Gaps in the spectrum of the Laplacian on $3N$-Gaskets

Abstract

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.

References in corpus (6)

• Heat Kernels and Zeta Functions on Fractals
• Laplace Operators on Fractals and Related Functional Equations
• Dirac and magnetic Schrödinger operators on fractals
• Spatial Log Periodic Oscillations of First-Passage Observables in Fractals
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Metrics and spectral triples for Dirichlet and resistance forms

Cited by in corpus (1)

• Singularly continuous spectrum of a self-similar Laplacian on the half-line