Self-Similarity of Volume Measures for Laplacians¶on P. C. F. Self-Similar Fractals

Abstract:

Our main goal in this paper is to obtain a precise analogue of Weyl's asymptotic formula for the eigenvalue distribution of Laplacians on a certain class of “finitely ramified” (or p.c.f.) self-similar fractals, building, in particular, on the work of [7, 9, 22, 24]. Our main result consists in precisely identifying (for the class of “decimable fractals”) the volume measures constructed by the second author in [24] for general p.c.f. fractals and showing that they are self-similar.

From a physical point of view, our results should be relevant to the study of the density of states for diffusions and wave propagation in fractal media.