121 research outputs found
Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series
We study the analog of power series expansions on the Sierpinski gasket, for
analysis based on the Kigami Laplacian. The analog of polynomials are
multiharmonic functions, which have previously been studied in connection with
Taylor approximations and splines. Here the main technical result is an
estimate of the size of the monomials analogous to x^n/n!. We propose a
definition of entire analytic functions as functions represented by power
series whose coefficients satisfy exponential growth conditions that are
stronger than what is required to guarantee uniform convergence. We present a
characterization of these functions in terms of exponential growth conditions
on powers of the Laplacian of the function. These entire analytic functions
enjoy properties, such as rearrangement and unique determination by infinite
jets, that one would expect. However, not all exponential functions
(eigenfunctions of the Laplacian) are entire analytic, and also many other
natural candidates, such as the heat kernel, do not belong to this class.
Nevertheless, we are able to use spectral decimation to study exponentials, and
in particular to create exponentially decaying functions for negative
eigenvalues
Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets
In this paper we study the boundary limit properties of harmonic functions on
, the solutions to the Poisson equation where is a p.c.f. set
and its Laplacian given by a regular harmonic structure. In
particular, we prove the existence of nontangential limits of the corresponding
Poisson integrals, and the analogous results of the classical Fatou theorems
for bounded and nontangentially bounded harmonic functions.Comment: 22 page
Hydrodynamic limit for a zero-range process in the Sierpinski gasket
We prove that the hydrodynamic limit of a zero-range process evolving in
graphs approximating the Sierpinski gasket is given by a nonlinear heat
equation. We also prove existence and uniqueness of the hydrodynamic equation
by considering a finite-difference scheme.Comment: 24 pages, 1 figur
Spectral analysis on infinite Sierpinski fractafolds
A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions. In this paper we
combine the ideas from these earlier works to obtain a description of the
spectral resolution of the Laplacian for noncompact fractafolds. Our main
abstract results enable us to obtain a completely explicit description of the
spectral resolution of the fractafold Laplacian. For some specific examples we
turn the spectral resolution into a "Plancherel formula". We also present such
a formula for the graph Laplacian on the 3-regular tree, which appears to be a
new result of independent interest. In the end we discuss periodic fractafolds
and fractal fields
Vibration Spectra of the -Tree Fractal
We introduce a family of post-critically finite fractal trees indexed by the
number of branches they possess. Then we produce a Laplacian operator on graph
approximations to these fractals and use spectral decimation to describe the
spectrum of the Laplacian on these trees. Lastly we consider the behavior of
the spectrum as the number of branches increases.Comment: 21 pages, 4 figure
A renormalisation approach to excitable reaction-diffusion waves in fractal media
Of fundamental importance to wave propagation in a wide range of physical phenomena is the structural geometry of the supporting medium. Recently, there have been several investigations on wave propagation in fractal media. We present here a renormalization approach to the study of reaction-diffusion (RD) wave propagation on finitely ramified fractal structures. In particular we will study a Rinzel-Keller (RK) type model, supporting travelling waves on a Sierpinski gasket (SG), lattice
Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry
The aim of the present work is to show how, using the differential calculus
associated to Dirichlet forms, it is possible to construct Fredholm modules on
post critically finite fractals by regular harmonic structures. The modules are
d-summable, the summability exponent d coinciding with the spectral dimension
of the generalized laplacian operator associated with the regular harmonic
structures. The characteristic tools of the noncommutative infinitesimal
calculus allow to define a d-energy functional which is shown to be a
self-similar conformal invariant.Comment: 16 page
From non-symmetric particle systems to non-linear PDEs on fractals
We present new results and challenges in obtaining hydrodynamic limits for
non-symmetric (weakly asymmetric) particle systems (exclusion processes on
pre-fractal graphs) converging to a non-linear heat equation. We discuss a
joint density-current law of large numbers and a corresponding large deviations
principle.Comment: v2: 10 pages, 1 figure. To appear in the proceedings for the 2016
conference "Stochastic Partial Differential Equations & Related Fields" in
honor of Michael R\"ockner's 60th birthday, Bielefel
Upper estimate of martingale dimension for self-similar fractals
We study upper estimates of the martingale dimension of diffusion
processes associated with strong local Dirichlet forms. By applying a general
strategy to self-similar Dirichlet forms on self-similar fractals, we prove
that for natural diffusions on post-critically finite self-similar sets
and that is dominated by the spectral dimension for the Brownian motion
on Sierpinski carpets.Comment: 49 pages, 7 figures; minor revision with adding a referenc
Physical Consequences of Complex Dimensions of Fractals
It has recently been realized that fractals may be characterized by complex
dimensions, arising from complex poles of the corresponding zeta function, and
we show here that these lead to oscillatory behavior in various physical
quantities. We identify the physical origin of these complex poles as the
exponentially large degeneracy of the iterated eigenvalues of the Laplacian,
and discuss applications in quantum mesoscopic systems such as oscillations in
the fluctuation of the number of levels, as a correction to
results obtained in Random Matrix Theory. We present explicit expressions for
these oscillations for families of diamond fractals, also studied as
hierarchical lattices.Comment: 4 pages, 3 figures; v2: references added, as published in Europhysics
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