Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps
Printed: Jun 2011
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.
spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities
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]