Laplace operators on fractals and related functional equations

We give an overview over the application of functional equations, namely the classical Poincare and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the Euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to compute the Casimir energy of fractals. We give numerical values for this energy for the Sierpinski gasket.

[1]  Barry Simon,et al.  Notes on infinite determinants of Hilbert space operators , 1977 .

[2]  Michel L. Lapidus,et al.  Fractal Geometry, Complex Dimensions and Zeta Functions , 2006 .

[3]  Kenneth Falconer,et al.  The Hausdorff dimension of self-affine fractals , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  M. Vergne,et al.  Heat Kernels and Dirac Operators: Grundlehren 298 , 1992 .

[5]  S. Havlin,et al.  Diffusion in disordered media , 2002 .

[6]  Masayoshi Hata,et al.  On the structure of self-similar sets , 1985 .

[7]  LENGTH OF JULIA CURVES , 1995 .

[8]  Naotaka Kajino.  On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals , 2013 .

[9]  Peter J. Grabner,et al.  Functional iterations and stopping times for Brownian motion on the Sierpiński gasket , 1997 .

[10]  Elmar Teufl,et al.  On the asymptotic behaviour of analytic solutions of linear iterative functional equations , 2007 .

[11]  Elmar Teufl,et al.  Asymptotics of the transition probabilities of the simple random walk on self-similar graphs , 2002 .

[12]  D. Dawson,et al.  Probability and mathematical physics : a volume in honor of Stanislav Molchanov , 2007 .

[13]  Robert T. Seeley,et al.  Complex powers of an elliptic operator , 1967 .

[14]  Michel L. Lapidus,et al.  The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums , 1993 .

[15]  Richard F. Bass,et al.  Brownian Motion and Harmonic Analysis on Sierpinski Carpets , 1999, Canadian Journal of Mathematics.

[16]  M. Talagrand,et al.  Lectures on Probability Theory and Statistics , 2000 .

[17]  Michel L. Lapidus,et al.  Counterexamples to the modified Weyl–Berry conjecture on fractal drums , 1996, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  T. Kumagai.  Recent developments of analysis on fractals , 2008 .

[19]  S. Krantz.  Fractal geometry , 1989 .

[20]  A. Kirillov,et al.  A Tale of Two Fractals , 2013 .

[21]  Naotaka Kajino.  Spectral asymptotics for Laplacians on self-similar sets , 2010 .

[22]  Handbook of Geometric Analysis , 2010 .

[23]  Heat invariants of Riemannian manifolds , 1999, math/9905073.

[24]  Estimating the characteristic exponents of polynomials , 1997 .

[25]  Stephen W. Hawking.  Zeta function regularization of path integrals in curved spacetime , 1977 .

[26]  S. Karlin,et al.  Embedding iterates of analytic functions with two fixed points into continuous groups , 1968 .

[27]  V. Metz.  Renormalization contracts on nested fractals. , 1996 .

[28]  Peter B. Gilkey,et al.  Asymptotic Formulae in Spectral Geometry , 2003 .

[29]  Y. Oshima,et al.  Dirichlet forms, diffusion processes and spectral dimensions for nested fractals , 2010 .

[30]  A. Grigor’yan,et al.  Heat kernel upper bounds for jump processes and the first exit time , 2009 .

[31]  M. Solomyak,et al.  THE EIGENVALUE BEHAVIOUR FOR THE BOUNDARY VALUE PROBLEMS RELATED TO SELF-SIMILAR MEASURES ON R , 2004 .

[32]  S. Fulling,et al.  The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbol , 1988 .

[33]  A. Teplyaev,et al.  VIBRATION SPECTRA OF FINITELY RAMIFIED, SYMMETRIC FRACTALS , 2008 .

[34]  Alexander Teplyaev,et al.  Spatial log-periodic oscillations of first-passage observables in fractals. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  I. Gel'fand,et al.  REPRESENTATIONS OF THE GROUP OF DIFFEOMORPHISMS , 1975 .

[36]  Hans Triebel,et al.  Fractals and Spectra: Related to Fourier Analysis and Function Spaces , 1997 .

[37]  Martin T. Barlow,et al.  Diffusions on fractals , 1998 .

[38]  P. Grabner,et al.  The zeta function of the Laplacian on certain fractals , 2005, math/0508315.

[39]  G. Dunne.  Functional determinants in quantum field theory , 2007, 0711.1178.

[40]  C. Pommerenke.  On conformal mapping and iteration of rational functions , 1986 .

[41]  Fractal Geometry and Stochastics Ii , 2012 .

[42]  P. Grabner,et al.  Asymptotics of the Poincare Functions , 2007 .

[43]  Michel L. Lapidus,et al.  Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings , 2012 .

[44]  Michèle Vergne,et al.  Heat Kernels and Dirac Operators: Grundlehren 298 , 1992 .

[45]  S. Kusuoka,et al.  Dirichlet forms on fractals and products of random matrices , 1989 .

