  Self-similarity and fractals have fascinated researchers across various
disciplines. In graphene placed on boron nitride and subjected to a magnetic
field, self-similarity appears in the form of numerous replicas of the original
Dirac spectrum, and their quantization gives rise to a fractal pattern of
Landau levels, referred to as the Hofstadter butterfly. Here we employ
capacitance spectroscopy to probe directly the density of states (DoS) and
energy gaps in this spectrum. Without a magnetic field, replica spectra are
seen as pronounced DoS minima surrounded by van Hove singularities. The
Hofstadter butterfly shows up as recurring Landau fan diagrams in high fields.
Electron-electron interactions add another twist to the self-similar behaviour.
We observe suppression of quantum Hall ferromagnetism, a reverse Stoner
transition at commensurable fluxes and additional ferromagnetism within replica
spectra. The strength and variety of the interaction effects indicate a large
playground to study many-body physics in fractal Dirac systems.
