, Volume 104, Issue 1, pp 103-122

Can one hear the dimension of a fractal?

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Abstract

We consider the spectrum of the Laplacian in a bounded open domain of ℝ n with a rough boundary (i.e. with possibly non-integer dimension) and we discuss a conjecture by M. V. Berry generalizing Weyl's conjecture. Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function. The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Haussdorff ones.

Dedicated to the memory of Mark Kac
Communicated by B. Simon