This is a description of all code files that have been used to compute the eigenvalue spectra. The main constants are d , the degree of the polynomial, and k, the number of preimages of the starting point of the graph approximation. For our purposes d=3 or d=6 and k=2.
The first part is to compute the points on the unit circle. The unit circle can be identified as numbers between 0 and 1. Points are noted by their numerators as rational numbers. At level m of the graph approximation the denominator is (d^k-1)d^m. The files initial_a.m and ray_num.m have a matrix as output. The first column shows the numerators, if two points have the same value on the second column, then they are identified on the Julia set.
The file lazar_num.m is the central part. Given the numerators and identifications as outputs of the previous files for the current level and the level before, it computes a matrix of all eigenvectors and a diagonal matrix containing the eigenvalues of the Laplacian. Currantly, the program is restricted to the case k=2 .
Given an array of eigenvalues eigen and a linspace array, one can compute the eigenvalue counting function.
Helper function, two values are almost equal up to computer precision:
A simple function to plot Julia sets:
We also recommend to look at Taryn Flock's matlab code for degree 2, which also provide code for plotting eigenfunctions on the Julia set.