  We study the de-Haas van Alphen oscillations in the magnetization of the
Hofstadter model. Near a split band the magnetization is a rapidly oscillating
function of the Fermi energy with lip shaped envelopes. For generic magnetic
fields this structure appears on all scales and provides a thermodynamic
fingerprint of the fractal properties of the model. The analysis applies
equally well to the two dual interpretations of the Hofstadter model and the
nature of the duality transformation is elucidated.
