  Topological phases have been reported on self-similar structures in the
presence of a perpendicular magnetic field. Here, we present an understanding
of these phases from a perspective of spectral flow and charge pumping. We
study the Harper-Hofstadter model on self-similar structures constructed from
the Sierpinski gasket. We numerically investigate the spectral flow and the
associated charge pumping when a flux tube is inserted through the structure
and the flux through the tube is varied adiabatically. We find that the nature
of the spectral flow is qualitatively different from that of translationally
invariant non-interacting systems with a perpendicular magnetic field. We show
that the instantaneous eigenspectra can be used to understand the quantization
of the charge pumped over a cycle, and hence to understand the topological
character of the system. We show the correspondence between the local
contributions to the Hall conductivity and the spectral flow of the edge-like
states. We also show that the edge-like states can be approximated by
eigenstates of the discrete angular-momentum operator, their chiral nature
being a consequence of this.
