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

TL;DR

This paper investigates the spectral properties of a modified Vicsek fractal, VNLE, demonstrating spectral decimation and providing a complete eigenfunction description, with eigenfunctions restricted to circles shown to be Lipschitz.

Contribution

It introduces a new fractal VNLE derived from the Vicsek set and establishes spectral decimation for its Laplacian, offering a comprehensive eigenfunction analysis.

Findings

Spectral decimation applies to VNLE with the same polynomial as VS.

Complete eigenfunction characterization for the Laplacian on VNLE.

Eigenfunctions restricted to circles are Lipschitz functions.

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.