  We construct a one-parameter family of Laplacians on the Sierpinski Gasket
that are symmetric and self-similar for the 9-map iterated function system
obtained by iterating the standard 3-map iterated function system. Our main
result is the fact that all these Laplacians satisfy a version of spectral
decimation that builds a precise catalog of eigenvalues and eigenfunctions for
any choice of the parameter. We give a number of applications of this spectral
decimation. We also prove analogous results for fractal Laplacians on the unit
Interval, and this yields an analogue of the classical Sturm-Liouville theory
for the eigenfunctions of these one-dimensional Laplacians.