[46]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[47]  Physical consequences of complex dimensions of fractals , 2009, 0903.3681.

[48]  J. Bellissard.  Stability and instability in quantum mechanics , 1985 .

[49]  Spectral functions in mathematics and physics , 2000, hep-th/0007251.

[50]  Gerald V. Dunne,et al.  Heat kernels and zeta functions on fractals , 2012, 1205.2723.

[51]  G. Tenenbaum.  Introduction to Analytic and Probabilistic Number Theory , 1995 .

[52]  S. Minakshisundaram,et al.  Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds , 1949, Canadian Journal of Mathematics.

[53]  S. Dubuc.  étude théorique et numérique de la fonction de Karlin-McGregor , 1982 .

[54]  P. Gilkey.  The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary , 1975 .

[55]  M. V. Berry,et al.  Distribution of Modes in Fractal Resonators , 1979 .

[56]  Robert S. Strichartz,et al.  Differential Equations on Fractals: A Tutorial , 2006 .

[57]  M. Lapidus,et al.  Fractal Geometry and Number Theory , 1999 .

[58]  Casimir effect in de Sitter and Anti-de Sitter braneworlds , 2002, hep-th/0209242.

[59]  A. Teplyaev.  Diffusions and spectral analysis on fractals: an overview , 2010 .

[60]  A. Grigor’yan.  Heat kernels on metric measure spaces with regular volume growth , 2009 .

[61]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: The Heat Equation Approach to the Atiyah-Singer Index Theorem , 1997 .

[62]  K. Falconer.  Techniques in fractal geometry , 1997 .

[63]  Kenneth Falconer,et al.  The dimension of self-affine fractals II , 1992, Mathematical Proceedings of the Cambridge Philosophical Society.

[64]  V. Metz.  Hilbert′s Projective Metric on Cones of Dirichlet Forms , 1995 .

[65]  William Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 2 , 1967 .

[66]  T. Lindstrøm.  Brownian Motion On Nested Fractals , 1990 .

[67]  Institute for Mathematical Physics Complex Dimensions of Fractal Strings and Oscillatory Phenomena in Fractal Geometry and Arithmetic Complex Dimensions of Fractal Strings and Oscillatory Phenomena in Fractal Geometry and Arithmetic , 2009 .

[68]  M. Lapidus,et al.  Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions , 2005 .

[69]  Heat kernel coefficients of the Laplace operator on the D‐dimensional ball , 1995, hep-th/9503023.

[70]  R. Bass,et al.  Uniqueness of Brownian motion on Sierpinski carpets , 2010 .

[71]  W. Coffey,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2002 .

[72]  G. Hardy,et al.  The general theory of Dirichlet's series , 1916, The Mathematical Gazette.

[73]  Martin T. Barlow,et al.  Brownian motion on the Sierpinski gasket , 1988 .

[74]  Distributions of Localized Eigenvalues of Laplacians on Post Critically Finite Self-Similar Sets☆ , 1998 .

[75]  Michel L. Lapidus,et al.  Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture , 1991 .

[76]  Dmitri Vassiliev,et al.  Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals , 1996 .

[77]  E. Elizalde,et al.  Zeta Regularization Techniques with Applications , 1994 .

[78]  E. Akkermans,et al.  Thermodynamics of photons on fractals. , 2010, Physical review letters.

[79]  S. Goldstein.  Random Walks and Diffusions on Fractals , 1987 .

[80]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: The Laplacian on a Riemannian Manifold , 1997 .

[81]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .

[82]  H. Weldon.  Proof of zeta-function regularization of high-temperature expansions , 1986 .

[83]  I. Singer,et al.  R-Torsion and the Laplacian on Riemannian manifolds , 1971 .

[84]  I. Kocyigit,et al.  On the Dimension of Self-Affine Fractals , 2013 .

[85]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[86]  T. Kumagai.  Estimates of transition densities for Brownian motion on nested fractals , 1993 .

[87]  Jay Jorgenson,et al.  Basic Analysis of Regularized Series and Products , 1993 .

[88]  Claudio Perez Tamargo.  Can one hear the shape of a drum , 2008 .

[89]  Richard F. Bass,et al.  The construction of brownian motion on the Sierpinski carpet , 1989 .

[90]  W. H. Young,et al.  Lectures On The General Theory Of Integral Functions , 1923 .

[91]  P. Greiner.  An asymptotic expansion for the heat equation , 1971 .

[92]  M. Solomyak,et al.  On a Spectral Problem Related to Self‐Similar Measures , 1995 .

[93]  K. Falconer.  The geometry of fractal sets , 1985 .

[94]  Steven G. Krantz,et al.  Invariance Theory Heat Equation and Atiyah Singer Index Theorem , 1995 .

[95]  G. Levin.  On Pommerenke's inequality for the eigenvalues of fixed points , 1991 .

[96]  V. K. Patodi,et al.  On the heat equation and the index theorem , 1973 .

