ∈P_k,`:B /∈P¹_k,` . (2.20) Definition 2.1. We say that a graph-directed self-similar measure µ = Pq

i=1µi on R^d is essentially of finite type (EFT) if there exist a family of bounded open subsets {Ω_i}^q_i=1 with Ωi ⊆R^d, supp(µi)⊆Ωi and µ(Ωi)>0, and a finite family B:={B_1,`}_`∈Γ of measure disjoint

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cells,B_1,`⊆Ω_i_` for some i_`= 1, . . . , q, such that for any`∈Γ, there is a family of µ-partitions {P_k,`}_k≥1 of B_1,` satisfying the following conditions:

(1) P_1,`={B_1,`}, and there exists some B ∈P¹_2,` such that B6=B_1,`;

(2) if for some k≥2, there exists some B∈P¹_k,`, then B∈P¹_k+1,` and henceB ∈P¹_m,` for allm≥k;

(3) lim

k→∞

P

B∈P²_k,`

µ(B) = 0.

HereP¹_k,`andP²_k,`(k≥2) are defined as in (2.20). In this case, we call{Ω_i}^q_i=1anEFT-family, B a basic family of cells, and(B,P) := ({B_1,`},{P_k,`}_k≥1)`∈Γ a basic pair.

For k≥2 and`∈Γ, let P_k,`={B_k,`,i, i= 1,2, . . .}.

Definition 2.2. Assume that a graph-directed self-similar measureµ=Pq

i=1µ_isatisfies (EFT) with {Ω_i}^q_i=1 being an EFT-family and(B,P) := ({B_1,`},{P_k,`}_k≥1)`∈Γ being a basic pair. We say that (B,P) is regular if each cell B ∈S

k≥1,`∈ΓP_k,` is connected, and for any`∈Γ, there exist some similitude τ_`, some Ωj_` and some constant w(`) > 0 such that τ_`(Ωj_`) ⊆ B_1,` and µ≥w(`)µ◦τ_`⁻¹ onτ_`(Ωj_`). In this case, we call B a regular basic family of cells.

3. Eigenvalue counting function

3.1. Eigenvalue counting function on R. ( [20, Section 4.1] ) In this subsection, we only consider the one-dimensional eigenvalue counting function. Let (E,domE) be defined as in (1.3) with Ω = (a, b) and let −∆_µ be the associated Dirichlet Laplacian on L²((a, b), µ). Let P ={a_i}ⁿ⁺¹_i=0 be a partition of [a, b] satisfying

a0 :=a < a1 <· · ·< an+1 =:b fori∈ {0, . . . , n+ 1}.

Define F := F(P) = {u ∈ domE : u(ai) = 0 for alli = 0. . . , n+ 1}. Then F is a closed subspace of domE. Define a relation ∼_E on domE, induced by F, by u ∼_E v if and only if u−v∈ F. Then∼_E is an equivalence relation on domE. Define the quotient space

domE/F :={[u]_E :u∈domE},

where [u]_E is the equivalence class ofu. Define addition and scalar multiplication on domE/F as usual. For each i= 1, . . . , n, letfi be a function in domE that satisfies

fi(aj) =δij, i, j= 1, . . . ,2n,

whereδ_ij is the Kronecker delta. Such anf_i clearly exists. It is easy to prove that domE/F = span{[f_i]_E :i= 1, . . . , n} and dim(domE/F) =n.

Let −∆^F_µ|

(a,b) be the Laplacian defined by the Dirichlet form (1.3) with domE =F, and let N(λ,−∆^F_µ|

(a,b)) := #{n:λ_n(−∆^F_µ|

(a,b))≤λ} be the associated eigenvalue counting function. If

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F =N^⊥, whereN is defined as in Section 1, then N(λ,−∆^F_µ|

(a,b)) reduces toN(λ,−∆_µ|

(a,b)).

It follows from the variational formula that N(λ,−∆^F_µ|

(a,b))≤N(λ,−∆_µ|

(a,b))≤N(λ,−∆^F_µ|

(a,b)) + #P−2. (3.21)

If supp(µ) = [a, b], then N(λ,−∆^F_µ|

(a,b)) = Pn

i=0N(λ,−∆_µ|

(ai,ai+1)). Next, we state a similar formula. A proof can be found in [20, Proposition 4.1].

