  We demonstrate, by explicit construction, that a single band tight binding
Hamiltonian defined on a class of deterministic fractals of the b = 3N
Sierpinski type can give rise to an infinity of dispersionless, flat-band like
states which can be worked out analytically using the scale invariance of the
underlying lattice. The states are localized over clusters of increasing sizes,
displaying the existence of a multitude of localization areas. The onset of
localization can, in principle, be delayed in space by an appropriate choice of
the energy of the electron. A uniform magnetic field threading the elementary
plaquettes of the network is shown to destroy this staggered localization and
generate absolutely continuous sub-bands in the energy spectrum of these
non-translationally invariant networks.
