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Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

  Published:2011-02-25
 Printed: Jun 2011
  • Sze-Man Ngai,
    Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, U.S.A.
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Abstract

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.
Keywords: spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities
MSC Classifications: 28A80, 35P20, 35J05, 43A05, 47A75 show english descriptions Fractals [See also 37Fxx]
Asymptotic distribution of eigenvalues and eigenfunctions
Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
Measures on groups and semigroups, etc.
Eigenvalue problems [See also 47J10, 49R05]
28A80 - Fractals [See also 37Fxx]
35P20 - Asymptotic distribution of eigenvalues and eigenfunctions
35J05 - Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
43A05 - Measures on groups and semigroups, etc.
47A75 - Eigenvalue problems [See also 47J10, 49R05]
 

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