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Heat Kernel
From the eigenvalues and eigenfunctions we can construct the heat kernel
on the standard Vicsek set. This is especially easy when one of the
arguments is the center point of the Vicsek set, since then we only need
to consider 0-series eigenfunctions. Here are plots of this for
m=4 (diagonal plots have been normalized so that the value at the
center is 1)
The trace of the heat kernel and its value at the center, when
multiplied by t&alpha are both periodic in log
t. Here are plots for m=7:
Wave Propagator
From the eigenvalues and eigenfunctions we can also construct the wave
propagator on the standard Vicsek set. As with the heat kernel,
this is easiest to compute when on of the arguments is the center point
of the Vicsek set, since then we only need to consider 0-series
eigenfunctions. Here are plots of this for m=4:
