Can you hear the fractal dimension of a drum?
Abstract
Electromagnetics and Acoustics on a bounded domain are governed by the Helmholtz's equation; when such a domain is a [pre-]fractal described by means of a `just-touching' Iterated Function System (IFS) spectral decomposition of the Helmholtz's operator is self-similar as well. Renormalization of the Green's function proves this feature and isolates a subclass of eigenmodes, called ``diaperiodic'', whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator's eigenspaces. The definition of ``spectral dimension'' is given and proven to depend on diaperiodic modes only for a wide class of IFSs. Finally, asymptotic equivalence between box-counting and spectral dimensions in the fractal limit is proven. As the `self-similar' spectrum of the fractal is enough to compute box-counting dimension, positive answer is given to title question.
Keywords
Cite
@article{arxiv.math/0503748,
title = {Can you hear the fractal dimension of a drum?},
author = {W. Arrighetti and G. Gerosa},
journal= {arXiv preprint arXiv:math/0503748},
year = {2007}
}
Comments
[11 pages, 2 figures] To appear in ``Applied and Industrial Mathematics in Italy'', World Scientific, 2005. Authors are with the Electronic Engineering Department, ``LaSapienza'' University of Rome, at http://www.die.uniroma1.it/strutture/labcem/