SPECTRAL ASYMPTOTICS OF ONE-DIMENSIONAL FRACTAL LAPLACIANS IN THE ABSENCE OF SECOND-ORDER IDENTITIES

. We observe that some self-similar measures deﬁned by ﬁnite or inﬁnite iterated function systems with overlaps are in certain sense essentially of ﬁnite type, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian deﬁned by a self-similar measure satisfying this condition. For Laplacians deﬁned by fractal measures with over- laps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu [24].


1.
Introduction. The origin of spectral asymptotics can be traced back to the work of Weyl. Let U ⊂ R d be a bounded domain with smooth boundary and with volume vol(U ), ∆ be the Dirichlet Laplacian on U , {λ n } be the eigenvalues of −∆, and N (λ, −∆) be the number of eigenvalues less than or equal to λ. In a seminal work started in 1911, Weyl [40] proved the following asymptotic formula, known as the Weyl law :

SZE-MAN NGAI, WEI TANG AND YUANYUAN XIE
where ω d is the volume of the unit ball in R d . Using this formula he proved a conjecture posed independently by A. Sommerfeld and physicist H. A. Lorentz, which states that the density of standing electromagnetic waves in a bounded cavity U is, at high frequencies, independent of the shape of U . This formula has generated an enormous amount of work, both in Euclidean domains and manifolds, most notably the work concerning Weyl's conjecture on the remainder estimate [3,7,15,25,37]. For domains with fractal boundaries, remainder estimate, in terms of the Minkowski dimension of the boundary, was obtained by Lapidus [20]. For Dirichlet Laplacians −∆ µ on domains defined by a measure, we would like to obtain a crude analogue of (1.1) of the form C 1 λ ds/2 ≤ N (λ, −∆ µ ) ≤ C 2 λ ds/2 for all sufficiently large λ, where d s = d s (−∆ µ ) is the spectral dimension of −∆ µ (or simply of µ) (see definition below). Spectral dimension has been computed by McKean and Ray [29] for the Cantor measure, by Fujita [10] and Naimark and M. Solomyak [30] for selfsimilar measures satisfying the open set condition (OSC) (see [14] and definition below), and by Freiberg [9] for generalized measure geometric Laplacians on Cantor like sets. Kigami and Lapidus [19] computed the spectral dimension of Laplacians on postcritically finite self-similar sets with a harmonic structure. Hambly and Nyberg [12] obtained spectral dimension for a finitely ramified graph-directed fractal that admits a Laplacian. Spectral dimension on random fractals have also been studied by Croyden and Hambly [4,11]. Eigenvalue asymptotics of the Hanoi attractors have been obtained recently by Alonso-Ruiz and Freiberg [1]. Asymptotics of spectral partition functions associated with self-similar sets have been studied by Kajino [17,18].
Throughout this paper, unless stated otherwise, an iterated function system (IFS) refers to a finite or countably infinite family of contractive similitudes {S i } i∈Λ defined on a compact subset X of R d . To avoid triviality, we assume throughout this paper that the cardinality of the limit set (see Section 2) is at least 2. If necessary, we use FIFS and IIFS respectively to distinguish between finite and infinite IFSs. IFSs are used to generate fractal sets and measures. It is well-known that an FIFS and a probability vector together determine a unique probability measure, called a self-similar measure. However, for an IIFS, the existence of self-similar measures needs some additional assumptions (see [27] and Proposition 6.14). Throughout this paper, we assume that to each IIFS and each probability vector, there corre- (OSC) is a separation condition. If it fails, we say that the IFS, as well as any associate self-similar measure, has overlaps. In this case, it is much harder to compute the spectral dimension. The first author [31] computed the spectral dimension of a class of one-dimensional self-similar measures satisfying second-order identities. These identities were first introduced by Strichartz and are used in [38] to approximate the density of the infinite Bernoulli convolution associated with the golden ratio. However, very few self-similar measures are known to satisfy second-order identities. In fact, for the class of symmetric infinite Bernoulli convolutions with overlaps, only the one associated with the golden ratio has been verified rigorously to satisfy this condition. Other examples are all defined by IFSs with contraction ratios equal to the reciprocal of an integer. This includes a class of convolutions of the Cantor measure. To the best of the authors' knowledge, in the absence of second-order identities, the spectral dimension of Laplacians defined by IFSs with overlaps has not been obtained before, and this is a main motivation of this paper.
For convenience, we summarize the definition of the Dirichlet Laplacian on a bounded domain defined by a measure; details can be found in [13]. Let U ⊆ R d be a bounded open subset and µ be a positive finite Borel measure with supp(µ) ⊆ U and µ(U ) > 0. We assume that µ satisfies the Poincaré inequality (PI) for measures: There exists a constant C > 0 such that (see, e.g., [13,26,30]). (PI) implies that each equivalence class u ∈ H 1 0 (U ) contains a unique (in the L 2 (U, µ) sense) memberû that belongs to L 2 (U, µ) and satisfies both conditions below: (1) there exists a sequence {u n } in C ∞ c (U ) such that u n →û in H 1 0 (U ) and u n →û in L 2 (U, µ); (2)û satisfies inequality (1.2).
