Analysis on the Projective Octagasket

TL;DR

This paper investigates the existence of a self-similar Laplacian on the Projective Octagasket fractal, providing experimental, algorithmic, and spectral analysis to support the conjecture.

Contribution

It introduces a recursive algorithm for the discrete Laplacian and categorizes its spectrum and eigenfunctions on the fractal.

Findings

Experimental results support the conjecture of a self-similar Laplacian.

A recursive algorithm for the discrete Laplacian is developed.

Spectral analysis aids in solving the heat equation on the fractal.

Abstract

The existence of a self similar Laplacian on the Projective Octagasket, a non-finitely ramified fractal is only conjectured. We present experimental results using a cell approximation technique originally given by Kusuoka and Zhou. A rigorous recursive algorithm for the discrete Laplacian is given. Further, the spectrum and eigenfunctions of the Laplacian together with its symmetries are categorized and utilized in the construction of solutions to the heat equation.