November  2015, 14(6): 2509-2533. doi: 10.3934/cpaa.2015.14.2509

Gaps in the spectrum of the Laplacian on $3N$-Gaskets

1. 

Department of Mathematics, Purdue University, West Lafayette, IN, 47907, United States

2. 

Department of Mathematics, University of Connecticut, Storrs, CT 06269, United States, United States, United States, United States, United States

Received  June 2014 Revised  March 2015 Published  September 2015

This article develops analysis on fractal $3N$-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for $N=1$, specifically properties of the Laplacian $\Delta$ on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum.
Citation: D. Kelleher, N. Gupta, M. Margenot, J. Marsh, W. Oakley, A. Teplyaev. Gaps in the spectrum of the Laplacian on $3N$-Gaskets. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2509-2533. doi: 10.3934/cpaa.2015.14.2509
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show all references

References:
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S. Aaron, Z. Conn, R. S. Strichartz and H. Yu, Hodge-de Rham theory on fractal graphs and fractals, Commun. Pure Appl. Anal., 13 (2014), 903-928. doi: 10.3934/cpaa.2014.13.903.

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E. Akkermans, G. V. Dunne and A. Teplyaev, Physical consequences of complex dimensions of fractals, EPL (Europhysics Letters), 88 (2009), 40007.

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E. Akkermans, G. V. Dunne and A. Teplyaev, Thermodynamics of photons on fractals, Phys. Rev. Lett., 105 (2010), 230407.

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[5]

S. Alexander, Some properties of the spectrum of the sierpinski gasket in a magnetic field, Phys. Rev. B, 29 (1984), 5504-5508.

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J. Ambjørn, R. Loll, J. L. Nielsen and J. Rolf, Euclidean and Lorentzian quantum gravity-lessons from two dimensions, Chaos Solitons Fractals, 10 (1999), 177-195. doi: 10.1016/S0960-0779(98)00197-0.

[7]

N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev, Vibration modes of $3n$-gaskets and other fractals, J. Phys. A, 41 (2008), 015101, 21. doi: 10.1088/1751-8113/41/1/015101.

[8]

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[9]

J. R. Banavar, L. Kadanoff and A. M. Pruisken, Energy spectrum for a fractal lattice in a magnetic field, Phys. Rev., B Condens. Matter, 31 (1985), 1388-1395.

[10]

M. T. Barlow and D. Nualart, Lectures on Probability Theory and Statistics, vol. 1690 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998, Lectures from the 25th Saint-Flour Summer School held July 10-26, 1995 (P. Bernard ed.).

[11]

M. T. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS), 12 (2010), 655-701.

[12]

M. Begue, D. J. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and the Strichartz hexacarpet, Exp. Math., 21 (2012), 402-417.

[13]

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[14]

B. Boyle, K. Cekala, D. Ferrone, N. Rifkin and A. Teplyaev, Electrical resistance of $N$-gasket fractal networks, Pacific J. Math., 233 (2007), 15-40. doi: 10.2140/pjm.2007.233.15.

[15]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.

[16]

Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, vol. 35 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2012.

[17]

G. Derfel, P. J. Grabner and F. Vogl, The zeta function of the Laplacian on certain fractals, Trans. Amer. Math. Soc., 360 (2008), 881-897 (electronic). doi: 10.1090/S0002-9947-07-04240-7.

[18]

G. Derfel, P. J. Grabner and F. Vogl, Laplace operators on fractals and related functional equations, J. Phys. A, 45 (2012), 463001, 34. doi: 10.1088/1751-8113/45/46/463001.

[19]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets, Illinois J. Math., 53 (2009), 915-937 (2010).

[20]

G. V. Dunne, Heat kernels and zeta functions on fractals, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 374016. doi: 10.1088/1751-8113/45/37/374016.

[21]

F. Englert, J.-M. Frère, M. Rooman and P. Spindel, Metric space-time as fixed point of the renormalization group equations on fractal structures, Nuclear Phys. B, 280 (1987), 147-180. doi: 10.1016/0550-3213(87)90142-8.

[22]

D. J. Ford and B. Steinhurst, Vibration spectra of the m-tree fractal,, \emph{Fractals-complex Geometry Patterns and Scaling in Nature and Society}, 18 ().  doi: 10.1142/S0218348X1000483X.

[23]

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

[24]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric MArkov Processes, vol. 19 of de Gruyter Studies in Mathematics,, extended edition, (). 

[25]

B. M. Hambly, V. Metz and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc. (2), 74 (2006), 93-112. doi: 10.1112/S002461070602312X.

[26]

K. E. Hare, B. A. Steinhurst, A. Teplyaev and D. Zhou, Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals, Math. Res. Lett., 19 (2012), 537-553. doi: 10.4310/MRL.2012.v19.n3.a3.

[27]

K. E. Hare and D. Zhou, Gaps in the ratios of the spectra of Laplacians on fractals, Fractals, 17 (2009), 523-535. doi: 10.1142/S0218348X0900451X.

[28]

M. Hinz, 1-forms and polar decomposition on harmonic spaces, Potential Anal., 38 (2013), 261-279. doi: 10.1007/s11118-012-9272-2.

[29]

M. Hinz, Magnetic energies and feynman-kac-ito formulas for symmetric markov processes,, \emph{arXiv:1409.7743}., (). 

[30]

M. Hinz, D. J. Kelleher and A. Teplyaev, Measures and Dirichlet forms under the Gelfand transform, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 408 (2012), 303-322, 329-330.

[31]

M. Hinz, D. J. Kelleher and A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms, Journal of Noncommutative Geometry, vol. 9, no. 2 (2015). doi: 10.4171/JNCG/195.

[32]

M. Hinz, M. Röckner and A. Teplyaev, Vector analysis for Dirichlet forms and quasilinear {PDE} and SPDE on metric measure spaces, Stochastic Process. Appl., 123 (2013), 4373-4406. doi: 10.1016/j.spa.2013.06.009.

[33]

M. Hinz and L. G. Rogers, Magnetic fields on resistance spaces,, \emph{arXiv:1501.01100}., (). 

[34]

M. Hinz and A. Teplyaev, Dirac and magnetic Schrödinger operators on fractals, J. Funct. Anal., 265 (2013), 2830-2854. doi: 10.1016/j.jfa.2013.07.021.

[35]

M. Hinz and A. Teplyaev, Vector analysis on fractals and applications,, in \emph{Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in applied mathematics, ().  doi: 10.1090/conm/601/11960.

[36]

M. Hinz and A. Teplyaev, Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals, Trans. Amer. Math. Soc., 367 (2015), 1347-1380. doi: 10.1090/S0002-9947-2014-06203-X.

[37]

M. Ionescu, E. P. J. Pearse, L. G. Rogers, H.-J. Ruan and R. S. Strichartz, The resolvent kernel for PCF self-similar fractals, Trans. Amer. Math. Soc., 362 (2010), 4451-4479. doi: 10.1090/S0002-9947-10-05098-1.

[38]

M. Ionescu, L. G. Rogers and R. S. Strichartz, Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190. doi: 10.4171/RMI/752.

[39]

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