Physicists and mathematicians have used the self-similar nature of certain fractals to develop and study analytical structures on fractal spaces. We examine the analytical structure of a class of fractals that arise as limit sets of the Schreier graphs of the action of self-similar groups on infinite n-ary trees. In particular, we consider how the spectrum of a Laplacian operator on one level of a Schreier graph relates to the spectrum on the next level, a technique known as spectral decimation.
Grigorchuk and collaborators have developed a method to spectrally decimate Schreier graphs of several
important self-similar groups, and have derived significant consequences about the structure of amenable groups. Their method is related to a notion of spectral similarity arising from the work of Fukushima-Shima and Malozemov-Teplyaev. In the latter, a sufficient condition for spectral decimation for fractal graphs is obtained. We consider the analogous question for Schreier graphs of self-similar groups with the goal of understanding the class to which Grigorchuk’s approach is applicable.