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

* Corresponding author

Received  April 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is partially supported by a Feodor Lynen Fellowship from the Humboldt Foundation

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/.

Citation: P. Alonso Ruiz, Y. Chen, H. Gu, R. S. Strichartz, Z. Zhou. Analysis on hybrid fractals. Communications on Pure & Applied Analysis, 2020, 19 (1) : 47-84. doi: 10.3934/cpaa.2020004
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. Google Scholar

[2]

E. AkkermansO. 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. Google Scholar

[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.Google Scholar

[4]

P. Alonso RuizU. 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. Google Scholar

[5]

P. Alonso-RuizD. 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. Google Scholar

[6]

Patricia Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys., 58 (2017), 073503, 16. doi: 10.1063/1.4994197. Google Scholar

[7]

N. BajorinT. ChenA. DaganC. EmmonsM. HusseinM. KhalilP. ModyB. 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. Google Scholar

[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. Google Scholar

[9]

O. Ben-BassatR. 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. Google Scholar

[10]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. Google Scholar

[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. Google Scholar

[12]

Y. Chen, H. Gu, R. S. Strichartz and Z. Zhou, Hybrid fractals, Available at http://www.math.cornell.edu/~harry970804/, 2017.Google Scholar

[13]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784. Google Scholar

[14]

B. M. HamblyV. 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. Google Scholar

[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. Google Scholar

[16]

M. Hata, On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 99-102. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

M. IonescuL. 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. Google Scholar

[20]

J. Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar

[21]

_____, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721–755. doi: 10.2307/2154402. Google Scholar

[22]

_____, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar

[23]

_____, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), ⅵ+132.Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

J. Murai, Diffusion processes on mandala, Osaka J. Math., 32 (1995), 887-917. Google Scholar

[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. Google Scholar

[28]

R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127. doi: 10.1006/jfan.2000.3580. Google Scholar

[29]

_____, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial.Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

E. AkkermansO. 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. Google Scholar

[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.Google Scholar

[4]

P. Alonso RuizU. 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. Google Scholar

[5]

P. Alonso-RuizD. 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. Google Scholar

[6]

Patricia Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys., 58 (2017), 073503, 16. doi: 10.1063/1.4994197. Google Scholar

[7]

N. BajorinT. ChenA. DaganC. EmmonsM. HusseinM. KhalilP. ModyB. 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. Google Scholar

[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. Google Scholar

[9]

O. Ben-BassatR. 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. Google Scholar

[10]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. Google Scholar

[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. Google Scholar

[12]

Y. Chen, H. Gu, R. S. Strichartz and Z. Zhou, Hybrid fractals, Available at http://www.math.cornell.edu/~harry970804/, 2017.Google Scholar

[13]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784. Google Scholar

[14]

B. M. HamblyV. 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. Google Scholar

[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. Google Scholar

[16]

M. Hata, On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 99-102. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

M. IonescuL. 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. Google Scholar

[20]

J. Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar

[21]

_____, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721–755. doi: 10.2307/2154402. Google Scholar

[22]

_____, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar

[23]

_____, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), ⅵ+132.Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

J. Murai, Diffusion processes on mandala, Osaka J. Math., 32 (1995), 887-917. Google Scholar

[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. Google Scholar

[28]

R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127. doi: 10.1006/jfan.2000.3580. Google Scholar

[29]

_____, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial.Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  The Hanoi attractor or stretched Sierpinski gasket
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 10.  Approximating graphs $ \Gamma_0 $, $ \Gamma_1 $ and $ \Gamma_2 $
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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