# Resistance metric, and spectral asymptotics, on the graph of the Weierstrass function

## TL;DR

This paper investigates the resistance metric and spectral properties of the Laplacian on the graph of the Weierstrass function, extending previous work to analyze asymptotic spectral behavior and resistance metrics.

## Contribution

It provides an explicit resistance metric and explores spectral asymptotics of the Laplacian on the Weierstrass graph, extending known fractal spectral analysis results.

## Key findings

- Explicit resistance metric derived for the Weierstrass graph
- Spectral asymptotics of the Laplacian analyzed
- Connections to Weyl's law for fractal spectra established

## Abstract

Following our work on the graph of the Weierstrass function, in the spirit of those of J. Kigami and R. S. Strichartz, which enabled us to build a Laplacian on the aforementioned graph, it was natural to go further and give the related explicit resistance metric. The aim of this work is twofold. We had a special interest in the study of the spectral properties of the Laplacian. In our previous work, we have given the explicit the spectrum on the graph of the Weierstrass function. In the case of Laplacians on post-critically finite fractals, existing results of J. Kigami and M. Lapidus, R. S. Strichartz, make the link between resistance metric, and asymptotic properties of the spectrum of the Laplacian, by means of an analoguous of Weyl's formula. So we asked ourselves wether those results were still valid, for the graph of the Weierstrass function.