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 $s = 2/3$
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Reflected backward stochastic differential equations with perturbations
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 |
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[
Keywords:
Fractal,
Laplacian,
spectral dimension,
self-similar measure with overlaps,
essentially of finite type.
Mathematics Subject Classification:
Primary: 28A80, 35P20; Secondary: 35J05, 43A05, 47A75.
Citation:
Sze-Man Ngai, Wei Tang, Yuanyuan Xie. Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities. Discrete & 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. Google Scholar |
| [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. |
show all references
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. Google Scholar |
| [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. |
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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