Oscillations on the Stretched Sierpinski Gasket

TL;DR

This paper investigates the spectral properties of the Stretched Sierpinski Gasket, revealing that despite its high symmetry, it exhibits oscillations in the eigenvalue counting function's leading term, but without strict periodicity.

Contribution

It refines previous eigenvalue estimates for the Stretched Sierpinski Gasket and demonstrates the presence of oscillations in the spectral asymptotics without strict periodic behavior.

Findings

Existence of oscillations in the leading term of eigenvalue counting function.

High multiplicity localized eigenfunctions on the SSG.

Oscillations occur without strict periodicity in the spectral behavior.

Abstract

The Stretched Sierpinski Gasket is a non-self-similar set but it still exhibits very high symmetry. In an earlier work we calculated the leading term for the eigenvalue counting function for operators coming from resistance forms that were introduced by Alonso-Ruiz, Freiberg and Kigami. In this work we want to refine the results. The next question that arises is if there are oscillations in the leading term which are typical for highly symmetrical fractals. We have to distinguish between the existence of a periodic function in front of the leading term and oscillations in general. The first one is unlikely as we will see, however the second one still holds. This means there are oscillations in the leading term, but these will not have this very strict periodic behaviour that we know of the Sierpinski Gasket. We will show, that there exist localized eigenfunctions on the SSG which have eigenvalues with very high multiplicities. However in contrast to the self-similar case this fact alone is not enough to show that there can't be convergence. We need to use another method to show the existence of oscillations.