12 research outputs found
One-dimensional wave equations defined by fractal Laplacians
We study one-dimensional wave equations defined by a class of fractal
Laplacians. These Laplacians are defined by fractal measures generated by
iterated function systems with overlaps, such as the well-known infinite
Bernoulli convolution associated with the golden ratio and the 3-fold
convolution of the Cantor measure. The iterated function systems defining these
measures do not satisfy the post-critically finite condition or the open set
condition. By using second-order self-similar identities introduced by
Strichartz et al., we discretize the equations and use the finite element and
central difference methods to obtain numerical approximations to the weak
solutions. We prove that the numerical solutions converge to the weak solution,
and obtain estimates for the rate of convergence
Spectral Asymptotics of Laplacians Associated to One-Dimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition
Multifractal analysis via scaling zeta functions and recursive structure of lattice strings
The multifractal structure underlying a self-similar measure stems directly
from the weighted self-similar system (or weighted iterated function system)
which is used to construct the measure. This follows much in the way that the
dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity
dimension, is determined by the scaling ratios of the corresponding
self-similar system via Moran's theorem. The multifractal structure allows for
our definition of scaling regularity and scaling zeta functions motivated by
geometric zeta functions and, in particular, partition zeta functions. Some of
the results of this paper consolidate and partially extend the results
regarding a multifractal analysis for certain self-similar measures supported
on compact subsets of a Euclidean space based on partition zeta functions.
Specifically, scaling zeta functions generalize partition zeta functions when
the choice of the family of partitions is given by the natural family of
partitions determined by the self-similar system in question. Moreover, in
certain cases, self-similar measures can be shown to exhibit lattice or
nonlattice structure with respect to specified scaling regularity values.
Additionally, in the context provided by generalized fractal strings viewed as
measures, we define generalized self-similar strings, allowing for the
examination of many of the results presented here in a specific overarching
context and for a connection to the results regarding the corresponding complex
dimensions as roots of Dirichlet polynomials. Furthermore, generalized lattice
strings and recursive strings are defined and shown to be very closely related.Comment: 33 pages, no figures, in pres
One-Dimensional Fractal Wave Equations
We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition, and therefore the existing theory, introduced by Kigami and developed by many other mathematicians, cannot be appled. First, by using a weak formulation of the problem, we prove the existence, uniqueness and regularity of weak solutions of these wave equations. Second, we study numerical computations of the solutions. By using the second-order self-similar identities introduced by Strichartz et al., we discretize the equation and use the finite element method and central difference method to obtain numerical solutions. Last, we also prove that the numerical solutions converge to the weak solution, and obtain estimates for the convergence of this approximation scheme
Doubling properties of self-similar measures.
Yung Po-lam.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.Includes bibliographical references (leaves 62-64).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- The basics of doubling measures --- p.7Chapter 2.1 --- Existence of doubling measures --- p.7Chapter 2.2 --- More examples of doubling measures --- p.16Chapter 3 --- Doubling of self-similar measures --- p.20Chapter 3.1 --- Open set condition and doubling --- p.24Chapter 3.2 --- Examples of doubling with OSC --- p.31Chapter 3.3 --- Bernoulli convolution and golden ratio --- p.41Chapter 4 --- Applications of doubling measures --- p.48Chapter 4.1 --- Singular integral operators --- p.48Chapter 4.2 --- Poincare inequalities and local Sobolev embedding --- p.56Chapter 4.3 --- Remarks --- p.60Bibliography --- p.6