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Gaps in the spectrum of the Laplacian on \(3N\)-gaskets. (English) Zbl 1327.81211

Summary: This article develops analysis on fractal \(3N\)-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for \(N=1\), specifically properties of the Laplacian \(\Delta\) on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum.

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
60J35 Transition functions, generators and resolvents
28A80 Fractals
31C25 Dirichlet forms
31E05 Potential theory on fractals and metric spaces
35K08 Heat kernel
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