Can you hear the fractal dimension of a drum?

Details Can you hear the fractal dimension of a drum? Can you hear the fractal dimension of a drum?

2005-03-31

arxiv.org/abs/math/0503748v1

in ``Applied and Industrial Mathematics in Italy'' (ISBN 981-256-368-7; 600 pages), M.Primicerio, R.Spigler, V.Valente editors

Mathematics

Spectral Theory

[11 pages, 2 figures] To appear in ``Applied and Industrial Mathematics in Italy'', World Scientific, 2005. Authors are with t

Scientific paper

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.

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