Theorem 1.1. Let µ be a continuous positive finite Borel measure on R. Assume that µ satisfies (EFT) with Ω being an EFT-set on which ∆ µ is defined, and assume that there exists a regular pair with respect to Ω. 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 M α (∞) is equal to 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 is irreducible, then there exist positive constants C 1 and C 2 such that Let µ be a self-similar measure defined by a finite type IFS on R d . In Section 2, we define the set of all level-k islands (see Definition 2.6). Roughly speaking, a level-k island corresponds to a connected component of the level-k iterates of some fixed connected open set Ω. Two islands are of the same measure type (with respect to µ) if their corresponding connected components are µ-equivalent. For FIFSs, we give a sufficient condition for (EFT) with a regular basic family of cells (see Proposition 2.15 for details). In Section 5, we illustrate Theorem 1.1 by the following family of FIFSs: where the contraction ratios r 1 , r 2 ∈ (0, 1) satisfy r 1 + 2r 2 − r 1 r 2 ≤ 1, i.e., S 2 (1) ≤ S 3 (0) (see Figure 1). The Hausdorff dimension of each self-similar set in the family is computed in [23]. This family is also used as basic examples of IFSs of general finite type [16,22]. The multifractal properties of the corresponding self-similar measures are recently studied by Deng and two of the authors [6,34].
In Section 6, we study (EFT) for IIFSs, which is more complicated because of the presence of the so-called "tails" (see Definition 6.1). We give an analogous sufficient condition for a self-similar measure defined by an IIFS on R to satisfy (EFT) with a regular basic family of cells (see Proposition 6.4).
be a probability vector, µ be the associated self-similar measure, and ∆ µ be defined on Ω = (0, 1). Assume that there exists some integer L ≥ 2, which is chosen to be the minimal one, such that p 2i p 2i+1 : Let M α (∞) be defined as in (1.6).
We state some open problems in Section 7. Finally, we include a vector-valued renewal theorem proved by Lau, Wang and Chu [24] in the Appendix for convenience.
2. Self-similar measures and measures that are essentially of finite type.
2.1. Finite type condition and measure type. Let X be a compact subset of R d with nonempty interior, and {S i } i∈Λ be an IFS of contractive similitudes on X with limit set K ⊆ R d . If Λ is finite, K is the unique compact subset satisfying K = i∈Λ S i (K), called the attractor or self-similar set of the IFS. For IIFSs, K need not be compact (see [27]). If Λ is finite, then to each probability vector (p i ) i∈Λ (i.e., p i > 0 and i∈Λ p i = 1), there corresponds a unique probability measure, called a self-similar measure, satisfying the self-similar identity Moreover, supp(µ) = K. An analogous result, with supp(µ) = K, holds for IIFSs under additional assumptions (see [27] for IIFSs satisfying (OSC) and Proposition 6.14 for IIFSs studied in this paper). We assume that #K ≥ 2. It is well-known that in this case µ is continuous, and is in fact of pure type (see, e.g., [35]). We first extend the finite type condition [16,22,33] to IIFSs and then introduce the concept of measure type of an island. The finite type condition for FIFSs is a weaker notion of separation under which the dimension of the attractor can be computed in terms of the spectral radius of some weighted incidence matrix. We remark that the term island is adopted from [2].
We extend the finite type condition to include IIFSs. Define the following sets of indices . We call i = (i 1 , . . . , i k ) ∈ Λ k a word of length k, and denote its length by |i|. If no confusion is possible, we will denote i = (i 1 , . . . , i k ) simply by i := i 1 · · · i k ; in particular, if i j = i 1 for all j = 1, . . . , k, we write i =: i k 1 . For k ≥ 0 and i = (i 1 , . . . , i k ) ∈ Λ k , we use the standard notation We also let M 0 := {∅}.
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 lim k→∞ m k = lim k→∞ m k = ∞; (2) for each k ≥ 1, M k is an antichain in Λ * ; (3) for each j ∈ Λ * with |j| > m k or j ∈ M k+1 , there exists i ∈ M k such that i j; (4) for each j ∈ Λ * with |j| < m k or j ∈ M k−1 , there exists i ∈ M k such that j i; (5) there exists a positive integer L 0 , independent of k, such that for all i ∈ M k and j ∈ M k+1 with i j, we have |j| − |i| ≤ L 0 .
Condition (2) means that the indices in M k are incomparable. We also remark that (4) actually follows from (3). Clearly, by letting M k = Λ k for all k ≥ 0, we obtain an example of a sequence of nested index sets.
