Can you hear the fractal dimension of a drum.pdf

1 CAN YOU HEAR THE FRACTAL DIMENSION OF A DRUM?* WALTER ARRIGHETTI? Electronic Engineering Department, Università degli Studi di Roma “La Sapienza”, via Eudossiana 18, Rome, 00184, Italy, www.die.uniroma1.it/strutture/labcem/. GIORGIO GEROSA Electronic Engineering Department, Università degli Studi di Roma “La Sapienza”, via Eudossiana 18, Rome, 00184, ITALY, www.die.uniroma1.it/strutture/labcem/. Electromagnetics and Acoustics on a bounded domain is governed by the Helmholtz’s equation; when such a domain is [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. 1. Introduction 1.1. Between spectral and fractal geometry Marc Kac wondered in 1966 (in a famous paper entitled ‘Can you hear the shape of a drum?’, [1]) whether the shape of a plane domain could be inferred from the sole spectrum of its Laplace’s operator with Neumann/Dirichlet’s boundary conditions, i.e. whether the shape of a membrane can be inferred by just “hearing” all its vibrating modes. This conjecture was confuted in 1992, when two plane domains were found to have the same spectrum, but different shapes [2]. Thereinafter Euclidean domains are usually split into equivalence classes of isospectrality, whose members have the same laplacian spectrum but different shapes. Computationally