# American Institute of Mathematical Sciences

April  2018, 38(4): 1849-1887. doi: 10.3934/dcds.2018076

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

 1 College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China 2 Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA 3 Key Laboratory of High Performance Computing and Stochastic Information Processing, (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China

Received  November 2016 Revised  November 2017 Published  January 2018

Fund Project: 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[24].

Citation: Sze-Man Ngai, Wei Tang, Yuanyuan Xie. Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1849-1887. doi: 10.3934/dcds.2018076
##### References:
 [1] P. Alonso-Ruiz and U. R. Freiberg, Weyl asymptotics for Hanoi attractors, Forum Math., (2017), 1003-1021. [2] E. Ayer and R. S. Strichartz, Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Amer. Math. Soc., 351 (1999), 3725-3741.  doi: 10.1090/S0002-9947-99-01982-0. [3] R. Courant, Über die Schwinggungen eingespannter Platten, Math. Z., 15 (1922), 195-200.  doi: 10.1007/BF01494393. [4] D. Croydon and B. Hambly, Self-similarity and spectral asymptotics for the continuum random tree, Stochastic Process. Appl., 118 (2008), 730-754.  doi: 10.1016/j.spa.2007.06.005. [5] M. Das and S.-M. Ngai, Graph-directed iterated function systems with overlaps, Indiana Univ. Math. J., 53 (2004), 109-134.  doi: 10.1512/iumj.2004.53.2342. [6] G. Deng and S.-M. Ngai, Differentiability of $L^q$-spectrum and multifractal decomposition by using infinite graph-directed IFSs, Adv. Math., 311 (2017), 190-237.  doi: 10.1016/j.aim.2017.02.021. [7] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., (1975), 39-79.  doi: 10.1007/BF01405172. [8] K. J. Falconer, Techniques in Fractal Geometry, Wiley, 1997. [9] U. Freiberg, Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets, Forum Math., 17 (2005), 87-104. [10] T. Fujita, A fractional dimension, self-similarity and a generalized diffusion operator, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 83-90, Academic Press, Boston, MA, 1987. [11] B. M. Hambly, On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets, Probab. Theory Related Fields, 117 (2000), 221-247.  doi: 10.1007/s004400050005. [12] B. M. Hambly and S. O. G. Nyberg, Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem, Proc. Edinb. Math. Soc., 46 (2003), 1-34. [13] J. Hu, K.-S. Lau and S.-M. Ngai, Laplace operators related to self-similar measures on $\mathbb{R}^d$, J. Funct. Anal., 239 (2006), 542-565.  doi: 10.1016/j.jfa.2006.07.005. [14] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055. [15] V. Ivrii, Second term of the spectral asymptotic expansion of a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen, 14 (1980), 25-34. [16] N. Jin and S. S. T. Yau, General finite type IFS and $M$-matrix, Comm. Anal. Geom., 13 (2005), 821-843.  doi: 10.4310/CAG.2005.v13.n4.a8. [17] N. Kajino, Spectral asymptotics for Laplacians on self-similar sets, J. Funct. Anal., 258 (2010), 1310-1360.  doi: 10.1016/j.jfa.2009.11.001. [18] N. Kajino, Log-periodic asymptotic expansion of the spectral partition function for self-similar sets, Comm. Math. Phys., 328 (2014), 1341-1370.  doi: 10.1007/s00220-014-1922-3. [19] J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125.  doi: 10.1007/BF02097233. [20] M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc., 325 (1991), 465-529.  doi: 10.1090/S0002-9947-1991-0994168-5. [21] K.-S. Lau and S.-M. Ngai, $L^q$-spectrum of the Bernoulli convolution associated with the golden ratio, Studia Math., 131 (1998), 225-251. [22] K.-S. Lau and S.-M. Ngai, A generalized finite type condition for iterated function systems, Adv. Math., 208 (2007), 647-671.  doi: 10.1016/j.aim.2006.03.007. [23] K.-S. Lau and X.-Y. Wang, Iterated function systems with a weak separation condition, Studia Math., 161 (2004), 249-268.  doi: 10.4064/sm161-3-3. [24] K.-S. Lau, J. Wang and C.-H. Chu, Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures, Studia Math., 117 (1995), 1-28.  doi: 10.4064/sm-117-1-1-28. [25] B. M. Levitan, On a theorem of H. Weyl, Doklady Akad. Nauk SSSR (N.S.), 82 (1952), 673-676. [26] V. G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. [27] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154. [28] R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.  doi: 10.1090/S0002-9947-1988-0961615-4. [29] H. P. McKean and D. B. Ray, Spectral distribution of a differential operator, Duke Math. J., (1962), 281-292.  doi: 10.1215/S0012-7094-62-02928-9. [30] K. Naimark and M. Solomyak, The eigenvalue behaviour for the boundary value problems related to self-similar measures on $\mathbb{R}^ d$, Math. Res. Lett., 2 (1995), 279-298.  doi: 10.4310/MRL.1995.v2.n3.a5. [31] S.-M. Ngai, Spectral asymptotics of Laplacians associated with one-dimensional iterated function systems with overlaps, Canad. J. Math., 63 (2011), 648-688.  doi: 10.4153/CJM-2011-011-3. [32] S.-M. Ngai and J.-X. Tong, Infinite iterated function systems with overlaps, Ergodic Theory Dynam. Systems, 36 (2016), 890-907.  doi: 10.1017/etds.2014.86. [33] S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63 (2001), 655-672.  doi: 10.1017/S0024610701001946. [34] S. -M. Ngai and Y. Xie, $L^q$-spectrum of self-similar measures with overlaps in the absence of second-order identities, J. Aust. Math. Soc. , to appear. [35] Y. Peres, W. Schlag and B. Solomyak, Sixty years of Bernoulli convolutions, Fractal Geometry and Stochastics, Ⅱ, (Greifswald/Koserow, 1998), 39-65, Progr. Probab., 46, Birkhäuser, Basel, 2000. [36] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1. [37] R. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of $\mathbb{R}^3$, Adv. in Math., (1978), 244-269.  doi: 10.1016/0001-8708(78)90013-0. [38] R. S. Strichartz, A. Taylor and T. Zhang, Densities of self-similar measures on the line, Experiment. Math., 4 (1995), 101-128.  doi: 10.1080/10586458.1995.10504313. [39] T. Szarek and S. Wedrychowicz, The OSC does not imply the SOSC for infinite iterated function systems, Proc. Amer. Math. Soc., 133 (2005), 437-440.  doi: 10.1090/S0002-9939-04-07708-1. [40] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479.  doi: 10.1007/BF01456804.