To define neighborhood types, we fix a sequence of nested index sets {M k } ∞ k=0 . For each integer k ≥ 0, let V k be the set of level-k vertices (with respect to {M k }) defined as We call (id, 0) the root vertex and denote it by v root . Let V := k≥0 V k be the set of all vertices. For v = (S i , k) ∈ V k , we use the convenient notation S v := S i and More generally, for any k ≥ 0 and any subset A ⊂ V k , we use the notation Let Ω ⊆ X be a nonempty open set which is invariant under {S i } i∈Λ , i.e., i∈Λ S i (Ω) ⊆ Ω. Such an Ω exists by our assumption; in particular, X • is such a set. Two level-k vertices v, v ∈ V k (allowing v = v ) are said to be neighbors (with respect to Ω and {M k }) if S v (Ω) ∩ S v (Ω) = ∅. We call the set of vertices Define an equivalence relation on the set of vertices V. Let S : → X, the following conditions hold: (2) for u ∈ N(v) and u ∈ N(v ) such that S u = τ S u , and for any positive It is straightforward to show that ∼ is an equivalence relation. We define an infinite graph G with vertex set V and directed edges as follows.
Then we connect a directed edge l : v → u. We call v a parent of u and u an offspring of v. We write G = (V, E), where E is the set of all directed edges defined above. We call v = (S i , k) a predecessor of u = (S j , k ), and u a descendent of v, if i j and k ≥ k + 1. Proposition 2.3. For two equivalent vertices v ∈ V k and v ∈ V k , let {u i } i∈Λ1 and {u i } i∈Λ 1 be the offspring of v and v in G, respectively. Then, counting multiplicity, In particular, #Λ 1 = #Λ 1 .
Proof. The proof is similar to that of [14, Proposition 2.4(b)]; we give an outline as it is needed for the proof of Proposition 2.7.
Step 1. Let i, j ∈ Λ 1 and assume that i 1 , −→ w . One can show that u = w if and only if u = w , and u, w are neighbors if and only if u , w are.
Step 2. Let U and U be the sets of offspring of the vertices in N(v) and N(v ) respectively. Define a mapτ : U → U as follows. Suppose u is an offspring of v j in G by an edge i for some j ∈ Λ 2 . Then letτ (u) be the offspring of v j by the edge i. The definition of ∼ and Step 1 imply thatτ is well defined and bijective. Moreover, Step 1 also shows that u is an offspring of v in G if and only ifτ (u) is an offspring of v in G. Combining this with Remark 2.2, we have [u i ] = [τ (u i )], which completes the proof. Definition 2.4. 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 of finite type (or that it satisfies the finite type condition) if there exist 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 an Ω a finite type condition set (or FTC-set).   We say that two islands I ∈ I k and I ∈ I k are equivalent, and denote it by I ≈ τ I (or simply I ≈ I ), if there exists some τ ∈ S such that {S v : v ∈ I } = {τ S v : v ∈ I} and, moreover, v ∼ τ v for any v ∈ I and v ∈ I satisfying S v = τ S v . We denote the equivalence class of I by [I] and we call [I] the (island) type of I.
An island I ∈ I k is said to be a parent of an island I ∈ I k+1 , and I an offspring of I, if each v ∈ I has a parent in I. be the collection of all offspring of I. Analogously, we define predecessor and descendant of an island. 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 and v ∈ V k are µ-equivalent, As ∼ is an equivalence relation, so is ∼ µ . Denote the µ-equivalence class of v by [v] µ and call it the (neighborhood) measure type of v (with respect to Ω, {M k } and µ). Intuitively, v ∼ µ v means that the measures µ| S N(v) (Ω) and µ| S N(v ) (Ω) have the same structure. The following proposition shows that µ-equivalent vertices generate the same number of offspring of each neighborhood measure type.
Proof. Let u and u be offspring of v and v in G by an edge i, respectively. By the proof of Proposition 2.3, we have u ∼ τ u , where τ : Definition 2.8. Let µ be a self-similar measure defined by a finite type IFS {S i } i∈Λ on R d with Ω being an FTC-set. Two islands I ∈ I k and I ∈ I k are said to be µ-equivalent, denoted I ≈ µ,τ,w I (or simply I ≈ µ I ), if I ≈ τ I for some τ ∈ S and there exists some w > 0 such that (2.5) We remark that (2.5) holds if and only if v ∼ µ,τ,w v for any v ∈ I and v ∈ I satisfying S v = τ S v . We note that ≈ µ is an equivalence relation. We denote the µ-equivalence class of I by [I] µ , and call [I] µ the (island) measure type of I (with respect to Ω, {M k } and µ). From the definition of ≈ µ , we obtain an analog of Proposition 2.7 concerning ≈ µ . That is, µ-equivalent islands generate the same number of offspring of each island measure type. We remark that, by definition, for any ≥ 2, I is a level-nonbasic island with respect to B if and only if [I] µ / ∈ B µ , and there exists a finite sequence of islands I 1 , . . . , I −1 such that I 1 ∈ B, I i ∈ O(I i−1 ) is a level-i nonbasic island with respect to B for all i = 2, . . . , − 1, and I ∈ O(I −1 ).
