Explicit Spectral Decimation for a Class of Self--Similar Fractals

TL;DR

This paper develops an explicit spectral decimation method for a class of highly symmetric self-similar fractals, providing formulas for their Laplace eigenvalues.

Contribution

It introduces a new explicit construction for spectral decimation on nested fractals, enabling precise eigenvalue calculations.

Findings

Derived explicit formulas for Laplace eigenvalues on these fractals.

Extended spectral decimation techniques to a broad class of self-similar structures.

Enhanced understanding of spectral properties of symmetric fractals.

Abstract

The method of spectral decimation is applied to an infinite collection of self--similar fractals. The sets considered belong to the class of nested fractals, and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.