Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals

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.