Using the so-called “spectral decimation”, the authors establish a frame on Hambly’s homogeneous hierarchical gaskets. There are 5 sections in this paper:
1. Introduction – some definitions are given: a hierarchical fractal; the concept of finitely ramified: the homogeneous hierarchical gasket, denoted by ; a fractal realized as a limit of a sequence of graphs with vertices . Take , as vertices of a triangle, considered as the boundary of a , then the un-renormalized energy of a function on ; renormalized energy ; energy on ; ; a bilinear form as well as the standard Laplacian .
2. The spectral decimation on , the usual Sierpiński gasket.
3. Dirichlet and Neumann spectra for .
4. Spectral decimation on homogeneous hierarchical gaskets.
5. Spectral gaps.
As applications, the paper shows that these spectra have infinitely many large spectral gaps. And under certain restrictions, a computer-assisted proof that the set of ratios of eigenvalues has gaps, implying the existence of quasi-elliptic PDE’s on the product of two such fractals.