  It is known that Laplacian operators on many fractals have gaps in their
spectra. This fact precludes the possibility that a Weyl-type ratio can have a
limit and is also a key ingredient in proving that the Fourier series on such
fractals can have better convergence results than in the classical setting. In
this paper we prove that the existence of gaps is equivalent to the total
disconnectedness of the Julia set of the spectral decimation function for the
class of fully symmetric p.c.f. fractals, and for self-similar fully symmetric
finitely ramified fractals with regular harmonic structure. We also formulate
conjectures related to geometry of finitely ramified fractals with spectral
gaps, to complex spectral dimensions, and to convergence of Fourier series on
such fractals.
