Gaps in the spectrum of the Laplacian on $3N$-Gaskets

TL;DR

This paper investigates the spectral properties of the Laplacian on $3N$-gaskets, establishing heat kernel estimates, spectral decimation methods, and demonstrating the existence of spectral gaps due to complex dynamics.

Contribution

It introduces a self-similar geodesic metric on $3N$-gaskets, extends spectral decimation techniques to rational functions, and proves the presence of spectral gaps in these fractals.

Findings

Existence of a self-similar geodesic metric.

Heat kernel estimates for the Laplacian.

Spectral gaps due to rational spectral decimation dynamics.

Abstract

This article develops analysis on fractal $3N$-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for $N=1$, specifically properties of the Laplacian $\Delta$ on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum.