  Energy bands of electrons in a square lattice potential threaded by a uniform
magnetic field exhibit a fractal structure known as the Hofstadter butterfly.
Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a
tilt along the cavity axis. The hopping along the cavity axis is only induced
by resonant Raman scattering of transverse pump light into a standing wave
cavity mode. Choosing a suitable pump geometry allows to realize the
Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic
field, which appears at the onset of the superradiant phase transition. The
dynamical nature of this cavity-induced synthetic magnetic field arises from
the delicate interplay between collective superradiant scattering and the
underlying fractal band structure. Using a sixth-order expansion of the free
energy as function of the order parameter and by numerical simulations we show
that at low magnetic fluxes the superradiant ordering phase transition is first
order, while it becomes second order for higher flux. The dynamic nature of the
magnetic field induces a non-trivial deformation of the Hofstadter butterfly in
the superradiant phase. At strong pump far above the self-ordering threshold we
recover the Hofstadter butterfly one would obtain in a static magnetic field.
