
-
Previous Article
Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
- DCDS Home
- This Issue
-
Next Article
Reflected backward stochastic differential equations with perturbations
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[
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. |
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. |
[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. |









[1] |
Qingsong Gu, Jiaxin Hu, Sze-Man Ngai. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure and Applied Analysis, 2020, 19 (2) : 641-676. doi: 10.3934/cpaa.2020030 |
[2] |
Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003 |
[3] |
Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47 |
[4] |
Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002 |
[5] |
Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801 |
[6] |
Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323 |
[7] |
Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198 |
[8] |
Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks and Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401 |
[9] |
Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036 |
[10] |
D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685 |
[11] |
G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131 |
[12] |
Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101 |
[13] |
Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857 |
[14] |
F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91 |
[15] |
Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897 |
[16] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[17] |
Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313 |
[18] |
L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799 |
[19] |
Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703 |
[20] |
Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations and Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]