Analysis on hybrid fractals

P. Alonso Ruiz; Y. Chen; H. Gu; R. S. Strichartz; Z. Zhou

doi:10.3934/cpaa.2020004

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January 2020, 19(1): 47-84. doi: 10.3934/cpaa.2020004

Analysis on hybrid fractals

P. Alonso Ruiz ^1,, , Y. Chen ^2, , H. Gu ^3, , R. S. Strichartz ^4, and Z. Zhou ^5,

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/.

Keywords: Hybrid fractal, resistance form, energy, Laplacian, spectrum.

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 & 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. 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. 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 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. Google Scholar
[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. 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. 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. 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-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. 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. 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. 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. 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. 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. 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. 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 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. Google Scholar
[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. 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. 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. 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-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. 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. 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. 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. 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. 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

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 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 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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	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

American Institute of Mathematical Sciences

Analysis on hybrid fractals

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