Analogously, we define the equivalence and µ-equivalence of two subsets B ⊆ I k and B ⊆ I k , denoted B≈ τ B (or simply B≈ B ) and B≈ µ,τ,w B (or simply B≈ µ B ), respectively. Moreover, we denote the equivalence and µ-equivalence class of B by [B] and [B] µ , respectively.

Measures essentially of finite type.
Let Ω ⊆ R d be a bounded open subset and µ be a positive finite Borel measure with supp(µ) ⊆ Ω and µ(Ω) > 0. We call We say that two cells U and V are µ-equivalent, denoted U µ,τ,w V (or simply U µ V ), if there exist some similitude τ : U → V and some constant w > 0 such that τ (U ) = V and (2.6) It is easy to check that µ is an equivalence relation.
Let U ⊆ Ω be a cell. We call a finite family P of measure disjoint cells a µpartition of U if V ⊆ U for all V ∈ P, and µ(U ) = V ∈P µ(V ). A sequence of µ-partitions (P k ) k≥1 is refining if for any V ∈ P k and any W ∈ P k+1 , either W ⊆ V or they are measure disjoint, i.e., each member of P k+1 is a subset of some member of P k . 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 refining µ-partitions of B 1, with P 1, := {B 1, }.
We divide each P k, , k ≥ 2, into two (possibly empty) subcollections, P 1 k, and P 2 k, , with respect to B, defined as follows: (2.7) Definition 2.11. We say that a positive finite Borel measure µ on R d is essentially of finite type (EFT) if there exist a bounded open subset Ω ⊆ R d with supp(µ) ⊆ Ω and µ(Ω) > 0, and a finite family B := {B 1, : ∈ Γ} of measure disjoint cells in Ω such that for any ∈ Γ, there is a family of refining µ-partitions (P k, ) k≥1 of B 1, satisfying the following conditions: (1) P 1, = {B 1, }, and there exists some B ∈ P 1 2, such that B = B 1, ; (2) if for some k ≥ 2, there exists some B ∈ P 1 k, , then B ∈ P 1 k+1, and hence B ∈ P 1 m, for all m ≥ k; Here P 1 k, and P 2 k, (k ≥ 2) are defined as in (2.7). In this case, we call Ω an EFTset, B a basic family of cells (in Ω), and (B, P) := ({B 1, }, (P k, ) k≥1 ) ∈Γ a basic pair (with respect to Ω). Remark 2.12. (a) By definition, we see that if µ satisfies (EFT) with Ω being an EFT-set and with (B, P) = ({B 1, }, (P k, ) k≥1 ) ∈Γ being a basic pair (with respect to Ω), then for any Ω ⊆ R d with Ω ⊆ Ω , µ satisfies (EFT) with Ω being an EFT-set and (B, P) a basic pair (with respect to Ω ). (b) We remark that conditions (1) and (2) are needed in Section 4 to derive the vector-valued renewal equation, and error estimate forces condition (3) to hold. In fact, in Section 4, we only need condition (2) as well as (1') the existence of some B ∈ k≥2 P 1 k, such that B = B 1, . Since condition (3) implies that k≥2 P 1 k, = ∅, we have chosen to use the more convenient condition (1). (c) Let (B, P) := ({B 1, }, (P k, ) k≥1 ) ∈Γ be a basic pair. Then for any k ≥ 2, P 2 k, = ∅ if and only if P m, = P k, for all m ≥ k. (d) For any k ≥ 2 and any B ∈ P 2 k, , P k+1, \ P 1 k, contains a unique µ-partition of B.
We remark that in one-dimension, the regularity of B ensures that the eigenvalue counting function of the Laplacian on Ω behaves the same as that on each cell in B (see Proposition 4.5).
Proposition 2.14. Let {S i } i∈Λ be a finite type IFS on R d , (p i ) i∈Λ be a probability vector, and µ be the associated self-similar measure. Then for any I ∈ I, there exist some similitude τ and some constant w > 0 such that τ (Ω) ⊆ S I (Ω) and Thus the result follows. As measures studied in this paper are mainly self-similar, we give a sufficient condition for FIFSs to satisfy (EFT). An analog for IIFSs will be given in Proposition 6.4. Proposition 2.15. Let µ be a self-similar measure defined by a finite type FIFS on R d with a connected FTC-set Ω. Suppose there exists some m ≥ 0 such that the following two conditions hold.
(1) There exists a finite index set Γ such that I m = {I 1, : ∈ Γ}; moreover, for each ∈ Γ, there exist some constant c( ) ≥ 2 (chosen to be the minimum) and descendant J ∈ I m+c( )−1 of I 1, satisfying S J (Ω) = S I 1, (Ω) and J ≈ µ I 1,i for some i ∈ Γ. (2) For k ≥ 2, let I k be the collection of all level-k nonbasic islands with respect to I m . Then lim k→∞ I∈I k µ(S I (Ω)) = 0. Then µ satisfies (EFT) with Ω being an EFT-set and with B := {S I 1, (Ω) : ∈ Γ} being a regular basic family of cells in Ω.
