Spectra of self-similar Laplacians on the Sierpinski gasket with twists

Abstract

We study the spectra of a two-parameter family of self-similar Laplacians on the Sierpinski gasket (SG) with twists. By this we mean that instead of the usual IFS that yields SG as its invariant set, we compose each mapping with a reflection to obtain a new IFS that still has SG as its invariant set, but changes the definition of self-similarity. Using recent results of Cucuringu and Strichartz, we are able to approximate the spectra of these Laplacians by two different methods. To each Laplacian we associate a self-similar embedding of SG into the plane, and we present experimental evidence that the method of outer approximation, recently introduced by Berry, Goff and Strichartz, when applied to this embedding, yields the spectrum of the Laplacian (up to a constant multiple). © 2008 World Scientific Publishing Company.