Can you hear the fractal dimension of a drum?

W. Arrighetti; G. Gerosa

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Can you hear the fractal dimension of a drum?

W. Arrighetti·G. Gerosa

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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.

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BibTeX RIS

W. Arrighetti, G. Gerosa. 2005-03-31. Can you hear the fractal dimension of a drum?. https://arxiv.org/abs/math/0503748

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