Spectral Decimation for Families of Self-Similar Symmetric Laplacians on the Sierpinski Gasket

TL;DR

This paper introduces a family of symmetric, self-similar Laplacians on the Sierpinski Gasket, providing a spectral decimation method to explicitly determine eigenvalues and eigenfunctions, with applications to fractal analysis.

Contribution

It constructs a parameterized family of Laplacians on the Sierpinski Gasket and develops a spectral decimation technique for their spectral analysis, extending classical theories.

Findings

Spectral decimation applies to all Laplacians in the family.

Eigenvalues and eigenfunctions are explicitly characterized.

Analogous results are obtained for fractal Laplacians on the interval.

Abstract

We construct a one-parameter family of Laplacians on the Sierpinski Gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the fact that all these Laplacians satisfy a version of spectral decimation that builds a precise catalog of eigenvalues and eigenfunctions for any choice of the parameter. We give a number of applications of this spectral decimation. We also prove analogous results for fractal Laplacians on the unit Interval, and this yields an analogue of the classical Sturm-Liouville theory for the eigenfunctions of these one-dimensional Laplacians.