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 & Applied Analysis, 2015, 14 (6) : 2509-2533. doi: 10.3934/cpaa.2015.14.2509
References:
[1]

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

[2]

E. Akkermans, G. V. Dunne and A. Teplyaev, Physical consequences of complex dimensions of fractals,, \emph{EPL (Europhysics Letters)}, 88 (2009).   Google Scholar

[3]

E. Akkermans, G. V. Dunne and A. Teplyaev, Thermodynamics of photons on fractals,, \emph{Phys. Rev. Lett.}, 105 (2010).   Google Scholar

[4]

E. Akkermans, O. Benichou, G. V. Dunne, A. Teplyaev and R. Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals,, \emph{Phys. Rev. E}, 86 (2012).   Google Scholar

[5]

S. Alexander, Some properties of the spectrum of the sierpinski gasket in a magnetic field,, \emph{Phys. Rev. B}, 29 (1984), 5504.   Google Scholar

[6]

J. Ambjørn, R. Loll, J. L. Nielsen and J. Rolf, Euclidean and Lorentzian quantum gravity-lessons from two dimensions,, \emph{Chaos Solitons Fractals}, 10 (1999), 177.  doi: 10.1016/S0960-0779(98)00197-0.  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 $3n$-gaskets and other fractals,, \emph{J. Phys. A}, 41 (2008).  doi: 10.1088/1751-8113/41/1/015101.  Google Scholar

[8]

N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev, Vibration spectra of finitely ramified, symmetric fractals,, \emph{Fractals}, 16 (2008), 243.  doi: 10.1142/S0218348X08004010.  Google Scholar

[9]

J. R. Banavar, L. Kadanoff and A. M. Pruisken, Energy spectrum for a fractal lattice in a magnetic field,, \emph{Phys. Rev., 31 (1985), 1388.   Google Scholar

[10]

M. T. Barlow and D. Nualart, Lectures on Probability Theory and Statistics, vol. 1690 of Lecture Notes in Mathematics,, Springer-Verlag, (1998), 10.   Google Scholar

[11]

M. T. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets,, \emph{J. Eur. Math. Soc. (JEMS)}, 12 (2010), 655.   Google Scholar

[12]

M. Begue, D. J. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and the Strichartz hexacarpet,, \emph{Exp. Math.}, 21 (2012), 402.   Google Scholar

[13]

J. Bellissard, Renormalization group analysis and quasicrystals,, in \emph{Ideas and Methods in Quantum and Statistical Physics (Oslo, (1988).   Google Scholar

[14]

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

[15]

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

[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, (2012).   Google Scholar

[17]

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

[18]

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

[19]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets,, \emph{Illinois J. Math.}, 53 (2009), 915.   Google Scholar

[20]

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

[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,, \emph{Nuclear Phys. B}, 280 (1987), 147.  doi: 10.1016/0550-3213(87)90142-8.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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,, \emph{Math. Res. Lett.}, 19 (2012), 537.  doi: 10.4310/MRL.2012.v19.n3.a3.  Google Scholar

[27]

K. E. Hare and D. Zhou, Gaps in the ratios of the spectra of Laplacians on fractals,, \emph{Fractals}, 17 (2009), 523.  doi: 10.1142/S0218348X0900451X.  Google Scholar

[28]

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

[29]

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

[30]

M. Hinz, D. J. Kelleher and A. Teplyaev, Measures and Dirichlet forms under the Gelfand transform,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 408 (2012), 303.   Google Scholar

[31]

M. Hinz, D. J. Kelleher and A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms,, \emph{Journal of Noncommutative Geometry}, (2015).  doi: 10.4171/JNCG/195.  Google Scholar

[32]

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

[33]

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

[34]

M. Hinz and A. Teplyaev, Dirac and magnetic Schrödinger operators on fractals,, \emph{J. Funct. Anal.}, 265 (2013), 2830.  doi: 10.1016/j.jfa.2013.07.021.  Google Scholar

[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.  Google Scholar

[36]

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

[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,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4451.  doi: 10.1090/S0002-9947-10-05098-1.  Google Scholar

[38]

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

[39]

M. Ionescu, L. G. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals,, \emph{J. Funct. Anal.}, 263 (2012), 2141.  doi: 10.1016/j.jfa.2012.05.021.  Google Scholar

[40]

N. Kajino, Spectral asymptotics for Laplacians on self-similar sets,, \emph{J. Funct. Anal.}, 258 (2010), 1310.  doi: 10.1016/j.jfa.2009.11.001.  Google Scholar

[41]

N. Kajino, Non-regularly varying and non-periodic oscillation of the on-diagonal heat kernels on self-similar fractals,, in \emph{Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in Applied Mathematics}, ().  doi: 10.1090/conm/601/11935.  Google Scholar

[42]

N. Kajino, On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals,, \emph{Probab. Theory Related Fields}, 156 (2013), 51.  doi: 10.1007/s00440-012-0420-9.  Google Scholar

[43]

N. Kajino, Log-periodic asymptotic expansion of the spectral partition function for self-similar sets,, \emph{Comm. Math. Phys.}, 328 (2014), 1341.  doi: 10.1007/s00220-014-1922-3.  Google Scholar

[44]

C. J. Kauffman, R. M. Kesler, A. G. Parshall, E. A. Stamey and B. A. Steinhurst, Quantum mechanics on laakso spaces,, \emph{Journal of Mathematical Physics}, 53 (2012).  doi: 10.1063/1.3702099.  Google Scholar

[45]

D. J. Kelleher, B. A. Steinhurst and C.-M. M. Wong, From self-similar structures to self-similar groups, \emph{Internat. J. Algebra Comput.}, 22 (2012).  doi: 10.1142/S0218196712500567.  Google Scholar

[46]

J. Kigami, Harmonic calculus on p.c.f. self-similar sets,, \emph{Trans. Amer. Math. Soc.}, 335 (1993), 721.  doi: 10.2307/2154402.  Google Scholar

[47]

J. Kigami, Analysis on fractals, vol. 143 of Cambridge Tracts in Mathematics,, Cambridge University Press, (2001).  doi: 10.1017/CBO9780511470943.  Google Scholar

[48]

J. Kigami, Harmonic analysis for resistance forms,, \emph{J. Funct. Anal.}, 204 (2003), 399.  doi: 10.1016/S0022-1236(02)00149-0.  Google Scholar

[49]

J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates,, \emph{Mem. Amer. Math. Soc.}, 216 (2012).  doi: 10.1090/S0065-9266-2011-00632-5.  Google Scholar

[50]

J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals,, \emph{Comm. Math. Phys.}, 158 (1993), 93.   Google Scholar

[51]

N. Lal and M. L. Lapidus, Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals,, \emph{J. Phys. A}, 45 (2012).  doi: 10.1088/1751-8113/45/36/365205.  Google Scholar

[52]

M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings,, \emph{J. London Math. Soc. (2)}, 52 (1995), 15.  doi: 10.1112/jlms/52.1.15.  Google Scholar

[53]

M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions,, 2nd edition, (2013).  doi: 10.1007/978-1-4614-2176-4.  Google Scholar

[54]

O. Lauscher and M. Reuter, Asymptotic safety in quantum Einstein gravity: nonperturbative renormalizability and fractal spacetime structure,, in \emph{Quantum Gravity}, (2007), 293.  doi: 10.1007/978-3-7643-7978-0_15.  Google Scholar

[55]

L. Malozemov and A. Teplyaev, Pure point spectrum of the Laplacians on fractal graphs,, \emph{J. Funct. Anal.}, 129 (1995), 390.  doi: 10.1006/jfan.1995.1056.  Google Scholar

[56]

L. Malozemov and A. Teplyaev, Self-similarity, operators and dynamics,, \emph{Math. Phys. Anal. Geom.}, 6 (2003), 201.  doi: 10.1023/A:1024931603110.  Google Scholar

[57]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials,, Chapman & Hall/CRC, (2003).   Google Scholar

[58]

K. A. Okoudjou, L. G. Rogers and R. S. Strichartz, Szegö limit theorems on the Sierpiński gasket,, \emph{J. Fourier Anal. Appl.}, 16 (2010), 434.  doi: 10.1007/s00041-009-9102-0.  Google Scholar

[59]

K. A. Okoudjou, L. Saloff-Coste and A. Teplyaev, Weak uncertainty principle for fractals, graphs and metric measure spaces,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 3857.  doi: 10.1090/S0002-9947-08-04472-3.  Google Scholar

[60]

R. Peirone, Existence of eigenforms on nicely separated fractals,, in \emph{Analysis on Graphs and its Applications}, (2008), 231.  doi: 10.1090/pspum/077/2459872.  Google Scholar

[61]

R. Peirone, Existence of self-similar energies on finitely ramified fractals,, \emph{J. Anal. Math.}, 123 (2014), 35.  doi: 10.1007/s11854-014-013-x.  Google Scholar

[62]

R. Peirone, Uniqueness of eigenforms on fractals,, \emph{Math. Nachr.}, 287 (2014), 453.  doi: 10.1002/mana.201200247.  Google Scholar

[63]

O. Post, Spectral Analysis on Graph-like Spaces, vol. 2039 of Lecture Notes in Mathematics,, Springer, (2012).  doi: 10.1007/978-3-642-23840-6.  Google Scholar

[64]

R. Rammal, Spectrum of harmonic excitations on fractals,, \emph{J. Physique}, 45 (1984), 191.   Google Scholar

[65]

R. Rammal and G. Toulouse, Spectrum of the Schrödinger equation on a self-similar structure,, \emph{Phys. Rev. Lett.}, 49 (1982), 1194.  doi: 10.1103/PhysRevLett.49.1194.  Google Scholar

[66]

L. G. Rogers, Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups,, \emph{Trans. Amer. Math. Soc.}, 364 (2012), 1633.  doi: 10.1090/S0002-9947-2011-05551-0.  Google Scholar

[67]

L. G. Rogers and R. S. Strichartz, Distribution theory on P.C.F. fractals,, \emph{J. Anal. Math.}, 112 (2010), 137.  doi: 10.1007/s11854-010-0027-y.  Google Scholar

[68]

T. Shima, On eigenvalue problems for the random walks on the Sierpiński pre-gaskets,, \emph{Japan J. Indust. Appl. Math.}, 8 (1991), 127.  doi: 10.1007/BF03167188.  Google Scholar

[69]

B. A. Steinhurst and A. Teplyaev, Existence of a meromorphic extension of spectral zeta functions on fractals,, \emph{Lett. Math. Phys.}, 103 (2013), 1377.  doi: 10.1007/s11005-013-0649-y.  Google Scholar

[70]

R. S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series,, \emph{Math. Res. Lett.}, 12 (2005), 269.  doi: 10.4310/MRL.2005.v12.n2.a12.  Google Scholar

[71]

R. S. Strichartz, Differential Equations on Fractals, A tutorial,, Princeton University Press, (2006).   Google Scholar

[72]

R. S. Strichartz and A. Teplyaev, Spectral analysis on infinite Sierpiński fractafolds,, \emph{J. Anal. Math.}, 116 (2012), 255.  doi: 10.1007/s11854-012-0007-5.  Google Scholar

[73]

A. Teplyaev, Spectral analysis on infinite Sierpiński gaskets,, \emph{J. Funct. Anal.}, 159 (1998), 537.  doi: 10.1006/jfan.1998.3297.  Google Scholar

[74]

A. Teplyaev, Spectral zeta functions of fractals and the complex dynamics of polynomials,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 4339.  doi: 10.1090/S0002-9947-07-04150-5.  Google Scholar

[75]

A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure,, \emph{Canad. J. Math.}, 60 (2008), 457.  doi: 10.4153/CJM-2008-022-3.  Google Scholar

[76]

J. T. Tyson and J.-M. Wu, Characterizations of snowflake metric spaces,, \emph{Ann. Acad. Sci. Fenn. Math.}, 30 (2005), 313.   Google Scholar

[77]

J. T. Tyson and J.-M. Wu, Quasiconformal dimensions of self-similar fractals,, \emph{Rev. Mat. Iberoam.}, 22 (2006), 205.  doi: 10.4171/RMI/454.  Google Scholar

show all references

References:
[1]