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##### References:
 [1] P. Alonso-Ruiz and U. R. Freiberg, Weyl asymptotics for Hanoi attractors, Forum Math., (2017), 1003-1021. [2] E. Ayer and R. S. Strichartz, Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Amer. Math. Soc., 351 (1999), 3725-3741.  doi: 10.1090/S0002-9947-99-01982-0. [3] R. Courant, Über die Schwinggungen eingespannter Platten, Math. Z., 15 (1922), 195-200.  doi: 10.1007/BF01494393. [4] D. Croydon and B. Hambly, Self-similarity and spectral asymptotics for the continuum random tree, Stochastic Process. Appl., 118 (2008), 730-754.  doi: 10.1016/j.spa.2007.06.005. [5] M. Das and S.-M. Ngai, Graph-directed iterated function systems with overlaps, Indiana Univ. Math. J., 53 (2004), 109-134.  doi: 10.1512/iumj.2004.53.2342. [6] G. Deng and S.-M. Ngai, Differentiability of $L^q$-spectrum and multifractal decomposition by using infinite graph-directed IFSs, Adv. Math., 311 (2017), 190-237.  doi: 10.1016/j.aim.2017.02.021. [7] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., (1975), 39-79.  doi: 10.1007/BF01405172. [8] K. J. Falconer, Techniques in Fractal Geometry, Wiley, 1997. [9] U. Freiberg, Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets, Forum Math., 17 (2005), 87-104. [10] T. Fujita, A fractional dimension, self-similarity and a generalized diffusion operator, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 83-90, Academic Press, Boston, MA, 1987. [11] B. M. Hambly, On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets, Probab. Theory Related Fields, 117 (2000), 221-247.  doi: 10.1007/s004400050005. [12] B. M. Hambly and S. O. G. Nyberg, Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem, Proc. Edinb. Math. Soc., 46 (2003), 1-34. [13] J. Hu, K.-S. Lau and S.-M. Ngai, Laplace operators related to self-similar measures on $\mathbb{R}^d$, J. Funct. Anal., 239 (2006), 542-565.  doi: 10.1016/j.jfa.2006.07.005. [14] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055. [15] V. Ivrii, Second term of the spectral asymptotic expansion of a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen, 14 (1980), 25-34. [16] N. Jin and S. S. T. Yau, General finite type IFS and $M$-matrix, Comm. Anal. Geom., 13 (2005), 821-843.  doi: 10.4310/CAG.2005.v13.n4.a8. [17] N. Kajino, Spectral asymptotics for Laplacians on self-similar sets, J. Funct. Anal., 258 (2010), 1310-1360.  doi: 10.1016/j.jfa.2009.11.001. [18] N. Kajino, Log-periodic asymptotic expansion of the spectral partition function for self-similar sets, Comm. Math. Phys., 328 (2014), 1341-1370.  doi: 10.1007/s00220-014-1922-3. [19] J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125.  doi: 10.1007/BF02097233. [20] M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc., 325 (1991), 465-529.  doi: 10.1090/S0002-9947-1991-0994168-5. [21] K.-S. Lau and S.-M. Ngai, $L^q$-spectrum of the Bernoulli convolution associated with the golden ratio, Studia Math., 131 (1998), 225-251. [22] K.-S. Lau and S.-M. Ngai, A generalized finite type condition for iterated function systems, Adv. Math., 208 (2007), 647-671.  doi: 10.1016/j.aim.2006.03.007. [23] K.-S. Lau and X.-Y. Wang, Iterated function systems with a weak separation condition, Studia Math., 161 (2004), 249-268.  doi: 10.4064/sm161-3-3. [24] K.-S. Lau, J. Wang and C.-H. Chu, Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures, Studia Math., 117 (1995), 1-28.  doi: 10.4064/sm-117-1-1-28. [25] B. M. Levitan, On a theorem of H. Weyl, Doklady Akad. Nauk SSSR (N.S.), 82 (1952), 673-676. [26] V. G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985. [27] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154. [28] R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.  doi: 10.1090/S0002-9947-1988-0961615-4. [29] H. P. McKean and D. B. Ray, Spectral distribution of a differential operator, Duke Math. J., (1962), 281-292.  doi: 10.1215/S0012-7094-62-02928-9. [30] K. Naimark and M. Solomyak, The eigenvalue behaviour for the boundary value problems related to self-similar measures on $\mathbb{R}^ d$, Math. Res. Lett., 2 (1995), 279-298.  doi: 10.4310/MRL.1995.v2.n3.a5. [31] S.-M. Ngai, Spectral asymptotics of Laplacians associated with one-dimensional iterated function systems with overlaps, Canad. J. Math., 63 (2011), 648-688.  doi: 10.4153/CJM-2011-011-3. [32] S.-M. Ngai and J.-X. Tong, Infinite iterated function systems with overlaps, Ergodic Theory Dynam. Systems, 36 (2016), 890-907.  doi: 10.1017/etds.2014.86. [33] S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63 (2001), 655-672.  doi: 10.1017/S0024610701001946. [34] S. -M. Ngai and Y. Xie, $L^q$-spectrum of self-similar measures with overlaps in the absence of second-order identities, J. Aust. Math. Soc. , to appear. [35] Y. Peres, W. Schlag and B. Solomyak, Sixty years of Bernoulli convolutions, Fractal Geometry and Stochastics, Ⅱ, (Greifswald/Koserow, 1998), 39-65, Progr. Probab., 46, Birkhäuser, Basel, 2000. [36] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115.  doi: 10.1090/S0002-9939-1994-1191872-1. [37] R. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of $\mathbb{R}^3$, Adv. in Math., (1978), 244-269.  doi: 10.1016/0001-8708(78)90013-0. [38] R. S. Strichartz, A. Taylor and T. Zhang, Densities of self-similar measures on the line, Experiment. Math., 4 (1995), 101-128.  doi: 10.1080/10586458.1995.10504313. [39] T. Szarek and S. Wedrychowicz, The OSC does not imply the SOSC for infinite iterated function systems, Proc. Amer. Math. Soc., 133 (2005), 437-440.  doi: 10.1090/S0002-9939-04-07708-1. [40] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479.  doi: 10.1007/BF01456804.
First iteration of an IIFS $\{S_i\}_{i = 1}^\infty$ defined in (1.11). The figure is drawn with $r = 1/4$ and $s = 2/3$
First iteration of an IFS $\{S_i\}_{i = 1}^3$ in (1.9), drawn by using $r_1 = 1/3$ and $r_2 = 2/7$
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 $r_2 = 2/7$
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 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
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 $\ell\in \Gamma$
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 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
$\mu$-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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