Data
Here tables of eigenvalues, listed in increasing order, for various
values of n (the order of the Vicsek set) and m (the
level of the graph approximation). The first column shows the
approximate fractal eigenvalue, the second column is the series (either
0 or 4/3), the last column is the multiplicity, and the remaining colums
the letters in the "word" associated with each eigenvalue, which tell
which inverses of the spectral decimation functions were applied. We
approximate fractal eigenvalues by taking graph eigenvalues, continuing
them with the lowest inverse function several times (until a certain
error is small enough) and then multiplying by the appropriate power of
the renormalization constant.
Eigenvalue counting function, Weyl ratios
Here are plots of the eigenvalue counting function
N(x) for the same values of n and
m.
Since the eigenvalue counting function above grows as
x&alpha where
&alpha =
log(4n-3)/log((4n-3)(2n-1)), it's useful
to look at the Weyl ratio
N(x)/x&alpha.