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

[2]

E. Akkermans, G. V. Dunne and A. Teplyaev, Physical consequences of complex dimensions of fractals,, \emph{EPL (Europhysics Letters)}, 88 (2009).   Google Scholar

[3]

E. Akkermans, G. V. Dunne and A. Teplyaev, Thermodynamics of photons on fractals,, \emph{Phys. Rev. Lett.}, 105 (2010).   Google Scholar

[4]

E. Akkermans, O. Benichou, G. V. Dunne, A. Teplyaev and R. Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals,, \emph{Phys. Rev. E}, 86 (2012).   Google Scholar

[5]

S. Alexander, Some properties of the spectrum of the sierpinski gasket in a magnetic field,, \emph{Phys. Rev. B}, 29 (1984), 5504.   Google Scholar

[6]

J. Ambjørn, R. Loll, J. L. Nielsen and J. Rolf, Euclidean and Lorentzian quantum gravity-lessons from two dimensions,, \emph{Chaos Solitons Fractals}, 10 (1999), 177.  doi: 10.1016/S0960-0779(98)00197-0.  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 $3n$-gaskets and other fractals,, \emph{J. Phys. A}, 41 (2008).  doi: 10.1088/1751-8113/41/1/015101.  Google Scholar

[8]

N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev, Vibration spectra of finitely ramified, symmetric fractals,, \emph{Fractals}, 16 (2008), 243.  doi: 10.1142/S0218348X08004010.  Google Scholar

[9]

J. R. Banavar, L. Kadanoff and A. M. Pruisken, Energy spectrum for a fractal lattice in a magnetic field,, \emph{Phys. Rev., 31 (1985), 1388.   Google Scholar

[10]

M. T. Barlow and D. Nualart, Lectures on Probability Theory and Statistics, vol. 1690 of Lecture Notes in Mathematics,, Springer-Verlag, (1998), 10.   Google Scholar

[11]

M. T. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets,, \emph{J. Eur. Math. Soc. (JEMS)}, 12 (2010), 655.   Google Scholar

[12]

M. Begue, D. J. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and the Strichartz hexacarpet,, \emph{Exp. Math.}, 21 (2012), 402.   Google Scholar

[13]

J. Bellissard, Renormalization group analysis and quasicrystals,, in \emph{Ideas and Methods in Quantum and Statistical Physics (Oslo, (1988).   Google Scholar

[14]

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

[15]

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

[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, (2012).   Google Scholar

[17]

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

[18]

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

[19]

S. Drenning and R. S. Strichartz, Spectral decimation on Hambly's homogeneous hierarchical gaskets,, \emph{Illinois J. Math.}, 53 (2009), 915.   Google Scholar

[20]

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

[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,, \emph{Nuclear Phys. B}, 280 (1987), 147.  doi: 10.1016/0550-3213(87)90142-8.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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,, \emph{Math. Res. Lett.}, 19 (2012), 537.  doi: 10.4310/MRL.2012.v19.n3.a3.  Google Scholar

[27]

K. E. Hare and D. Zhou, Gaps in the ratios of the spectra of Laplacians on fractals,, \emph{Fractals}, 17 (2009), 523.  doi: 10.1142/S0218348X0900451X.  Google Scholar

[28]

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

[29]

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

[30]

M. Hinz, D. J. Kelleher and A. Teplyaev, Measures and Dirichlet forms under the Gelfand transform,, \emph{Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, 408 (2012), 303.   Google Scholar

[31]

M. Hinz, D. J. Kelleher and A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms,, \emph{Journal of Noncommutative Geometry}, (2015).  doi: 10.4171/JNCG/195.  Google Scholar

[32]

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

[33]

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

[34]

M. Hinz and A. Teplyaev, Dirac and magnetic Schrödinger operators on fractals,, \emph{J. Funct. Anal.}, 265 (2013), 2830.  doi: 10.1016/j.jfa.2013.07.021.  Google Scholar

[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.  Google Scholar

[36]

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

[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,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4451.  doi: 10.1090/S0002-9947-10-05098-1.  Google Scholar

[38]

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

[39]