Similarly, Remark 2.10(c) implies that (P k, ) k≥1 is a family of refining µ-partitions of B 1, . By (2.9), condition (2) of (EFT) holds for . If B ∈ P 2 k, for some k ≥ 2, then by the definition of P k, , there exists a sequence of islands (I i ) k i=2 such that S Ii (Ω) ∈ P 2 i, and I i+1 ∈ O(I i ) for all i = 2, . . . , k − 1, and B = S I k (Ω). Thus for any i = 2, . . . , k, I i is not µ-equivalent to any island in I m . By the minimality of c( ), I 2 is a level-c( ) nonbasic island with respect to I m . It follows that for all i = 2, . . . , k, I i is a level-(c( ) + i − 2) nonbasic island with respect to I m .
3. Examples for (EFT). We illustrate (EFT) by the following four classes of examples. It is well known that for any FIFS {S i } i∈Λ on R d satisfying (OSC), there always exists an OSC-set Ω that has nonempty intersection with the corresponding self-similar set (see [36]), and thus any associated self-similar measure µ satisfies µ(Ω) > 0. This is not true for IIFSs (see [39]).
Our second example is the IFS defining the infinite Bernoulli convolution associated with the golden ratio, defined as: The corresponding self-similar identity is Strichartz et al. [38] showed that µ satisfies a family of second-order identities with respect to the following auxiliary IFS: The spectral dimension of µ is computed in [31].
be a probability vector, and µ be the associated self-similar measure. Then µ satisfies (EFT) with Ω = (0, 1) being an EFT-set and there exists a regular basic pair with respect to Ω.  and W k := {2 k−i 13 i : i = 0, 1, . . . , k} for all k ≥ 1 (see Figure 2). We remark that I 1 = {I 1,0 , I 1,1 }. Proposition 3.4(a) below implies that all multi-indices in W k correspond to the same vertex. We summarize without proof some elementary properties.
be a probability vector, and µ be the associated self-similar measure. Define We remark that for k ≥ 0, For any k ≥ 1, w 1 (k) denotes the sum of probability weights of all multi-indices in W k , as can be seen from part (b) of the lemma below.
Lemma 3.5. Assume the hypotheses of Example 3.3 and let Ω = (0, 1). Define where I 1,i is defined in (3.13). Let w 1 (k) be defined as in (3.14). Then (c) The proof is similar to that of (b).
Proof of Example 3.3. We show that all assumptions in Proposition 2.15 are sat- } is the only level-k nonbasic island with respect to I 1 for any k ≥ 2 (see Figure 2). Since I(v root ) ≈ µ I 1,0 , I is not a level-2 nonbasic island with respect to I 1 for any I ∈ O(I 1,0 ), and thus assumption (1) of Proposition 2.15 holds for = 0 with c(0) = 2. Upon iterating the IFS once, I 1,1 generates three islands: r r r r r r r r r r r r r r r r r r r r r r r r r r r I 3,1,2 Figure 2. Level-k islands I k for k = 0, 1, 2, 3 in Example 3.3. I 1 = {I 1,0 , I 1,1 } corresponds to the basic family of cells and I k,1,2 is the unique level-k nonbasic island with respect to I 1 for k ≥ 2. W k corresponds to those iterates in S I k+1,1,2 (Ω) that overlap exactly and hence give rise to the same vertex. Islands that are labeled consist of vertices enclosed by a box. The figure is drawn with r 1 = 1/3 and r 2 = 2/7.
We now give an example of a class of graph-directed self-similar measures with overlaps that satisfy (EFT).
A graph-directed iterated function system (GIFS) of contractive similitudes is an ordered pair G = (V, E) described as follows (see [28]): V := {1, . . . , q} is the set of vertices and E is the set of directed edges with each edge beginning and ending at a vertex. It is possible for any edge to begin and end at the same vertex and we allow more than one edge between two vertices. Let E ij denote the set of all edges that begin at vertex i and end at vertex j. We call e = e 1 . . . e k a path with length k, if the terminal vertex of each edge e i (1 ≤ i ≤ k − 1) equals the initial vertex of the edge e i+1 .
