Spectral decimation on Hambly’s homogeneous hierarchical gaskets

Abstract

We give a complete description of the Dirichlet and Neumann spectra of the Laplacian on a class of homogeneous hierarchical fractals introduced by Hambly. These fractals are finitely ramified but not self-similar. We use the method of spectral decimation. As applications, we show that these spectra always have infinitely many large spectral gaps, allowing for nice convergence results for eigenfunction expansions, and under certain restrictions we give a computer-assisted proof that the set of ratios of eigenvalues has gaps, implying the existence of quasielliptic PDE’s on the product of two such fractals. The computer programs used in this paper and more detailed explanations of the algorithms can be found at www.math.cornell.edu/˜sld32/ FractalAnalysis.html.