  Compact localized single particle eigenstates on a deterministic fractal
substrate, modelled by a triangular Sierpinski gasket of arbitrarily large
size, are unravelled and examined analytically. We prescribe an exact real
space renormalization group (RSRG) decimation scheme within a tight binding
formalism to discern these states, and argue that the number of such states can
be infinite if the fractal substrate is enlarged to its thermodynamic limit.
Interestingly, these localized states turn out to populate the non-dispersive,
flat bands in a periodic array of Sierpinski gasket motifs, however large they
may be. Our results match and corroborate the recently observed compact
localized, flat band states engineered on two dimensional photonic waveguide
networks with a fractal geometry, and provide a whole subset of them, which, in
principle, should be observable in fractal photonic lattice experiments.
