*(English)*Zbl 1211.28005

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 $HH\left(b\right)$; a fractal ${\Gamma}$ realized as a limit of a sequence of graphs ${{\Gamma}}_{0},{{\Gamma}}_{1},\cdots $ with vertices ${V}_{0}\subseteq {V}_{1}\subseteq \cdots $. Take ${V}_{0}=\{{q}_{0},{q}_{2},{q}_{2}\}$, as vertices of a triangle, considered as the boundary of a $HH\left(b\right)$, then the un-renormalized energy ${E}_{m}\left(u\right)$ of a function on ${V}_{m}$; renormalized energy ${\mathbb{E}}_{m}\left(u\right)$; energy on $HH\left(b\right)$; $E\left(u\right)={lim}_{m\to \infty}{\mathbb{E}}_{m}\left(u\right)$; a bilinear form $E(u,v)$ as well as the standard Laplacian ${\Delta}u$.

2. The spectral decimation on $S{G}_{3}$, the usual Sierpiński gasket.

3. Dirichlet and Neumann spectra for $S{G}_{3}$.

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.