Spectral asymptotics for Stretched Fractals

TL;DR

This paper investigates spectral properties of stretched fractals, specifically the Hanoi attractor, by constructing Dirichlet forms and analyzing eigenvalue asymptotics, extending understanding of non-self-similar fractal structures.

Contribution

It introduces a method to create non-self-similar fractals from self-similar sets and analyzes their spectral asymptotics using resistance metrics and eigenvalue counting functions.

Findings

Calculated Hausdorff dimension with respect to resistance metric

Determined leading term of eigenvalue counting function

Extended spectral analysis to non-self-similar fractals

Abstract

The Stretched Sierpinski Gasket (or Hanoi attractor) was subject of several prior works. In this work we use this idea of stretching self-similar sets to obtain non-self-similar ones. We are able to do this for a subset of the connected p.c.f. self-similar sets that fulfill a certain connectivity condition. We construct Dirichlet forms and study the associated self-adjoint operators by calculating the Hausdorff dimension w.r.t. the resistance metric as well as the leading term of the eigenvalue counting function.