[97]  Jiaxin Hu,et al.  Laplace operators related to self-similar measures on R d , 2006 .

[98]  Uses of zeta regularization in QFT with boundary conditions: a cosmo-topological Casimir effect , 2006, hep-th/0607185.

[99]  S. Karlin,et al.  Embeddability of discrete time simple branching processes into continuous time branching processes , 1968 .

[100]  Jun Kigami,et al.  A harmonic calculus on the Sierpinski spaces , 1989 .

[101]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[102]  Alexander Teplyaev,et al.  What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals , 1999 .

[103]  G. Toulouse,et al.  Random walks on fractal structures and percolation clusters , 1983 .

[104]  R. Strichartz.  Fractafolds based on the Sierpinski gasket and their spectra , 2003 .

[105]  A. Teplyaev,et al.  Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals , 2010, 1011.5485.

[106]  T. Shima.  On eigenvalue problems for Laplacians on P.C.F. self-similar sets , 1996 .

[107]  A. Teplyaev.  Spectral Zeta Function of Symmetric Fractals , 2004 .

[108]  I. Procaccia,et al.  Diffusion on fractals. , 1985, Physical review. A, General physics.

[109]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

[110]  Jiaxin Hu,et al.  Comparison inequalities for heat semigroups and heat kernels on metric measure spaces , 2010 .

[111]  Stanislav Molchanov,et al.  DIFFUSION PROCESSES AND RIEMANNIAN GEOMETRY , 1975 .

[112]  Spectral zeta functions of fractals and the complex dynamics of polynomials , 2005, math/0505546.

[113]  H. Fédérer.  Geometric Measure Theory , 1969 .

[114]  M. Solomyak,et al.  Eigenvalue distribution of some fractal semi-elliptic differential operators: Combinatorial approach , 2001 .

[115]  Alexander Teplyaev,et al.  Vibration modes of 3n-gaskets and other fractals , 2008 .

[116]  V. Metz.  The cone of diffusions on finitely ramified fractals , 2003 .

[117]  Wolfgang Woess,et al.  Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph , 1997 .

[118]  Michel L. Lapidus,et al.  Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function , 1997 .

[119]  N. Yanagihara,et al.  Borel and Julia directions of meromorphic Schröder functions , 2005, Mathematical Proceedings of the Cambridge Philosophical Society.

[120]  P. Grabner,et al.  Complex asymptotics of Poincaré functions and properties of Julia sets , 2007, Mathematical Proceedings of the Cambridge Philosophical Society.

[121]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[122]  Gian Paolo Leonardi,et al.  Topics in mathematical analysis , 2008 .

[123]  Tadashi Shima,et al.  On a spectral analysis for the Sierpinski gasket , 1992 .

[124]  A. Eremenko,et al.  Periodic points of polynomials , 1989 .

[125]  J. Milnor.  Dynamics in one complex variable , 2000 .

[126]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[127]  J. S. Dowker,et al.  Effective Lagrangian and energy-momentum tensor in de Sitter space , 1976 .

[128]  Jun Kigami,et al.  Harmonic calculus on p.c.f. self-similar sets , 1993 .

[129]  H. Triebel.  Fractals and spectra , 1997 .

[130]  A. Teplyaev,et al.  Self-Similarity, Operators and Dynamics , 2003 .

[131]  G. Derfel,et al.  Long-time properties of solutions of simplest nonlinear q-difference equations , 2000 .

[132]  Thierry Coulhon,et al.  Manifolds and graphs with slow heat kernel decay , 2001 .

[133]  Michel L. Lapidus,et al.  Spectral and Fractal Geometry: From the Weyl-Berry Conjecture for the Vibrations of Fractal Drums to the Riemann Zeta-Function , 1992 .

[134]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[135]  Alan F. Beardon,et al.  Iteration of Rational Functions , 1991 .

[136]  Bernhard Krön,et al.  Green functions on self-similar graphs and bounds for the spectrum of the laplacian , 2002 .

[137]  Spatially chaotic configurations and functional equations with rescaling , 1996 .

[138]  Iterations of rational functions and the distribution of the values of the Poincaré functions , 1992 .

[139]  畑 政義,et al.  On the structure of self-similar sets , 1985 .

[140]  S. Alexander,et al.  Density of states on fractals : « fractons » , 1982 .

[141]  J. Kigami,et al.  Analysis on Fractals , 2001 .

[142]  Thomas Ransford,et al.  Potential Theory in the Complex Plane: Bibliography , 1995 .

[143]  René Carmona,et al.  Can one hear the dimension of a fractal? , 1986 .

[144]  C. Sabot.  Existence and uniqueness of diffusions on finitely ramified self-similar fractals , 1997 .

[145]  E Bergshoeff,et al.  Ten Physical Applications of Spectral Zeta Functions , 1996 .

[146]  Jun Kigami,et al.  Weyl's problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals , 1993 .