Proposition 3.1. Let µbe a continuous positive finite Borel measure on [a, b] withsupp(µ)⊆ [a, b]. Suppose there exists a nonempty subset Λ ⊆ {0,1, . . . , n} such that µ(a_i, a_i+1) > 0 for anyi∈Λ and µ(aj, aj+1) = 0 for anyj /∈Λ. Then

N(λ,−∆^F_µ|

(a,b)) =X

i∈Λ

N(λ,−∆_µ|

(ai,ai+1)).

3.2. Unitarily equivalent operators. In this subsection, we state a slightly modified version of [18, Propositions 2.2 and 2.3] below.

Proposition 3.2. ( [18, Proposition 2.2])LetS :R→Rbe a similitude, with Lipschitz constant r, such that S(a, b) = (c, d). Let µ be a continuous positive finite Borel measure on [a, b] with supp(µ)⊆[a, b]. Then

(a) −∆_µ◦S⁻¹_|

(c,d) and r⁻¹· −∆µ|_(a,b)

are unitarily equivalent.

(b) If, in addition,µ|_(c,d) =wµ◦S⁻¹ on (c, d) for some constantw >0, then −∆_ν|

(c,d) and (rw)⁻¹· −∆_µ|_(a,b)

are unitarily equivalent.

Note that unitarily equivalent operators have the same set of eigenvalues.

Proposition 3.3. ( [18, Proposition 2.3])Let µ,ν be continuous positive finite Borel measures on [a, b] and assume that there exists some constant w >0 such that µ ≤wν on [a, b]. Then for anyn≥1, λn(−∆_µ)≥w⁻¹·λn(−∆_ν).

The following result follows by combining Propositions 3.2 and 3.3. The proof can be found in [20].

Proposition 3.4. Let µ be a continuous positive finite Borel measure on R and assume that there exist a similitudeS with Lipschitz constant r, and a constantw >0 such thatS([a, b]) = [c, d]and µ≥wµ◦S⁻¹ on[c, d]. Then N(wrλ,−∆_µ|

(a,b))≤N(λ,−∆_µ|

(c,d)).

4. Renewal equation and Proof of Theorem 1.1 and Theorem 1.3 4.1. Renewal equation. Let µ = Pq

i=1µ_i be a graph-directed self-similar measure defined by G = (V, E) on R. In the rest of this section, assume that µ satisfies (EFT) with {Ω_i}^q_i=1 being an EFT-family, with (B,P) := ({B_1,`},{P_k,`}_k≥1)`∈Γ being a regular basic pair. The regularity of (B,P) implies that each cell B ∈ S

k≥1,`∈ΓP_k,` is an interval. This allows us to apply Propositions 3.1–3.4. Let

Γi :={`∈Γ :B1,`⊆Ωi} and Bi :={B_1,`:`∈Γi} for i∈V. (4.22)

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Then Γ = Sq

i=1Γ_i and B = Sq

i=1B_i. Note that Γ_i and B_i maybe empty. The following Proposition has been modified from [20, Proposition 4.5] to suit our purpose. The proof is similar.

Proposition 4.1. Let µ=Pq

i=1µ_i be a graph-directed self-similar measure defined by a GIFS G = (V, E) on R. Assume that µ satisfies (EFT) with{Ω_i}^q_i=1 being an EFT-family and with B := {B_1,` :` ∈Γ} being a regular basic family of cells. Let Ω = Sq

i=1Ωi, and Γi, Bi defined as in (4.22). Then for i∈V and any `∈Γi, there exists some constantc_` >0 such that

N(λ,−∆_µ_i_|_B

1,`)≤N(λ,−∆_µ|_Ω)≤N c_`λ,−∆_µ_i_|_B

1,`

. (4.23)

Proposition 4.1 implies that the asymptotic behavior of N(λ,−∆_µ) is controlled by that of N(λ,−∆_µ

i|_B

1,`) for i∈V and `∈Γ_i.

Step 1. Derivation of functional equations. For `∈Γi and k≥2, letP¹_k,` and P²_k,` be defined as in (2.20) with respect to B, where i∈ V. Without loss of generality, we may assume that Γi can be partitioned into two (possibly empty) sub-collections, Γ⁰_i and Γ^∗_i, defined as follows.

An index ` ∈ Γ_i belongs to Γ⁰_i if there exists some integer k satisfying P²_k,` = ∅. Let κ_` ≥ 2 (depending on `) denote the smallest of suchk. Define Γ^∗_i := Γ_i\Γ⁰_i and letκ_` :=∞ for`∈Γ^∗_i. Let Γ⁰ =Sq

i=1Γ⁰_i and Γ^∗=Sq

i=1Γ^∗_i. Then Γ = Γ⁰∪Γ^∗.