M. Ionescu, L. G. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals,, \emph{J. Funct. Anal.}, 263 (2012), 2141.  doi: 10.1016/j.jfa.2012.05.021.  Google Scholar

[40]

N. Kajino, Spectral asymptotics for Laplacians on self-similar sets,, \emph{J. Funct. Anal.}, 258 (2010), 1310.  doi: 10.1016/j.jfa.2009.11.001.  Google Scholar

[41]

N. Kajino, Non-regularly varying and non-periodic oscillation of the on-diagonal heat kernels on self-similar fractals,, in \emph{Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in Applied Mathematics}, ().  doi: 10.1090/conm/601/11935.  Google Scholar

[42]

N. Kajino, On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals,, \emph{Probab. Theory Related Fields}, 156 (2013), 51.  doi: 10.1007/s00440-012-0420-9.  Google Scholar

[43]

N. Kajino, Log-periodic asymptotic expansion of the spectral partition function for self-similar sets,, \emph{Comm. Math. Phys.}, 328 (2014), 1341.  doi: 10.1007/s00220-014-1922-3.  Google Scholar

[44]

C. J. Kauffman, R. M. Kesler, A. G. Parshall, E. A. Stamey and B. A. Steinhurst, Quantum mechanics on laakso spaces,, \emph{Journal of Mathematical Physics}, 53 (2012).  doi: 10.1063/1.3702099.  Google Scholar

[45]

D. J. Kelleher, B. A. Steinhurst and C.-M. M. Wong, From self-similar structures to self-similar groups, \emph{Internat. J. Algebra Comput.}, 22 (2012).  doi: 10.1142/S0218196712500567.  Google Scholar

[46]

J. Kigami, Harmonic calculus on p.c.f. self-similar sets,, \emph{Trans. Amer. Math. Soc.}, 335 (1993), 721.  doi: 10.2307/2154402.  Google Scholar

[47]

J. Kigami, Analysis on fractals, vol. 143 of Cambridge Tracts in Mathematics,, Cambridge University Press, (2001).  doi: 10.1017/CBO9780511470943.  Google Scholar

[48]

J. Kigami, Harmonic analysis for resistance forms,, \emph{J. Funct. Anal.}, 204 (2003), 399.  doi: 10.1016/S0022-1236(02)00149-0.  Google Scholar

[49]

J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates,, \emph{Mem. Amer. Math. Soc.}, 216 (2012).  doi: 10.1090/S0065-9266-2011-00632-5.  Google Scholar

[50]

J. Kigami and M. L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals,, \emph{Comm. Math. Phys.}, 158 (1993), 93.   Google Scholar

[51]

N. Lal and M. L. Lapidus, Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals,, \emph{J. Phys. A}, 45 (2012).  doi: 10.1088/1751-8113/45/36/365205.  Google Scholar

[52]

M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings,, \emph{J. London Math. Soc. (2)}, 52 (1995), 15.  doi: 10.1112/jlms/52.1.15.  Google Scholar

[53]

M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions,, 2nd edition, (2013).  doi: 10.1007/978-1-4614-2176-4.  Google Scholar

[54]

O. Lauscher and M. Reuter, Asymptotic safety in quantum Einstein gravity: nonperturbative renormalizability and fractal spacetime structure,, in \emph{Quantum Gravity}, (2007), 293.  doi: 10.1007/978-3-7643-7978-0_15.  Google Scholar

[55]

L. Malozemov and A. Teplyaev, Pure point spectrum of the Laplacians on fractal graphs,, \emph{J. Funct. Anal.}, 129 (1995), 390.  doi: 10.1006/jfan.1995.1056.  Google Scholar

[56]

L. Malozemov and A. Teplyaev, Self-similarity, operators and dynamics,, \emph{Math. Phys. Anal. Geom.}, 6 (2003), 201.  doi: 10.1023/A:1024931603110.  Google Scholar

[57]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials,, Chapman & Hall/CRC, (2003).   Google Scholar

[58]

K. A. Okoudjou, L. G. Rogers and R. S. Strichartz, Szegö limit theorems on the Sierpiński gasket,, \emph{J. Fourier Anal. Appl.}, 16 (2010), 434.  doi: 10.1007/s00041-009-9102-0.  Google Scholar