To each edge e ∈ E, we associate a contractive similitude S e with contraction ratio r(e) ∈ (0, 1). It is known (see [8,28]) that there exists a unique family of non-empty compact sets F 1 , . . . , F q satisfying We call F the graph-directed self-similar set defined by G = (V, E). Suppose for each edge e ∈ E, there corresponds a transition probability p(e); that is, p(e) > 0 and the weights of all edges leaving a given vertex i sum to 1, namely, j∈V e∈Eij p(e) = 1. Then for each i ∈ V , there exists a unique Borel probability measure µ i such that for all measurable set E, and we call µ the graph-directed self-similar measure. A GIFS, as well as any associated graph-directed self-similar measure, are said to have overlaps if the graph open set condition (see [28]) fails. We say that a family of non-empty bounded open subsets For each path e = e 1 . . . e k of length k, we use the notation p(e) := p(e 1 ) · · · p(e k ), The GIFS G = (V, E) below is used as basic example of the graph finite type condition [5].  Figure 3). Let µ be the graph-directed self-similar measure defined by G = (V, E) and probability matrix (p(e)) e∈E . Then µ satisfies (EFT) with Ω := (0, 1) ∪ (2, 3) being an EFT-set and there exists a regular basic pair with respect to Ω. 0 1   If for some k ≥ 2 and ∈ Γ, P k, is a well-defined µ-partition of B 1, , then we let P 1 k, and P 2 k, be defined as in (2.7) with respect to B. We note that B 1,3 is not µ-equivalent to any cell in B (see Figure 4). Define P k,1 := {S e2 (B 1,0 ), S e2 (B 1,1 )} for k ≥ 2. Since Ω 1 µ B 1,1 , P 1 k,1 = P k,1 for all k ≥ 2. Thus (P k,1 ) k≥1 is a sequence of refining µ-partitions of B 1,1 satisfying all conditions of (EFT) with = 1. We note that S e1e2 = S e3e4 and S e1 (Ω 1 ) ∩ S e3 (Ω 2 ) = S e1e2 (Ω 1 ) = S e3e4 (Ω 1 ). Define Then the elements of P 2,0 are pairwise measure disjoint, with each being a subset of B 1,0 . Hence P 2,0 is a µ-partition of B 1,0 . It follows from (3.21) and the equality S e1e2 = S e3e4 that P 1 2,0 = {S e1 (B 1,0 ), S e1 (B 1,1 )} and P 2 2,0 = { S e3 (B 1,3 )}. Thus condition (1) of (EFT) holds with = 0. For k ≥ 2, define P k+1,0 := P 1 Similarly, we can show that B 1,1 µ S e3e k−1 Hence, conditions (2) and (3) of (EFT) hold with = 0. Consequently, the first assertion follows. Finally, the regularity of ({B 1, }, (P k, ) k≥1 ) ∈Γ is obvious. ) ≤ λ} be the associated eigenvalue counting function. If F = N ⊥ , where N is defined as in Section 1, then N (λ, −∆ F µ| (a,b) ) reduces to N (λ, −∆ µ| (a,b) ). It follows from the variational formula that ,a i+1 ) ). We prove a similar formula.  ,a i+1 ) ).
Proof. Let λ be an eigenvalue of −∆ µ| (a i ,a i+1 ) with an eigenfunction u i for i ∈ J. Define u i := u i on (a i , a i+1 ) and u i := 0 otherwise. Then u i ∈ F and for all It follows that λ is also an eigenvalue of −∆ F µ| (a,b) with u i being an eigenfunction. ,a i+1 ) ). To prove the reverse inequality, let λ be an eigenvalue of −∆ F µ| (a,b) with u being an eigenfunction. Define u i := u| (ai,ai+1) for all i ∈ J. Then for all v ∈ F,

In particular, for all
Now let i ∈ J such that u i = 0. Then λ is also an eigenvalue of −∆ µ| (a i ,a i+1 ) with u i being an eigenfunction, proving the reverse inequality.  Note that unitarily equivalent operators have the same set of eigenvalues.
Note that by Proposition 4.2(a), we have a,b) ). The assertion follows by combining this with (4.2).
Step 1. Derivation of functional equations. For ∈ Γ and k ≥ 2, let P 1 k, and P 2 k, be defined as in (2.7) with respect to B. Without loss of generality, we may assume that Γ can be partitioned into two (possibly empty) sub-collections, Γ 1 and Γ 2 , defined as follows. An index ∈ Γ belongs to Γ 1 if there exists some integer k satisfying P 2 k, = ∅. Let κ ≥ 2 (depending on ) denote the smallest of such k. Define Γ 2 := Γ \ Γ 1 and let κ := ∞ for ∈ Γ 2 .
Hence, (4.15) follows from our assumption lim α→ α + i F i (α) > 1. Moreover, it follows from the derivation of equations (4.10) and (4.13) that each column of M α is nondegenerate at 0. From Theorem 4.7, we have f = f * M α + z, where, by assumption, z is directly Riemann integrable on R.
We first consider the case M α (∞) is irreducible. Then all conditions of Theorem A.1 are satisfied. Hence there exist positive constants c 1 and c 2 such that It follows from the definition of f (t) in (4.9) that there exist positive constants C 1 and C 2 such that for all ∈ Γ, Moreover, there exists some 0 ∈ Γ such that lim t→∞ f In the rest of this section, we use the notation defined in Section 4.2. For I ∈ I, let S I (Ω) and O(I) be defined as in (2.2)    Thus P 2,0 = P 1 2,0 = {B 2,0,0 , B 2,0,1 } and P 2 2,0 = ∅ (see Figure 5). It follows that P k,0 = P 2,0 for all k ≥ 2; in particular, 0 ∈ Γ 1 , κ 0 = 2, G 2,0 = {0, 1}.
