  Gauge potentials with different configurations have been recently realized in
the optical lattice experiments. It is remarkable that one of the simplest
gauge can generate particle energy spectrum with the self-similar structure
known as a Hofstadter butterfly. We investigate theoretically the impact of
strong on-site interaction on such a spectrum. In particular, it is shown that
the fractal structure is encoded in the quasi-particle and hole bosonic
branches. A square lattice and other structures (brick-wall and staggered
magnetic flux lattice) with relativistic energy dispersions which are currently
accessible in the experiments are considered. Moreover, although in brick-wall
and staggered flux lattices the quasi-particle densities of states looks
qualitatively similar, the corresponding Hofstadter butterfly assumes different
forms. In particular, we use a superposition of two different synthetic gauge
fields which appears to be a generator of non-trivial phenomena in the optical
lattice systems. The analysis is carried out within the strong coupling
expansion method on the finite size lattices and also at finite temperatures
which are relevant for the currently made experiments.
