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 overlaps, spectral dimension has been obtained earlier only for a small class of onedimensional 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].2010 Mathematics Subject Classification. Primary: 28A80, 35P20; Secondary: 35J05, 43A05, 47A75.Key words and phrases. Fractal, Laplacian, spectral dimension, self-similar measure with overlaps, essentially of finite type. 1851 1852 SZE-MAN NGAI, WEI TANG AND YUANYUAN SPECTRAL ASYMPTOTICS OF FRACTAL LAPLACIANS, is called the spectral dimension of −∆ µ (or simply of µ); it measures the asymptotic growth rate of the eigenvalue counting function as well as the magnitude of the n-th eigenvalue. This paper studies measures that are essentially of finite type (EFT), a condition that we introduce to describe the finiteness of basic measure types, as defined below. (EFT) is a key assumption in computing spectral dimension and is formulated in Section 2.2.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. Roughly speaking, two cells (that is, subsets of Ω with positive µ measure), U and V are µ-equivalent if µ| V = wµ| U • τ −1 for some w > 0 and some surjective similitude τ : U → V , where µ| F denotes the restriction of the measure µ to F ⊆ R d . A µ-partition P of U is a finite family of measure disjoint sub-cells of U such that µ(U ) = V ∈P µ(V ). A sequence of µ-partitions (P k ) k≥1 is refining if each member of P k+1 is a subset of some member of P k .Intuitively, µ satisfies (EFT) if there exist some bounded open set Ω ⊂ R d with supp(µ) ⊆ Ω and µ(Ω) > 0, together with a finite family B := {B 1, : ∈ Γ} of measure disjoint cells in Ω such that for each ∈ Γ, there is a family of refining µ-partitions (P k, ) k≥1 of B 1, satisfying the following conditions: (1) for all k ≥ 2, P k+1, contains all cells in P k, that are µ-equivalent to some cell in B;(2) the sum of the µ-measures of those cells B ∈ P k, that are not µ-equivalent to any cell in B tends to 0 as k → ∞. In this case, we call Ω an EFT-set, 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. The precise statements are given in Definition 2.11. In particular, we say that (B, P) is regular if each cell B ∈ k≥1, ∈Γ P k, is connected, and for any ∈ Γ, there exist some similitude τ and some constant w( ) > 0 such that τ (Ω) ⊆ B 1, and µ ≥ w( )µ • τ −1 on τ (Ω).Let µ be a continuous positive finite Borel measure on R. Assume that...

For the class of self-similar measures in $\mathbb{R}^{d}$ with overlaps that are essentially of finite type, we set up a framework for deriving a closed formula for the $L^{q}$-spectrum of the measure for $q\geq 0$. This 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’s 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.

For the class of graph-directed self-similar measures on R, 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 the Laplacians defined by 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 [6]. The main novelty of our results is that the graphdirected self-similar measures we consider do not need to satisfy the graph open set condition.

The spectral dimension of a fractal Laplacian encodes important geometric, analytic, and measure-theoretic information. Unlike standard Laplacians on Euclidean spaces or Riemannian manifolds, the spectral dimension of fractal Laplacians are often non-integral and difficult to compute. The computation is much harder in higher-dimensions. In this paper, we set up a framework for computing the spectral dimension of the Laplacians defined by a class of graph-directed self-similar measures on R d (d ⩾ 2) satisfying the graph open set condition. The main ingredients of this framework include a technique of Naimark and Solomyak and a vector-valued renewal theorem of Lau et al.

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