No abstract

Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M. V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl's conjecture for the asymptotics of the eigenvalues of the Laplacian on a bounded open subset of W (see [16]). We solve here in the one-dimensional case (that is, when n = 1) this 'modified Weyl-Berry conjecture'. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. We therefore show that one can 'hear' (that is, recover from the spectrum) not only the Minkowski fractal dimension of the boundary-as was established previously by the first author-but also, under the stronger assumptions of the conjecture, its Minkowski content (a 'fractal' analogue of its 'length').We also prove (still in dimension one) a related conjecture of the first author, as well as its converse, which characterizes the situation when the error estimates of the aforementioned paper are sharp.

We establish an analogue of WeyΓs classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractal K, such as, for example, the Sierpinski gasket. We consider both Dirichlet and Neumann boundary conditions, as well as Laplacians associated with Bernoulli-type ("multifractal") measures on K. From a physical point of view, we study the density of states for diffusions or for wave propagation in fractal media. More precisely, let Q(X) be the number of eigenvalues less than x. Then we show that ρ(x) is of the order of x ds / 2 as x -» +00, where the "spectral exponent" d s is computed in terms of the geometric as well as analytic structures of K. Further, we give an effective condition that guarantees the existence of the limit of x~d s / 2 ρ(x) as x -> -hoc; this condition is, in some sense, "generic". In addition, we define in terms of the above "spectral exponents" and calculate explicitly the "spectral dimension" of K.

Abstract.Let Í2 be a bounded open set of E" (n > 1) with "fractal" boundary T . We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order 2m (m > 1) on Í2 . We consider both Dirichlet and Neumann boundary conditions. Our estimate-which is expressed in terms of the Minkowski rather than the Hausdorff dimension of Y-specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry's conjecture-which extends to "fractals" Weyl's conjecture-is closely related to Kac's question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension.The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and-to a lesser extent-geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log 3/ log 2.

No abstract

No abstract

No abstract