Figure 1.
The Hanoi attractor or stretched Sierpinski gasket
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January
2020, 19(1): 47-84.
doi: 10.3934/cpaa.2020004
Analysis on hybrid fractals
| 1. | University of Connecticut, 341 Mansfield Road, Storrs CT 06269-1009, USA |
| 2. | The University of Chicago, 5747 S. Ellis Avenue, Chicago, IL 60637, USA |
| 3. | University of California Berkeley, 2333 College Ave, Berkeley, CA 94704, USA |
| 4. | Cornell University, 310 Malott Hall, Ithaca NY 14853, USA |
| 5. | University of California Berkeley, 2301 Durant Avenue, Berkeley CA 94704, USA |
We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the $ 3 $-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the $ 3 $-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space is investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.
Mathematics Subject Classification:
Primary: 28A80, 35P20, 31C25; Secondary: 31C20.
Citation:
P. Alonso Ruiz, Y. Chen, H. Gu, R. S. Strichartz, Z. Zhou. Analysis on hybrid fractals. Communications on Pure and Applied Analysis,
2020, 19
(1)
: 47-84.
doi: 10.3934/cpaa.2020004
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References:
| [1] |
B. Adams, S. A. Smith, R. S. Strichartz and A. Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 1–24. |
| [2] |
E. Akkermans, O. Benichou and G. V. Dunne,
A. Teplyaev and R. Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals, Phys. Rev. E, 86 (2012), 061125.
|
| [3] |
E. Akkermans, G. V. Dunne and E. Levy, Wave propagation in one-dimension: Methods and applications to complex and fractal structures, Optics of Aperiodic Structures - Fundamentals and Device Applications, L. Dal Negro (Ed), Pan Stanford Press, 2014, 407–449. |
| [4] |
P. Alonso Ruiz, U. Freiberg and J. Kigami,
Completely symmetric resistance forms on the stretched Sierpiński gasket, J. of Fractal Geometry, 5 (2018), 227-277.
doi: 10.4171/JFG/61. |
| [5] |
P. Alonso-Ruiz, D. J. Kelleher and A. Teplyaev,
Energy and Laplacian on Hanoi-type fractal quantum graphs, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 165206.
doi: 10.1088/1751-8113/49/16/165206. |
| [6] |
Patricia Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys., 58 (2017), 073503, 16.
doi: 10.1063/1.4994197. |
| [7] |
N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev,
Vibration modes of 3 n -gaskets and other fractals, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 015101.
doi: 10.1088/1751-8113/41/1/015101. |
| [8] |
M. Begué, T. Kalloniatis and R. S. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet, Fractals, 21 (2013), 1350002, 32.
doi: 10.1142/S0218348X13500023. |
| [9] |
O. Ben-Bassat, R. S. Strichartz and A. Teplyaev,
What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.
doi: 10.1006/jfan.1999.3431. |
| [10] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. |
| [11] |
J. P. Chen, L. G. Rogers, L. Anderson, U. Andrews, A. Brzoska, A. Coffey, H. Davis, L. Fisher, M. Hansalik, S. Loew and A. Teplyaev, Power dissipation in fractal AC circuits, J. Phys. A, 50 (2017), 325205, 20.
doi: 10.1088/1751-8121/aa7a66. |
| [12] |
Y. Chen, H. Gu, R. S. Strichartz and Z. Zhou, Hybrid fractals, Available at http://www.math.cornell.edu/~harry970804/, 2017. |
| [13] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
| [14] |
B. M. Hambly, V. Metz and A. Teplyaev,
Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc., 74 (2006), 93-112.
doi: 10.1112/S002461070602312X. |
| [15] |
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.
doi: 10.1017/S0013091500000730. |
| [16] |
M. Hata,
On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 99-102.
|
| [17] |
M. Hinz,
Sup-norm-closable bilinear forms and Lagrangians, Ann. Mat. Pura Appl., 195 (2016), 1021-1054.
doi: 10.1007/s10231-015-0503-1. |
| [18] |
M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), no. Veroyatnost$\prime$ i Statistika. 22,299–317.
doi: 10.1007/s10958-016-3149-7. |
| [19] |
M. Ionescu, L. G. Rogers and A. Teplyaev,
Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169.
doi: 10.1016/j.jfa.2012.05.021. |
| [20] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
| [21] |
_____, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721–755.
doi: 10.2307/2154402. |
| [22] |
_____, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
| [23] |
_____, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), ⅵ+132. |
| [24] |
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.
|
| [25] |
R. D. Mauldin and S. C. Williams,
Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.
doi: 10.2307/2000940. |
| [26] |
J. Murai,
Diffusion processes on mandala, Osaka J. Math., 32 (1995), 887-917.
|
| [27] |
L. G. Rogers and A. Teplyaev,
Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal., 9 (2010), 211-231.
doi: 10.3934/cpaa.2010.9.211. |
| [28] |
R. S. Strichartz,
Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127.
doi: 10.1006/jfan.2000.3580. |
| [29] |
_____, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial. |
| [30] |
R. S. Strichartz and J. Zhu, Spectrum of the Laplacian on the Vicsek set "with no loose ends", Fractals, 25 (2017), 1750062, 15.
doi: 10.1142/S0218348X17500621. |
| [31] |
A. Teplyaev,
Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480.
doi: 10.4153/CJM-2008-022-3. |
show all references
References:
| [1] |
B. Adams, S. A. Smith, R. S. Strichartz and A. Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 1–24. |
| [2] |
E. Akkermans, O. Benichou and G. V. Dunne,
A. Teplyaev and R. Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals, Phys. Rev. E, 86 (2012), 061125.
|
| [3] |
E. Akkermans, G. V. Dunne and E. Levy, Wave propagation in one-dimension: Methods and applications to complex and fractal structures, Optics of Aperiodic Structures - Fundamentals and Device Applications, L. Dal Negro (Ed), Pan Stanford Press, 2014, 407–449. |
| [4] |
P. Alonso Ruiz, U. Freiberg and J. Kigami,
Completely symmetric resistance forms on the stretched Sierpiński gasket, J. of Fractal Geometry, 5 (2018), 227-277.
doi: 10.4171/JFG/61. |
| [5] |
P. Alonso-Ruiz, D. J. Kelleher and A. Teplyaev,
Energy and Laplacian on Hanoi-type fractal quantum graphs, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 165206.
doi: 10.1088/1751-8113/49/16/165206. |
| [6] |
Patricia Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys., 58 (2017), 073503, 16.
doi: 10.1063/1.4994197. |
| [7] |
N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev,
Vibration modes of 3 n -gaskets and other fractals, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 015101.
doi: 10.1088/1751-8113/41/1/015101. |
| [8] |
M. Begué, T. Kalloniatis and R. S. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet, Fractals, 21 (2013), 1350002, 32.
doi: 10.1142/S0218348X13500023. |
| [9] |
O. Ben-Bassat, R. S. Strichartz and A. Teplyaev,
What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.
doi: 10.1006/jfan.1999.3431. |
| [10] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. |
| [11] |
J. P. Chen, L. G. Rogers, L. Anderson, U. Andrews, A. Brzoska, A. Coffey, H. Davis, L. Fisher, M. Hansalik, S. Loew and A. Teplyaev, Power dissipation in fractal AC circuits, J. Phys. A, 50 (2017), 325205, 20.
doi: 10.1088/1751-8121/aa7a66. |
| [12] |
Y. Chen, H. Gu, R. S. Strichartz and Z. Zhou, Hybrid fractals, Available at http://www.math.cornell.edu/~harry970804/, 2017. |
| [13] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
| [14] |
B. M. Hambly, V. Metz and A. Teplyaev,
Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc., 74 (2006), 93-112.
doi: 10.1112/S002461070602312X. |
| [15] |
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.
doi: 10.1017/S0013091500000730. |
| [16] |
M. Hata,
On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 99-102.
|
| [17] |
M. Hinz,
Sup-norm-closable bilinear forms and Lagrangians, Ann. Mat. Pura Appl., 195 (2016), 1021-1054.
doi: 10.1007/s10231-015-0503-1. |
| [18] |
M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), no. Veroyatnost$\prime$ i Statistika. 22,299–317.
doi: 10.1007/s10958-016-3149-7. |
| [19] |
M. Ionescu, L. G. Rogers and A. Teplyaev,
Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169.
doi: 10.1016/j.jfa.2012.05.021. |
| [20] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
| [21] |
_____, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721–755.
doi: 10.2307/2154402. |
| [22] |
_____, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
| [23] |
_____, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), ⅵ+132. |
| [24] |
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.
|
| [25] |
R. D. Mauldin and S. C. Williams,
Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.
doi: 10.2307/2000940. |
| [26] |
J. Murai,
Diffusion processes on mandala, Osaka J. Math., 32 (1995), 887-917.
|
| [27] |
L. G. Rogers and A. Teplyaev,
Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal., 9 (2010), 211-231.
doi: 10.3934/cpaa.2010.9.211. |
| [28] |
R. S. Strichartz,
Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127.
doi: 10.1006/jfan.2000.3580. |
| [29] |
_____, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial. |
| [30] |
R. S. Strichartz and J. Zhu, Spectrum of the Laplacian on the Vicsek set "with no loose ends", Fractals, 25 (2017), 1750062, 15.
doi: 10.1142/S0218348X17500621. |
| [31] |
A. Teplyaev,
Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480.
doi: 10.4153/CJM-2008-022-3. |
Figure 2.
3-level Sierpinski gasket, interval and regular inverted Sierpinski gasket
Figure 3.
The $ 3 $ -level Sierpinski gasket
Figure 4.
Line segments and one inverted $ \operatorname{SG} $ join the six triangular $ 1 $ -cells of the first level approximation of $ \operatorname{SG}_3 $
Figure 5.
Generators of the hybrid fractal $ \rm H $ based on the the upright triangle
Figure 6.
Graph subdivision and corresponding resistance scaling factors
Figure 7.
As networks, each triangular cell must be equivalent to its (n + 1)th-level subdivision
Figure 8.
The bonds $ \operatorname{I}\in\{K_j\}_{j\in B} $
Figure 9.
Graph subdivision and corresponding resistance scaling factors at level n
Figure 11.
Localized D-N eigenfunctions, see [12] for further examples
Figure 12.
Eigenvalue counting functions of discrete Dirichlet Laplacians, see [12] for further examples
Figure 13.
'$ y $ -$ \log x $ ' plots of Weyl ratios corresponding to different choices of $ a $ and $ r $ at level 6, see [12] for further examples
Figure 14.
Restriction of the eigenfunction corresponding to the lowest eigenvalue to one edge, $ r = a = \frac{1}{6} $ , levels $ 1 $ through $ 6 $
Figure 15.
Restriction of the eigenfunction corresponding to the lowest eigenvalue to one edge, $ r = a = \frac{1}{10} $ , levels $ 1 $ through $ 6 $
Figure 17.
Eigenfunction corresponding to the lowest eigenvalue, $ a = \frac{1}{6} $ , level 6, increasing $ r $
Figure 18.
Eigenfunction corresponding to the lowest eigenvalue, $ r = \frac{1}{3} $ , level $ 6 $ , increasing $ a $ , see [12] for further examples
Figure 19.
Plot of $ N_{Q_m}(\sqrt{x}) $ for level 1 (blue) and level 2 (yellow) quantum hybrid SG-based graph with scaling factor $ r = \frac{1}{6} $
Figure 20.
Eigenfunctions for level 1 quantum graph approximation of $ \rm H $ , see [12] for further examples
Figure 16.
Eigenfunction corresponding to the lowest eigenvalue, $ r = a = \frac{1}{6} $ , see [12] for further examples
Figure 21.
Contraction property of the resistance
Figure 22.
Contraction property of measure
Figure 23.
Level 4 eigenfunction, see [12] for further examples
Figure 25.
Directed graph associated with the hybrid $ \rm H $ with base $ \operatorname{SG}_3 $ from Section 2
Table 1.
Bottom part of the Dirichlet spectrum, $ r = a = 1/6 $ , levels $ 1-7 $ . Eigenvalues are shown with their multiplicity and listed in increasing order
| level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 |
| 43.20, 1 | 33.98, 1 | 33.74, 1 | 33.69, 1 | 33.68, 1 | 33.68, 1 | 33.68, 1 |
| 57.19, 2 | 47.86, 2 | 49.44, 2 | 49.79, 2 | 49.88, 2 | 49.90, 2 | 49.91, 2 |
| 135.55, 2 | 104.23, 2 | 117.75, 2 | 121.25, 2 | 122.12, 2 | 122.34, 2 | 122.40, 2 |
| 149.54, 1 | 126.28, 1 | 184.79, 1 | 202.29, 1 | 206.86, 1 | 208.02, 1 | 208.31, 1 |
| 257.04, 1 | 207.30, 1 | 217.81, 1 | 220.21, 1 | 220.80, 1 | 220.95, 1 | |
| 265.30, 2 | 289.42, 2 | 331.89, 2 | 341.94, 2 | 344.43, 2 | 345.05, 2 | |
| 1881.37, 2 | 397.92, 2 | 474.25, 2 | 495.84, 2 | 501.41, 2 | 502.81, 2 | |
| 1881.41, 1 | 466.78, 1 | 584.16, 1 | 615.23, 1 | 623.24, 1 | 625.26, 1 | |
| 3103.35, 2 | 476.77, 1 | 700.76, 1 | 768.64, 1 | 786.02, 1 | 790.39, 1 | |
| 3103.74, 1 | 514.87, 2 | 787.80, 2 | 869.46, 2 | 890.54, 2 | 895.85, 2 | |
| 3259.84, 1 | 1607.46, 2 | 1089.04, 2 | 1264.93, 2 | 1309.12, 2 | 1320.11, 2 | |
| 3260.17, 2 | 1607.46, 1 | 1130.98, 1 | 1328.95, 1 | 1379.76, 1 | 1392.49, 1 | |
| 4883.03, 2 | 2150.98, 1 | 1455.32, 1 | 1653.59, 1 | 1662.94, 1 | 1664.78, 1 | |
| 4883.03, 1 | 2150.99, 2 | 1481.31, 2 | 1679.55, 2 | 1686.86, 2 | 1688.66, 2 | |
| 4889.90, 1 | 2314.95, 2 | 1666.05, 2 | 1861.84, 1 | 1948.37, 1 | 1970.24, 1 | |
| 4889.90, 2 | 2314.96, 1 | 1675.70, 1 | 1870.76, 2 | 1961.04, 2 | 1983.08, 2 | |
| 5383.38, 3 | 4046.57, 2 | 1812.72, 2 | 2405.81, 1 | 2492.48, 1 | 2510.96, 1 | |
| 4046.57, 1 | 1814.01, 1 | 2447.98, 2 | 2565.41, 2 | 2591.25, 2 | ||
| 6047.56, 1 | 2042.00, 1 | 2740.18, 2 | 2847.39, 2 | 2876.06, 2 | ||
| 6047.56, 2 | 2042.44, 2 | 2949.41, 1 | 3148.07, 1 | 3178.77, 1 | ||
| 6264.63, 2 | 2557.42, 1 | 3012.62, 1 | 3148.76, 1 | 3203.72, 1 | ||
| 6264.63, 1 | 2557.63, 2 | 3256.00, 2 | 3438.70, 2 | 3485.04, 2 | ||
| 9253.61, 3 | 2967.71, 2 | 3571.97, 2 | 3954.97, 2 | 4055.68, 2 | ||
| 9551.29, 3 | 2967.75, 1 | 3674.09, 1 | 4019.24, 1 | 4108.50, 1 | ||
| 9552.10, 3 | 5053.44, 3 | 4260.92, 1 | 4843.37, 2 | 4972.61, 2 | ||
| 67729.76, 3 | 7462.73, 3 | 4270.27, 2 | 4923.03, 1 | 5092.29, 1 | ||
| 83789.93, 3 | 7514.45, 3 | 4962.44, 1 | 5503.63, 1 | 5592.27, 1 | ||
| 83790.87, 3 | 9156.79, 3 | 5022.27, 2 | 5738.82, 2 | 5866.46, 2 | ||
| 111725.04, 3 | 11142.55, 3 | 5527.03, 2 | 6202.97, 2 | 6437.89, 2 |
| level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 |
| 43.20, 1 | 33.98, 1 | 33.74, 1 | 33.69, 1 | 33.68, 1 | 33.68, 1 | 33.68, 1 |
| 57.19, 2 | 47.86, 2 | 49.44, 2 | 49.79, 2 | 49.88, 2 | 49.90, 2 | 49.91, 2 |
| 135.55, 2 | 104.23, 2 | 117.75, 2 | 121.25, 2 | 122.12, 2 | 122.34, 2 | 122.40, 2 |
| 149.54, 1 | 126.28, 1 | 184.79, 1 | 202.29, 1 | 206.86, 1 | 208.02, 1 | 208.31, 1 |
| 257.04, 1 | 207.30, 1 | 217.81, 1 | 220.21, 1 | 220.80, 1 | 220.95, 1 | |
| 265.30, 2 | 289.42, 2 | 331.89, 2 | 341.94, 2 | 344.43, 2 | 345.05, 2 | |
| 1881.37, 2 | 397.92, 2 | 474.25, 2 | 495.84, 2 | 501.41, 2 | 502.81, 2 | |
| 1881.41, 1 | 466.78, 1 | 584.16, 1 | 615.23, 1 | 623.24, 1 | 625.26, 1 | |
| 3103.35, 2 | 476.77, 1 | 700.76, 1 | 768.64, 1 | 786.02, 1 | 790.39, 1 | |
| 3103.74, 1 | 514.87, 2 | 787.80, 2 | 869.46, 2 | 890.54, 2 | 895.85, 2 | |
| 3259.84, 1 | 1607.46, 2 | 1089.04, 2 | 1264.93, 2 | 1309.12, 2 | 1320.11, 2 | |
| 3260.17, 2 | 1607.46, 1 | 1130.98, 1 | 1328.95, 1 | 1379.76, 1 | 1392.49, 1 | |
| 4883.03, 2 | 2150.98, 1 | 1455.32, 1 | 1653.59, 1 | 1662.94, 1 | 1664.78, 1 | |
| 4883.03, 1 | 2150.99, 2 | 1481.31, 2 | 1679.55, 2 | 1686.86, 2 | 1688.66, 2 | |
| 4889.90, 1 | 2314.95, 2 | 1666.05, 2 | 1861.84, 1 | 1948.37, 1 | 1970.24, 1 | |
| 4889.90, 2 | 2314.96, 1 | 1675.70, 1 | 1870.76, 2 | 1961.04, 2 | 1983.08, 2 | |
| 5383.38, 3 | 4046.57, 2 | 1812.72, 2 | 2405.81, 1 | 2492.48, 1 | 2510.96, 1 | |
| 4046.57, 1 | 1814.01, 1 | 2447.98, 2 | 2565.41, 2 | 2591.25, 2 | ||
| 6047.56, 1 | 2042.00, 1 | 2740.18, 2 | 2847.39, 2 | 2876.06, 2 | ||
| 6047.56, 2 | 2042.44, 2 | 2949.41, 1 | 3148.07, 1 | 3178.77, 1 | ||
| 6264.63, 2 | 2557.42, 1 | 3012.62, 1 | 3148.76, 1 | 3203.72, 1 | ||
| 6264.63, 1 | 2557.63, 2 | 3256.00, 2 | 3438.70, 2 | 3485.04, 2 | ||
| 9253.61, 3 | 2967.71, 2 | 3571.97, 2 | 3954.97, 2 | 4055.68, 2 | ||
| 9551.29, 3 | 2967.75, 1 | 3674.09, 1 | 4019.24, 1 | 4108.50, 1 | ||
| 9552.10, 3 | 5053.44, 3 | 4260.92, 1 | 4843.37, 2 | 4972.61, 2 | ||
| 67729.76, 3 | 7462.73, 3 | 4270.27, 2 | 4923.03, 1 | 5092.29, 1 | ||
| 83789.93, 3 | 7514.45, 3 | 4962.44, 1 | 5503.63, 1 | 5592.27, 1 | ||
| 83790.87, 3 | 9156.79, 3 | 5022.27, 2 | 5738.82, 2 | 5866.46, 2 | ||
| 111725.04, 3 | 11142.55, 3 | 5527.03, 2 | 6202.97, 2 | 6437.89, 2 |
Table 2.
Convergence of the Dirichlet spectrum, with the estimated convergence order 1
| level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 | |
| Eigenvalue | 43.2000 | 33.9771 | 33.7412 | 33.6913 | 33.6794 | 33.6764 | 33.6756 |
| Difference | 0.2359 | 0.0499 | 0.0119 | 0.0030 | 0.0008 | ||
| Eigenvalue | 57.1907 | 47.8622 | 49.4401 | 49.7929 | 49.8791 | 49.9005 | 49.9058 |
| Difference | 1.5779 | 0.3528 | 0.0862 | 0.0214 | 0.0053 | ||
| Eigenvalue | 135.5477 | 104.2339 | 117.748 | 121.248 | 122.1248 | 122.3441 | 122.3989 |
| Difference | 13.5141 | 3.5000 | 0.8768 | 0.2193 | 0.0548 | ||
| Eigenvalue | 149.5385 | 126.2839 | 184.7948 | 202.2888 | 206.8625 | 208.0185 | 208.3083 |
| Difference | 58.5109 | 17.4940 | 4.5737 | 1.156 | 0.2853 | ||
| Eigenvalue | 257.0447 | 207.2981 | 217.8053 | 220.2108 | 220.8016 | 220.9487 | |
| Difference | 49.7466 | 10.5072 | 2.4055 | 0.5908 | 0.1471 | ||
| Eigenvalue | 265.302 | 289.4171 | 331.8853 | 341.9429 | 344.4302 | 345.0504 | |
| Difference | 42.4682 | 10.0576 | 2.4873 | 0.6202 |
| level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 | |
| Eigenvalue | 43.2000 | 33.9771 | 33.7412 | 33.6913 | 33.6794 | 33.6764 | 33.6756 |
| Difference | 0.2359 | 0.0499 | 0.0119 | 0.0030 | 0.0008 | ||
| Eigenvalue | 57.1907 | 47.8622 | 49.4401 | 49.7929 | 49.8791 | 49.9005 | 49.9058 |
| Difference | 1.5779 | 0.3528 | 0.0862 | 0.0214 | 0.0053 | ||
| Eigenvalue | 135.5477 | 104.2339 | 117.748 | 121.248 | 122.1248 | 122.3441 | 122.3989 |
| Difference | 13.5141 | 3.5000 | 0.8768 | 0.2193 | 0.0548 | ||
| Eigenvalue | 149.5385 | 126.2839 | 184.7948 | 202.2888 | 206.8625 | 208.0185 | 208.3083 |
| Difference | 58.5109 | 17.4940 | 4.5737 | 1.156 | 0.2853 | ||
| Eigenvalue | 257.0447 | 207.2981 | 217.8053 | 220.2108 | 220.8016 | 220.9487 | |
| Difference | 49.7466 | 10.5072 | 2.4055 | 0.5908 | 0.1471 | ||
| Eigenvalue | 265.302 | 289.4171 | 331.8853 | 341.9429 | 344.4302 | 345.0504 | |
| Difference | 42.4682 | 10.0576 | 2.4873 | 0.6202 |
Table 3.
(cont.) Top of the spectrum, eigenvalues in increasing order, $ r = a = 1/6 $ , level 7
| level 7 | |
| Dirichlet | Neumann |
| 113759984105.32153, 3 | 25411878638.230247, 3 |
| 114102841006, 6 | 114102841006, 6 |
| 114788072698, 18 | 114788072698, 18 |
| 116839632421, 54 | 116839632421, 54 |
| 122950383156,162 | 122950383156,162 |
| 140734901210,243 | 140734901210,243 |
| 140736477695,243 | 140736477626,243 |
| 187655171187, 3 | 186882957201, 3 |
| 187659146198, 6 | 187659146198, 6 |
| 187667196655, 18 | 187667196655, 18 |
| 187692182839, 54 | 187692182839, 54 |
| 187775569901,162 | 187775569901,162 |
| 188131568230,243 | 188131568230,243 |
| 188147174195, 3 | 188147026223, 3 |
| 188147176156, 6 | 188147176156, 6 |
| 188147180228, 18 | 188147180228, 18 |
| 188147193778, 54 | 188147193778, 54 |
| 188147252150,162 | 188147252150,162 |
| 197109853834,243 | 197109853834,243 |
| 197130271198,243 | 197130271198,243 |
| 197130271465,243 | 197130271465,243 |
| 295258340787, 3 | 294738682923, 3 |
| 295259492723, 6 | 295259492723, 6 |
| 295261805573, 18 | 295261805573, 18 |
| 295268816917, 54 | 295268816917, 54 |
| 295290531946,162 | 295290531946,162 |
| 295362581347,243 | 295362581347,243 |
| 295362609295,243 | 295362609295,243 |
| 295673500703,243 | 295673500703,243 |
| 295673630427,486 | 295673630427,486 |
| 325512681630,729 | 325512681630,729 |
| level 7 | |
| Dirichlet | Neumann |
| 113759984105.32153, 3 | 25411878638.230247, 3 |
| 114102841006, 6 | 114102841006, 6 |
| 114788072698, 18 | 114788072698, 18 |
| 116839632421, 54 | 116839632421, 54 |
| 122950383156,162 | 122950383156,162 |
| 140734901210,243 | 140734901210,243 |
| 140736477695,243 | 140736477626,243 |
| 187655171187, 3 | 186882957201, 3 |
| 187659146198, 6 | 187659146198, 6 |
| 187667196655, 18 | 187667196655, 18 |
| 187692182839, 54 | 187692182839, 54 |
| 187775569901,162 | 187775569901,162 |
| 188131568230,243 | 188131568230,243 |
| 188147174195, 3 | 188147026223, 3 |
| 188147176156, 6 | 188147176156, 6 |
| 188147180228, 18 | 188147180228, 18 |
| 188147193778, 54 | 188147193778, 54 |
| 188147252150,162 | 188147252150,162 |
| 197109853834,243 | 197109853834,243 |
| 197130271198,243 | 197130271198,243 |
| 197130271465,243 | 197130271465,243 |
| 295258340787, 3 | 294738682923, 3 |
| 295259492723, 6 | 295259492723, 6 |
| 295261805573, 18 | 295261805573, 18 |
| 295268816917, 54 | 295268816917, 54 |
| 295290531946,162 | 295290531946,162 |
| 295362581347,243 | 295362581347,243 |
| 295362609295,243 | 295362609295,243 |
| 295673500703,243 | 295673500703,243 |
| 295673630427,486 | 295673630427,486 |
| 325512681630,729 | 325512681630,729 |
Table 4.
