Approximation of fractals by discrete graphs: norm resolvent and spectral convergence

TL;DR

This paper proves that Laplacians on certain fractals can be approximated by graph Laplacians, ensuring convergence of spectra, eigenfunctions, and related operators, which advances understanding of fractal analysis and graph approximations.

Contribution

It establishes norm resolvent and spectral convergence of graph Laplacians to fractal Laplacians for a broad class of post-critically finite fractals with arbitrary measures.

Findings

Convergence of Laplacians in operator norm

Spectral convergence and eigenfunction approximation

Applicability to a wide class of fractals

Abstract

We show a norm convergence result for the Laplacian on a class of post-critically finite fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional graph Laplacians with corresponding discrete probability measures. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.