Open Access
Fall 2009 Spectral decimation on Hambly’s homogeneous hierarchical gaskets
Shawn Drenning, Robert S. Strichartz
Illinois J. Math. 53(3): 915-937 (Fall 2009). DOI: 10.1215/ijm/1286212923

Abstract

We give a complete description of the Dirichlet and Neumann spectra of the Laplacian on a class of homogeneous hierarchical fractals introduced by Hambly. These fractals are finitely ramified but not self-similar. We use the method of spectral decimation. As applications, we show that these spectra always have infinitely many large spectral gaps, allowing for nice convergence results for eigenfunction expansions, and under certain restrictions we give a computer-assisted proof that the set of ratios of eigenvalues has gaps, implying the existence of quasielliptic PDE’s on the product of two such fractals. The computer programs used in this paper and more detailed explanations of the algorithms can be found at www.math.cornell.edu/˜sld32/ FractalAnalysis.html.

Citation

Download Citation

Shawn Drenning. Robert S. Strichartz. "Spectral decimation on Hambly’s homogeneous hierarchical gaskets." Illinois J. Math. 53 (3) 915 - 937, Fall 2009. https://doi.org/10.1215/ijm/1286212923

Information

Published: Fall 2009
First available in Project Euclid: 4 October 2010

zbMATH: 1211.28005
MathSciNet: MR2727362
Digital Object Identifier: 10.1215/ijm/1286212923

Subjects:
Primary: 28A80 , 31C99

Rights: Copyright © 2009 University of Illinois at Urbana-Champaign

Vol.53 • No. 3 • Fall 2009
Back to Top