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

D.  Kelleher; N.  Gupta; M.  Margenot; J.  Marsh; W.  Oakley; A.  Teplyaev

doi:10.3934/cpaa.2015.14.2509

Article Contents

2015, Volume 14, Issue 6: 2509-2533. Doi: 10.3934/cpaa.2015.14.2509

This issue Previous Article Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion Next Article Dynamics of a host-pathogen system on a bounded spatial domain

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

1.
Department of Mathematics, Purdue University, West Lafayette, IN, 47907
2.
Department of Mathematics, University of Connecticut, Storrs, CT 06269

Received: May 31, 2014

Revised: February 28, 2015

Published: October 2015

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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.

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Mathematics Subject Classification: Primary: 81Q35, 60J35; Secondary: 28A80, 31C25, 31E05, 35K08.

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