Now we need to show that there exists some σ > 0 such that as t → ∞ for = 0, 1. To this end, we will first show that N (e t , −∆ µ| B n t ,1,2 ) is bounded.
6. IIFSs with overlaps. In this section we study (EFT) for IIFSs and prove Theorem 1.3. We will only consider IIFSs on R. Let µ be a self-similar measure defined by a finite type IIFS {S i } i∈Λ on R with Ω being an FTC-set. In this section, we use the notation introduced in Section 2.
6.1. A sufficient condition for IIFSs to satisfy (EFT). We first introduce the definition of a tail. Definition 6.1. Let µ be a self-similar measure defined by a finite type IIFS {S i } i∈Λ on R with Ω := (a, b) being an FTC-set. For k ≥ 1, let T ⊆ I k be a countably infinite sequence of islands, and let U := (c, d) ⊆ Ω be the minimal open interval containing S T (Ω).
(a) We call T a level-k semi-tail if it satisfies the following two conditions: (1) for any I ∈ I k \ T , S I (Ω) ∩ U = ∅; (2) for any c 1 > c and d 1 < d, either #{I ∈ T : T is called a level-k tail if it is a level-k semi-tail and the following two additional conditions hold: (3) there exists a finite subset B ⊆ I k \ T containing all island measure types in T such that B ∪ T is a maximal level-k semi-tail; (4) T is a maximal level-k semi-tail satisfying condition (3).
Intuitively, condition (1) means that the interior of the convex hull of S T (Ω) does not contain any cell that corresponds to some level-k island I / ∈ T . Thus, µ| U and µ| S T (Ω) have similar measure structures; in particular, the closures of their supports with respect to Ω are the same. Condition (2) implies that T can be expressed as a sequence of islands  We denote the collection of all level-k tails by T k for k ≥ 1, and define T := k≥1 T k . For T ∈ T, [T ] µ is said to be the (tail) measure type of T . Since any tail consists of countably infinitely many islands, if #I k < ∞ for some k ≥ 1, then T k = ∅.
i=0 is an IIFS on R satisfying (OSC) with Ω = (a, b) being an OSC-set. Let µ be an associated self-similar measure. Define v i := (S i , 1) for any i ≥ 0. Assume that there exist some (possibly empty) finite subset Λ 1 ⊆ Λ and some x 0 ∈ Ω such that Λ x 0 ) is the minimal open interval containing S T (Ω) and thus T is a semitail. Let B := {I(v i0 )}. Then B ∪ T is the maximal semi-tail containing T . Since I/ ≈µ contains only one element, conditions (3) and (4) hold, which completes the proof.
We will give another example of a tail in Lemma 6.12.
Proposition 6.4. Let µ be a self-similar measure defined by a finite type IIFS {S i } i∈Λ on R with Ω = (a, b) being an FTC-set. Suppose that there exists some m ≥ 1 such that the following conditions hold: (1) both T m and I m := I m \ ( T ∈Tm T ) are nonempty and finite; (2) for any k ≥ m and I ∈ I k , O(I) can be expressed as the disjoint union of a nonempty finite family {I i } i∈Λ1 of islands and a nonempty finite family {T i } i∈Λ2 of semi-tails with the property that for any i ∈ Λ 2 , there exists some T ∈ T m such that [T ] µ = [T i ] µ ; (3) the sum of the µ-measures of all level-k nonbasic islands with respect to I m tends to 0 as k → ∞.
Then µ satisfies (EFT) with Ω being an EFT-set and there exists a regular basic pair with respect to Ω.
Fix any ∈ Γ * . By the definition of a tail, T 1, can be uniquely expressed as otherwise, define P n, := P k−1, for all n ≥ k. We remark that (P k, ) k≥1 is a family of refining µ-partitions of B 1, . For any k ≥ 2, by combining Remark 6.2(b) and the definition of I m , we have I 1,i ≈ µ I k, ,0 for some i ∈ Γ * . Hence Remark 2.10(b) implies that for any k ≥ 2, B k, ,0 is µ-equivalent to some cell in B. Thus {B i, ,0 : 2 ≤ i ≤ k} ⊆ P 1 k, and P 2 k, ⊆ {B k, ,1 } for all k ≥ 2. It follows that conditions (1) and (2) of (EFT) hold for . By condition (2) of the definition of a tail, the closure of B k, ,1 converges to a point. Hence, B∈P 2 k, µ(B) ≤ µ(B k, ,1 ) → 0 as k → ∞, i.e., condition (3) of (EFT) holds for . Therefore, for each ∈ Γ, all conditions of (EFT) hold. Since Ω = (a, b), each cell B ∈ k≥1, ∈Γ P k, is connected, which, together with Proposition 2.14, yields the regularity of (B, P).