Bottom of spectrum for quantum graph compared to the spectrum of the discrete level 6 graph approximation of the Hanoi attractor. Ev. = eigenvalue, Renorm. ev. = renormalized eigenvalue, Mult. = multiplicity
| Level 0(Q) | Level 1(Q) | Level 2(Q) | Hanoi Attractor | ||||
| Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Mult. |
| 10.247 | 44.402 | 8.578 | 37.173 | 7.896 | 34.216 | 33.676 | 1 |
| 13.627 | 59.051 | 12.266 | 53.153 | 11.424 | 49.506 | 49.906 | 2 |
| 41.306 | 178.992 | 32.951 | 142.786 | 30.030 | 130.132 | 122.399 | 2 |
| 59.750 | 258.918 | 54.613 | 236.657 | 51.955 | 225.139 | 208.308 | 1 |
| 75.686 | 327.975 | 57.438 | 248.897 | 52.592 | 227.897 | 220.949 | 1 |
| 107.259 | 464.788 | 89.685 | 388.635 | 83.999 | 363.995 | 345.050 | 2 |
| 156.406 | 677.761 | 132.033 | 572.143 | 122.324 | 530.069 | 502.813 | 2 |
| 213.693 | 926.002 | 172.604 | 747.953 | 156.876 | 679.794 | 625.255 | 1 |
| 217.180 | 941.113 | 192.661 | 834.863 | 186.323 | 807.398 | 790.386 | 1 |
| 280.562 | 1215.767 | 232.571 | 1007.807 | 218.448 | 946.610 | 895.853 | 2 |
| 358.903 | 1555.247 | 320.370 | 1388.268 | 1320.110 | 2 | ||
| 400.372 | 1734.945 | 343.876 | 1490.131 | 1392.494 | 1 | ||
| Level 0(Q) | Level 1(Q) | Level 2(Q) | Hanoi Attractor | ||||
| Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Mult. |
| 10.247 | 44.402 | 8.578 | 37.173 | 7.896 | 34.216 | 33.676 | 1 |
| 13.627 | 59.051 | 12.266 | 53.153 | 11.424 | 49.506 | 49.906 | 2 |
| 41.306 | 178.992 | 32.951 | 142.786 | 30.030 | 130.132 | 122.399 | 2 |
| 59.750 | 258.918 | 54.613 | 236.657 | 51.955 | 225.139 | 208.308 | 1 |
| 75.686 | 327.975 | 57.438 | 248.897 | 52.592 | 227.897 | 220.949 | 1 |
| 107.259 | 464.788 | 89.685 | 388.635 | 83.999 | 363.995 | 345.050 | 2 |
| 156.406 | 677.761 | 132.033 | 572.143 | 122.324 | 530.069 | 502.813 | 2 |
| 213.693 | 926.002 | 172.604 | 747.953 | 156.876 | 679.794 | 625.255 | 1 |
| 217.180 | 941.113 | 192.661 | 834.863 | 186.323 | 807.398 | 790.386 | 1 |
| 280.562 | 1215.767 | 232.571 | 1007.807 | 218.448 | 946.610 | 895.853 | 2 |
| 358.903 | 1555.247 | 320.370 | 1388.268 | 1320.110 | 2 | ||
| 400.372 | 1734.945 | 343.876 | 1490.131 | 1392.494 | 1 | ||
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