Sierpi\'nski fractals and the dimension of their Laplacian spectrum

TL;DR

This paper provides rigorous estimates for the Hausdorff dimension of the spectra of Laplacians on Sierpiński fractals, advancing understanding of their spectral properties and geometric complexity.

Contribution

It introduces new mathematical estimates for the Hausdorff dimension of Laplacian spectra on various Sierpiński fractals, including lattices and gaskets.

Findings

Established bounds for the Hausdorff dimension of spectra

Applied to Sierpiński lattices and gaskets

Enhanced understanding of spectral geometry of fractals

Abstract

We estabish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated to Sierpi\'nski lattices and infinite Sierpi\'nski gaskets and other post-critically finite self-similar sets.