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We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals.
Lectures on the General Theory of Integral Functions
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Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds
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Fonctions Analytiques
G. Valiron · 1954
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The Theory of Branching Processes
T. E. Harris · 1963
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The general theory of Dirichlet’s series
G. H. Hardy and M. Riesz · 1964
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Can one hear the shape of a drum?
M. Kac · 1966
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An Introduction to Probability Theory and Its Applications
W. Feller · 1966
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Geometric measure theory
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Functional Analysis
K. Yosida · 1971
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La percolation: un concept unifacateur
P. G. de Gennes · 1976
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Distribution of modes in fractal resonators
M. V. Berry · 1979
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The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary
V. Ja. Ivriĭ · 1980
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Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals
M. V. Berry · 1980
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Fractals and self-similarity
J. E. Hutchinson · 1981
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The fractal geometry if nature
B. B. Mandelbrot · 1982
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Density of states on fractals: “fractons”
S. Alexander and R. Orbach · 1982
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Random walks on fractal structures and percolation clusters
R. Rammal and G. Toulouse · 1983
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On the structure of self-similar sets
M. Hata · 1985
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The Geometry of Fractal Sets
K. J. Falconer · 1986
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Can one hear the dimension of a fractal?
J. Brossard and R. Carmona · 1986
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On conformal mapping and iteration of rational functions
Ch. Pommerenke · 1986
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Diffusion on disordered media
S. Havlin and D. Ben-Avraham · 1987
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Random walks and diffusions on fractals
S. Goldstein · 1987
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A diffusion process on a fractal
S. Kusuoka · 1987
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Brownian motion on the Sierpiński gasket
M. T. Barlow and E. A. Perkins · 1988
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The Hausdorff dimension of self-affine fractals
K. J. Falconer · 1988
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A harmonic calculus on the Sierpiński spaces
J. Kigami · 1989
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Dirichlet forms on fractals and products of random matrices
S. Kusuoka · 1989
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The construction for Brownian motion on the Sierpiński carpet
M. T. Barlow and R. F. Bass · 1989
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Periodic points of polynomials (Russian)
A. È. Erëmenko and G. M. Levin · 1989
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Brownian Motion on Nested Fractals
T. Lindstrøm · 1990
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Iteration of rational functions
A. F. Beardon · 1991
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Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture
M. L. Lapidus · 1991
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On Pommerenke’s inequality for the eigenvalues of fixed points
G. M. Levin · 1991
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On the spectral analysis for the Sierpiński gasket
M. Fukushima and T. Shima · 1992
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Spectral and fractal geometry: from the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function
M. L. Lapidus · 1992
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The dimension of self-affine fractals. II
K. J. Falconer · 1992
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Dirichlet forms, diffusion processes and spectral dimensions for nested fractals
M. Fukushima · 1992
Cited alongside, same era.
Heat kernels and Dirac operators
N. Berline, E. Getzler, and M. Vergne · 1992
Cited alongside, same era.
Iterations of rational functions and the distribution of the values of Poincaré functions (Russian
A. È. Erëmenko and M. L. Sodin · 1992
Cited alongside, same era.
Harmonic calculus on p.c.f. self-similar sets
J. Kigami · 1993
Cited alongside, same era.
The eigenvalue problem for the Laplacian on the Sierpiński gasket
T. Shima · 1993
Cited alongside, same era.
The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums
M. L. Lapidus and C. Pomerance · 1993
Cited alongside, same era.
Analysis on fractals
J. Kigami · 2001
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Manifolds and graphs with slow heat kernel decay
M. Barlow, T. Coulhon, and A. Grigor’yan · 2001
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Eigenvalue distribution of some fractal semi-elliptic differential operators: combinatorial approach
K. Naimark and M. Solomyak · 2001
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Green functions on self-similar graphs and bounds for the spectrum of the Laplacian
B. Krön · 2002
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Spectral Functions in Mathematics and Physics
K. Kirsten · 2002
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Self-similarity, operators and dynamics
L. Malozemov and A. Teplyaev · 2003
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Estimates of transition densities for Brownian motion on nested fractals
T. Kumagai · 1993
Cited alongside, same era.
Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals
J. Kigami and M. L. Lapidus · 1993
Cited alongside, same era.
On the eigenvalue behaviour for a class of operators related to self-similar measures on 𝐑 d {\bf R}^{d}
K. Naimark and M. Solomyak · 1994
Cited alongside, same era.
The eigenvalue behaviour for the boundary value problems related to self-similar measures on 𝐑 d {\bf R}^{d}
K. Naimark and M. Solomyak · 1995
Cited alongside, same era.
On a spectral problem related to self-similar measures
M. Solomyak and E. Verbitsky · 1995
Cited alongside, same era.
Hilbert’s projective metric on cones of Dirichlet forms
V. Metz · 1995
Cited alongside, same era.
Fractafolds based on the Sierpiński gasket and their spectra
R. S. Strichartz · 2003
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The cone of diffusions on finitely ramified fractals
V. Metz · 2003
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Fractal Geometry
K. J. Falconer · 2003
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On the Bieberbach conjecture and holomorphic dynamics
X. Buff · 2003
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Spectral zeta function of symmetric fractals
A. Teplyaev · 2004
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Asymptotics of the transition probabilities of the simple random walk on self-similar graphs
B. Krön and E. Teufl · 2004
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Borel and Julia directions of meromorphic Schröder functions
K. Ishizaki and N. Yanagihara · 2005
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Dynamics in one complex variable
J. Milnor · 2006
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Fractal geometry, complex dimensions and zeta functions, Geometry and spectra of fractal strings
M. L. Lapidus and M. van Frankenhuysen · 2006
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Differential equations on fractals
R. S. Strichartz · 2006
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Laplace operators related to self-similar measures on ℝ d \mathbb{R}^{d}
J. Hu, K.-S. Lau, and S.-M. Ngai · 2006
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Diffusion on Simply and Doubly Symmetric Fractal Graphs
G. Schweitzer · 2006
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On the asymptotic behaviour of analytic solutions of linear iterative functional equations
E. Teufl · 2007
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Spectral zeta functions of fractals and the complex dynamics of polynomials
A. Teplyaev · 2007
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Asymptotics of the Poincaré functions
G. Derfel, P. J. Grabner, and F. Vogl · 2007
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Recent developments of analysis on fractals
T. Kumagai · 2008
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Fractal analysis, an approach via function spaces
H. Triebel · 2008
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The Zeta function of the Laplacian on certain fractals
G. Derfel, P. J. Grabner, and F. Vogl · 2008
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Complex asymptotics of Poincaré functions and properties of Julia sets
G. Derfel, P. J. Grabner, and F. Vogl · 2008
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Physical consequences of complex dimensions of fractals
E. Akkermans, G. Dunne, and A. Teplyaev · 2009
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A tale of two fractals (Russian)
A. A. Kirillov · 2009
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Heat kernel upper bounds for jump processes and the first exit time
M. T. Barlow and T. Grigor’yan, A.and Kumagai · 2009
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Thermodynamics of photons on fractals
E. Akkermans, G. Dunne, and A. Teplyaev · 2010
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Diffusions and spectral analysis on fractals: an overview
A. Teplyaev · 2010
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Uniqueness of Brownian motion on Sierpiński carpets
M. T. Barlow, R. F. Bass, T. Kumagai, and A. Teplyaev · 2010
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Heat kernels on metric measure spaces with regular volume growth
A. Grigor’yan · 2010
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Comparison inequalities for heat semigroups and heat kernels on metric measure spaces
A. Grigor’yan, J. Hu, and K.-S. Lau · 2010
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Spectral asymptotics for Laplacians on self-similar sets
N. Kajino · 2010
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Dirichlet forms and symmetric Markov processes
M. Fukushima, Y. Oshima, and M. Takeda · 2011
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On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals
N. Kajino · 2011
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