We remark that in condition (2), it is possible that for some i ∈ Λ 2 , T i ∈ T. (See Figure 7; T 2,1,2 is a semi-tail but not a tail.) Compared with that for FIFSs (Proposition 2.15), the above sufficient condition for (EFT) for IIFSs includes one additional assumption, namely, condition (2). Thus assumption (2) of Proposition 6.4 holds if for any I ∈ I m , any n ≥ 2, as well as any level-n nonbasic island I with respect to I m , O(I) satisfies the property in assumption (2) of Proposition 6.4.
Using Proposition 6.4, we can prove that the measures in the following two classes of examples satisfy (EFT). Example 6.6. Let µ be a self-similar measure defined by an IIFS {S i } i∈Λ on R satisfying (OSC) with Ω = (a, b). If µ(Ω) > 0 and assumption (1) of Proposition 6.4 holds for m = 1, then µ satisfies (EFT) with Ω being an EFT-set and, moreover, a regular basic pair exists.
be an IIFS as in (1.11) (see Figure 6), (p i ) ∞ i=1 be a probability vector, and µ be the associated self-similar measure. Assume that (1.12) holds. Then µ satisfies (EFT) with Ω = (0, 1) being an EFT-set and there exists a regular basic pair with respect to Ω. In order to prove Example 6.7, we need to establish some preliminary results. We first summarize without proof some elementary properties. Define We remark that Figure 8). Proposition 6.8(a) below shows that all multi-indices in W k, correspond to the same vertex. We will see that for all i ∈ W k, , S i (Ω) ⊆ S I 1, (Ω) and (S i , k + 1) belongs to a level-(k + 1) island of a new measure type with two vertices (see Figure 8).
Proof. We first note that condition (1) holds with (S 2L (0), s) being the minimal open interval. Also, note that lim k→∞ S 2k (x) = lim k→∞ S 2k+1 (x) = s for any x ∈ (0, 1). Thus condition (2) of Definition 6.1 holds, and hence T 1,L is a semi-tail. q q I 1,2 I 1,3 · · · q q T 2,1,1 T 2,1,2 q q q q q q q q q q q q q q Figure 7. Islands, semi-tails, and tails for an IIFS in (1.11). The figure is drawn by using r = 1/4 and s = 2/3 and by assuming that (1.12) holds with L = 2. T 1,2 , T 2,1,1 , and T 2,1,2 are defined in Lemma 6.12 and the proof of Example 6.7. They consist of islands enclosed by a box. T 1,2 is the only level-1 tail (Lemma 6.12). One can verify directly that T 2,1,1 is a tail with the set B in Definition 6.1 consisting of the island on its left. T 2,1,2 is a semi-tail but not a tail; an analogous B cannot be found, and thus condition (3) of Definition 6.1 is not satisfied.
The following corollary follows from Example 6.6 and Theorem 1.1.
6.2. The family of IIFSs with overlaps in (1.11). We first state a result concerning the existence of self-similar measures associated with an IIFS {S i } i∈Λ . The proof is similar to that of [8, Theorem 2.8].
Proposition 6.14. Let {S i } i∈Λ be an IIFS of contractions (not necessarily similitudes) on R d and let r i be the Lipschitz constant of S i . Assume there exists c > 0 such that r i ≤ c < 1 for all i ∈ Λ. Then for any probability vector (p i ) i∈Λ , there exists a unique probability measure µ satisfying the self-similar identity (2.1).
In this subsection, we consider the family of IIFSs defined as in (1.11) and fix the FTC-set Ω = (0, 1). Proposition 6.14 implies the existence of a self-similar measure µ for any probability vector (p i ) ∞ i=1 . Assume that (1.12) holds and L is given as in (1.12) in the rest of this section. Let I 1,i be defined as in (6.3) for i ≥ 0, and T 1,L be defined as in Lemma 6.12. Then By the proof of Examples 6.7, we see that all assumptions in Proposition 6.4 hold with m = 1. Hence µ satisfies (EFT) with Ω = (0, 1) being an EFT-set. For each ∈ Γ, let (P k, ) k≥1 be the family of refining µ-partitions of B 1, defined as in the proof of Proposition 6.4. Then (B, P) := ({B 1, }, (P 1, ) k≥1 ) ∈Γ is a regular basic pair.
Proof. The proof is similar to that of Proposition 5.4.
It is of interest to express the eigenvalue counting function in terms of the properties of the measure and the domain, as in the original Weyl law. Also, in view of Weyl's conjecture stated in the introduction, it is of interest to study the second order term in the asymptotic expansion of the eigenvalue counting function.
In this case, γ is called a cycle if i 1 = i k ; in particular, if it is a cycle and i 1 , . . . , i k−1 are distinct, then γ is said to be a simple cycle. We denote by R F the closed subgroup of (R, +) generated by supp(µ γ ) : γ is a simple cycle on {1, . . . , n} .