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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 $\Gamma$ realized as a limit of a sequence of graphs $\Gamma_0,\Gamma_1,\dots$ 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 $HH(b)$, then the un-renormalized energy $E_m(u)$ of a function on $V_m$; renormalized energy $\bbfE_m(u)$; energy on $HH(b)$; $E(u)= \lim_{m\to\infty} \bbfE_m(u)$; a bilinear form $E(u,v)$ as well as the standard Laplacian $\Delta u$. 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
Full Text: Euclid
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