For i∈ V, fix any ` ∈Γ_i. The definition of (EFT) implies that for any 2≤ k ≤ κ_`, there exist two finite disjoint G¹_k,`, G²_k,`⊆Nsuch that

P¹_k,`=

k

[

m=2

B_m,`,p:p∈G¹_m,` and P²_k,`=

B_k,`,p:p∈G²_k,` .

Condition (1) of (EFT) implies that G¹_2,` 6= ∅. If `∈ Γ^∗, condition (3) of (EFT) implies that

k→∞lim P

p∈G²_k,`

µ(Bk,`,p) = 0.

Proposition 4.2. Assume that µsatisfies (EFT). Let `∈Γi, and J_`:=

j∈V :S_e(Ω_j)⊆B_1,` for e∈E^i,j , (4.24) where i∈V. Let2≤k≤κ_`. IfG¹_k,`6=∅, then for each p∈G¹_k,`, there exist someξ(k, `, p)>0 and c(k, `, p)∈Γj,j ∈J_`, such that

N(λ,−∆_µ_i_|

Bk,`,p) =N(ξ(k, `, p)λ,−∆_µ_j_|_B

1,c(k,`,p)). (4.25)

Proof. For any p ∈ G¹_k,`, by the definition of P¹_k,`, there exist some similitude S_e(k,`,p) with Lipschitz constant r_e(k,`,p), as well as constants w(k, `, p) > 0 and c(k, `, p) ∈ Γ_j such that µ_i|_B_k,`,p =w(k, `, p)µ_j|_B

1,c(k,`,p)◦S_e(k,`,p)⁻¹ , wherej∈J_`. Combining this with Proposition 3.2(b),

we obtain (4.25) with ξ(k, `, p) :=w(k, `, p)r_e(k,`,p).

For all i∈V, each `∈Γi, and 1≤n≤κ_`, we define a partitionP_n,` of B_1,` as follows:

P_n,` :=

x:x is an end-point of some interval inPn,` ,

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and let F_n,` := F(P_n,`). Note that for all i ∈ V, any ` ∈ Γ_i and 2 ≤ n ≤ κ_`, we have

#P_n,`≤2#P_n,`. It follows from Proposition 3.1 that for i∈V,`∈Γ_i, and 2≤n≤κ_`, N(λ,−∆^F_µ^n,`

i|_B

1,`

) =

n

X

k=2

X

p∈G¹_k,`

N(λ,−∆_µ_i_|

Bk,`,p) + X

p∈G²_n,`

N(λ,−∆_µ_i_|

Bn,`,p).

Combining this with (3.21) and Proposition 4.2, for any`∈Γ⁰_i, N(λ,−∆_µ

i|_B

1,`) =

κ_`

X

k=2

X

p∈G¹_k,`

N(ξ(k, `, p)λ,−∆_µ

j|_B

1,c(k,`,p)) +(κ_`, `), (4.26) where 0≤(κ`, `)≤2#Pκ_`,`−2. Similarly, for `∈Γ^∗_i and n≥2, we have

N(λ,−∆_µ_i_|_B

1,`) =

n

X

k=2

X

p∈G¹_k,`

N(ξ(k, `, p)λ,−∆_µ_j_|_B

1,c(k,`,p))

+ X

p∈G²_n,`

N(λ,−∆_µ_i_|

Bn,`,p) +(n, `),

(4.27)

where 0≤(n, `)≤2#P_n,`−2.

Step 2. Derivation of the vector-valued equation.

Case (1). G is strongly connected. For all i∈V, each `∈Γi, and α >0, define f_`(t) =f_`^(α)(t) :=e^−αtN(e^t,−∆_µ_i_|

B1,`), t∈R. (4.28)

Let λ=e^t. Then e^−αtN(βλ,−∆_µ_i_|_B

1,`) =β^αf`(t+ lnβ) for anyβ >0. Now, multiply both sides of (4.26) and (4.27) bye^−αt. Then for`∈Γ⁰_i, we have

f_`(t) =

κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^αf_c(k,`,p) t+ lnξ(k, `, p)

+z^(α)_` (t), (4.29)

wherez_`^(α)(t) :=e^−αt(κ_`, `). For `∈Γ^∗_i andn≥2, we obtain f_`(t) =

∞

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^αf_c(k,`,p) t+ lnξ(k, `, p)

