Spectrum of the Laplacian on the Vicsek Set "with no loose ends"

Abstract

We study the spectral properties of a fractal VNLE obtained from the standard Vicsek set VS by making a countable number of identifications of points so that all the line segments in VS become circles in VNLE. We show that the standard Laplacian on VNLE satisfies spectral decimation with the same cubic renormalization polynomial as for VS, and thereby give a complete description of all eigenfunctions of the Laplacian. We then study the restrictions of eigenfunctions to the large circles in VNLE and prove that these are Lipschitz functions.