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# Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities

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

• 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 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.

Mathematics Subject Classification: Primary: 28A80, 35P20; Secondary: 35J05, 43A05, 47A75.

 Citation: • • Figure 6.  First iteration of an IIFS $\{S_i\}_{i = 1}^\infty$ defined in (1.11). The figure is drawn with $r = 1/4$ and $s = 2/3$

Figure 1.  First iteration of an IFS $\{S_i\}_{i = 1}^3$ in (1.9), drawn by using $r_1 = 1/3$ and $r_2 = 2/7$

Figure 2.  Level-$k$ islands $\mathbb{I}_k$ for $k = 0, 1, 2, 3$ in Example 3.3. $\mathbb I_1 = \{\mathcal{I}_{1, 0}, \mathcal{I}_{1, 1}\}$ corresponds to the basic family of cells and $\mathcal{I}_{k, 1, 2}$ is the unique level-$k$ nonbasic island with respect to $\mathbb I_1$ for $k\ge 2$. $W_k$ corresponds to those iterates in $S_{\mathcal{I}_{k+1, 1, 2}}(\Omega)$ 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 $r_2 = 2/7$

Figure 3.  The first iteration of the GIFS $G = (V, E)$ defined in Example 3.6, where $\Omega_1 = (0, 1)$ and $\Omega_2 = (2, 3)$

Figure 4.  Figure showing some cells in a $\mu$-partition ${\mathbf P}_{k, \ell}$ of $B_{1, \ell}$ for $k = 1, 2$ and $\ell \in \Gamma$, as defined in the proof of Example 3.6. ${\mathbf B}: = \{B_{1, \ell}:\ell\in \Gamma\}$ is a basic family of cells. Cells are represented by line segments with dots if they originate from $\Omega_1$, or circles if they originate from $\Omega_2$. Overlapping cells are separated vertically to show distinction and multiplicity

Figure 5.  Iterates of the IFS $\{S_i\}_{i = 1}^3$ with $r_1 = 1/3$ and $r_2 = 2/7$. $({\mathbf P}_{k, \ell})_{k\ge 1}$ is the family of $\mu$-partitions of $B_{1, \ell}$ given as in Section 5 for $\ell\in \Gamma$

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 $L = 2$. $\mathcal{T}_{1, 2}$, $\mathcal{T}_{2, 1, 1}$, and $\mathcal{T}_{2, 1, 2}$ are defined in Lemma 6.12 and the proof of Example 6.7. They consist of islands enclosed by a box. $\mathcal{T}_{1, 2}$ is the only level-$1$ tail (Lemma 6.12). One can verify directly that $\mathcal{T}_{2, 1, 1}$ is a tail with the set $\mathbb{B}$ in Definition 6.1 consisting of the island on its left. $\mathcal{T}_{2, 1, 2}$ is a semi-tail but not a tail; an analogous $\mathbb{B}$ cannot be found, and thus condition (3) of Definition 6.1 is not satisfied

Figure 8.  Figure showing some iterates of the IIFS $\{S_i\}_{i = 1}^\infty$ in (1.11), drawn by using $r = 1/4$ and $s = 2/3$ and by assuming that (1.12) holds with $L = 2$. Using the notation in the proof of Example 6.7, we see that $\{\mathcal{I}_{1, 0}, \mathcal{I}_{1, 1}, \mathcal{T}_{1, 2}\}$ corresponds to a basic family of cells, and $\mathcal{I}_{k, 1, 1}^{2}$ is the only level-$k$ nonbasic island with respect to $\mathbf{I}_1$. $W_{k, 1}$ corresponds to those iterates in $S_{\mathcal{I}_{{k+1}, 1, 1}^2}(\Omega)$ that overlap exactly and hence give rise to the same vertex. All nonbasic islands are boxed

Figure 9.  $\mu$-partitions ${\mathbf P}_{k, \ell}$ of $B_{1, \ell}$, as defined in Section 6.2, for an IIFS $\{S_i\}_{i = 1}^\infty$ in (1.11). The figure is drawn by using $r = 1/4$ and $s = 2/3$. Here we assume that (1.12) holds with $L = 2\in \Gamma_{1}$ and $\kappa_L = 2$

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