+z_`^(α)(t)

−

∞

X

k=n+1

X

p∈G¹_k,`

ξ(k, `, p)^αf_c(k,`,p) t+ lnξ(k, `, p) ,

(4.30)

where

z_`^(α)(t) :=e^−αt

X

p∈G²_n,`

N(λ,−∆_µ_i_|

Bn,`,p) +(n, `)

. (4.31)

Since λ1(−∆_µ_i_|_B

1,`) >0 for any i∈V and any `∈Γi, there exists t0 ∈R such that f_`(t) = 0 for anyt < t0 and any `∈Γi. For each t ∈R, all i∈V and all `∈Γ^∗_i, let nt := nt(`) be the positive integer such that

t+ max

lnξ(k, `, p) :p∈G¹_k,` < t₀ for allk > n_t. (4.32)

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Letn=n_t in (4.30). Then f`(t) =

∞

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^αf_c(k,`,p) t+ lnξ(k, `, p)

+z^(α)_` (t) for`∈Γ^∗_i, (4.33)

wherez_`^(α)(t) is obtained from that in (4.31) by replacingnwith nt. Fori∈V,`∈Γi, letµ^(α)_`0`

be the discrete measure such that µ^(α)_`0` −lnξ(k, `, p)

:=ξ(k, `, p)^α for 2≤k≤κ_`, p∈G¹_k,`, `⁰=c(k, `, p)∈Γj and j∈J_`. (4.34) Then (see (1.7))

µ^(α)_`0`(R) =

κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α and

F_`(α) =X

`⁰∈Γ κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α.

Case (2). G is not strongly connected. If G = (V, E) is not strongly connected, then there exists somei, j ∈V satisfying E^i,j =∅. That is,S

`∈Γ_iJ_` =∅. Assume thatG hasγ strongly connected components. For m= 1, . . . , γ, letSC_m be defined as in (1.11).

For m= 1, . . . , γ, each i∈SCm and each `∈Γi, define f_`(t) =f_`^(α^m⁾(t) :=e^−α^m^tN(e^t,−∆_µ

i|_B

1,`), α_m>0, t∈R. (4.35) Let λ= e^t. Then e^−α^m^tN(βλ,−∆_µ_i_|_B

1,`) = β^α^mf_`(t+ lnβ) for any β > 0. Now, multiply both sides of (4.26) by e^−α^m^t. Then for`∈Γ⁰_i, we have

f_`(t) =

κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α^mf_c(k,`,p) t+ lnξ(k, `, p)

+z^(α_` ^m⁾(t), (4.36)

where z_`^(α^m⁾(t) :=e^−α^m^t(κ_`, `). Similarly, for`∈Γ^∗_i and n≥2, we obtain f`(t) =

∞

X

k=2

X

p∈G¹_k,`

+z_`^(α^m⁾(t)

−

∞

X

k=n+1

X

p∈G¹_k,`

ξ(k, `, p)^α^mf_c(k,`,p) t+ lnξ(k, `, p) ,

(4.37)

where

z_`^(α^m⁾(t) :=e^−α^m^t X

p∈G²_n,`

N(λ,−∆_µ_i_|

Bn,`,p) +(n, `)

. (4.38)

Since λ1(−∆_µ_i_|_B

1,`) > 0 for all i∈ V and all ` ∈ Γi, there exists t0 ∈ R such that f`(t) = 0 for any t < t0 and any `∈Γi. For each t∈ R, all i∈V and all `∈Γ^∗_i, let nt := nt(`) be the positive integer such that

t+ max

lnξ(k, `, p) :p∈G¹_k,` < t₀ for allk > n_t. (4.39)

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Letn=n_t in (4.37). Then for m= 1, . . . , γ, eachi∈SC_m and `∈Γ^∗_i, we have

f_`(t) =

∞

X

k=2

X

p∈G¹_k,`

+z_`^(α^m⁾(t), (4.40)

where z^(α_` ^m⁾(t) is obtained from that in (4.38) by replacing n with n_t. For m = 1, . . . , γ, i∈SC_m,`∈Γ_i, let µ^(α_`0`^m⁾ be the discrete measure such that

µ^(α_`0`^m⁾ −lnξ(k, `, p)

:=ξ(k, `, p)^α^m for 2≤k≤κ_`, p∈G¹_k,`, `⁰ =c(k, `, p)∈Γ_j and j∈J_`. (4.41) Then (see (1.7))

µ^(α_`0`^m⁾(R) =

κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α^m

and

F_`(α_m) = X

`⁰∈Γ κ_`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α^m.