[59]

K. A. Okoudjou, L. Saloff-Coste and A. Teplyaev, Weak uncertainty principle for fractals, graphs and metric measure spaces,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 3857.  doi: 10.1090/S0002-9947-08-04472-3.  Google Scholar

[60]

R. Peirone, Existence of eigenforms on nicely separated fractals,, in \emph{Analysis on Graphs and its Applications}, (2008), 231.  doi: 10.1090/pspum/077/2459872.  Google Scholar

[61]

R. Peirone, Existence of self-similar energies on finitely ramified fractals,, \emph{J. Anal. Math.}, 123 (2014), 35.  doi: 10.1007/s11854-014-013-x.  Google Scholar

[62]

R. Peirone, Uniqueness of eigenforms on fractals,, \emph{Math. Nachr.}, 287 (2014), 453.  doi: 10.1002/mana.201200247.  Google Scholar

[63]

O. Post, Spectral Analysis on Graph-like Spaces, vol. 2039 of Lecture Notes in Mathematics,, Springer, (2012).  doi: 10.1007/978-3-642-23840-6.  Google Scholar

[64]

R. Rammal, Spectrum of harmonic excitations on fractals,, \emph{J. Physique}, 45 (1984), 191.   Google Scholar

[65]

R. Rammal and G. Toulouse, Spectrum of the Schrödinger equation on a self-similar structure,, \emph{Phys. Rev. Lett.}, 49 (1982), 1194.  doi: 10.1103/PhysRevLett.49.1194.  Google Scholar

[66]

L. G. Rogers, Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups,, \emph{Trans. Amer. Math. Soc.}, 364 (2012), 1633.  doi: 10.1090/S0002-9947-2011-05551-0.  Google Scholar

[67]

L. G. Rogers and R. S. Strichartz, Distribution theory on P.C.F. fractals,, \emph{J. Anal. Math.}, 112 (2010), 137.  doi: 10.1007/s11854-010-0027-y.  Google Scholar

[68]

T. Shima, On eigenvalue problems for the random walks on the Sierpiński pre-gaskets,, \emph{Japan J. Indust. Appl. Math.}, 8 (1991), 127.  doi: 10.1007/BF03167188.  Google Scholar

[69]

B. A. Steinhurst and A. Teplyaev, Existence of a meromorphic extension of spectral zeta functions on fractals,, \emph{Lett. Math. Phys.}, 103 (2013), 1377.  doi: 10.1007/s11005-013-0649-y.  Google Scholar

[70]

R. S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series,, \emph{Math. Res. Lett.}, 12 (2005), 269.  doi: 10.4310/MRL.2005.v12.n2.a12.  Google Scholar

[71]

R. S. Strichartz, Differential Equations on Fractals, A tutorial,, Princeton University Press, (2006).   Google Scholar

[72]

R. S. Strichartz and A. Teplyaev, Spectral analysis on infinite Sierpiński fractafolds,, \emph{J. Anal. Math.}, 116 (2012), 255.  doi: 10.1007/s11854-012-0007-5.  Google Scholar

[73]

A. Teplyaev, Spectral analysis on infinite Sierpiński gaskets,, \emph{J. Funct. Anal.}, 159 (1998), 537.  doi: 10.1006/jfan.1998.3297.  Google Scholar

[74]

A. Teplyaev, Spectral zeta functions of fractals and the complex dynamics of polynomials,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 4339.  doi: 10.1090/S0002-9947-07-04150-5.  Google Scholar

[75]

A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure,, \emph{Canad. J. Math.}, 60 (2008), 457.  doi: 10.4153/CJM-2008-022-3.  Google Scholar

[76]

J. T. Tyson and J.-M. Wu, Characterizations of snowflake metric spaces,, \emph{Ann. Acad. Sci. Fenn. Math.}, 30 (2005), 313.   Google Scholar

[77]

J. T. Tyson and J.-M. Wu, Quasiconformal dimensions of self-similar fractals,, \emph{Rev. Mat. Iberoam.}, 22 (2006), 205.  doi: 10.4171/RMI/454.  Google Scholar

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