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Spectral decimation on Hambly’s homogeneous hierarchical gaskets. (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(b); a fractal Γ realized as a limit of a sequence of graphs Γ 0 ,Γ 1 , with vertices V 0 V 1 . Take V 0 ={q 0 ,q 2 ,q 2 }, as vertices of a triangle, considered as the boundary of a HH(b), then the un-renormalized energy E m (u) of a function on V m ; renormalized energy 𝔼 m (u); energy on HH(b); E(u)=lim m 𝔼 m (u); a bilinear form E(u,v) as well as the standard Laplacian Δu.

2. The spectral decimation on SG 3 , the usual Sierpiński gasket.

3. Dirichlet and Neumann spectra for SG 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.

31C99Generalizations in potential theory
35H99Close-to-elliptic equations