We summarize the above derivations in the following theorem.

Theorem 4.3. Let µ= Pq

i=1µi be a graph-directed self-similar measure on R. Assume that µ satisfies (EFT) with{Ω_i}^q_i=1 being an EFT-family and there exists a regular basic pair. Let Ω =Sq

i=1Ωi and ∆µ is defined on Ω. Let f,Mα andz be defined as in (1.5). Thenf satisfies the vector-valued renewal equation f =f∗Mα+z.

4.2. Proof of Theorem 1.1 and Theorem 1.3.

Proof of Theorem 1.1. (a) Since eachF_`(α) is a strictly decreasing positive continuous function ofα, lim

α→∞F_`(α) = 0, and lim

α→αe⁺_`

F_`(α)>1, there exists a uniqueαsuch that the spectral radius ofM_α(∞) is 1.

(b) Let α be the unique number in part (a). Letm:= [m^(α)_`0`] = [R∞

0 x dµ^(α)_`0`] be the moment matrix. Following the proof of [18, Theorem 1.1(b)], we need to show that some moment condition holds, and it suffices to show that 0 < P

`⁰∈Γ

m^(α)_`0` < ∞. It is easy to check that for

`∈Γ, P

`⁰∈Γ

m^(α)_`0` takes the following values:

X

`⁰∈Γ κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α

ln(ξ(k, `, p)) .

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It follows from lim

α→αe⁺_`

F_`(α)>1 that there exists >0 such that 0< F_`(α−)<∞. Thus

0<X

`⁰∈Γ κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α

ln(ξ(k, `, p))

=X

`⁰∈Γ κ`

X

k=2

X

p∈G¹_k,`

ξ(k, `, p)^α−ξ(k, `, p)

ln(ξ(k, `, p))

<∞.

The last inequality follows from the fact lim

t→0⁺tlnt= 0. By (4.34), we have P

`⁰∈Γ

µ^(α)_`0`(0) = 0<

P

`⁰∈Γ

µ^(α)_`0`(∞), i.e., each column ofMα is nondegenerate at 0. From Theorem 4.3,f =f∗Mα+z, where, by assumption,z is directly Riemann integrable onR.

We first consider the case M_α(∞) is irreducible. It follows from the above observations and [18, Theorem 4.1] that there exist positive constants C₁ and C₂ such that 0 < C₁ ≤

lim

t→∞

f_`(t) ≤ lim

t→∞f_`(t) ≤ C₂ < ∞ for all ` ∈ Γ. The definition in (4.28) implies that C₁ ≤ λ^−αN(λ,−∆_µ_i_|

B1,`) ≤ C₂, which, together with (4.23), yields C₁λ^α ≤ N(λ,−∆_µ|

Ω) ≤ C₂λ^α. Combining this, part (a), and the definition of d_s(µ), we get d_s(µ) = 2α.

It remains to consider the case M_α(∞) is reducible. As in the proof of [18, Theorem 1.1(b), Case 2], we have

t→∞lim f_`^(β)(t) = 0 for all `∈Γi, i∈ {1, . . . , q} and allβ < α.

Moreover, there exists some `₀ ∈Γ_i such that lim

t→∞

f_`^(α)

0 (t) >0. The definition of f_`(t) implies that N(λ,−∆_µ

i|_B

1,`) = o(λ^β) for ` ∈ Γ_i and β < α and lim

λ→∞

λ^−αN(λ,−∆_µ

i|_B

1,`0

) > 0. Thus N(λ,−∆_µ) = o(λ^β) for any β < α and lim

λ→∞

λ^−αN(λ,−∆_µ) > 0. Hence ds(−∆_µ) ≤ 2α and d_s(−∆_µ)≥2α, which completes the proof.

Proof of Theorem 1.3. (a) We observe that each F_`(α_m) is a strictly decreasing positive continuous function ofα_m, lim

αm→∞F_`(α_m) = 0, and lim

αm→αe⁺_`

F_`(α_m)>1. Thus there exists a unique α= max{α_m:m= 1, . . . , γ}such that the spectral radius of Mαm(∞) is 1.

(b) The proof is similar to that of Theorem 1.1(b).

5. Strongly connected GIFSs on R

In this section, we compute the spectral dimension of some graph-directed self-similar measures defined by strongly connected GIFSsG= (V, E), which have overlaps.