  We propose a scheme to determine the energy-band dispersion of quasicrystals
which does not require any periodic approximation and which directly provides
the correct structure of the extended Brillouin zones. In the gap labelling
viewpoint, this allow to transpose the measure of the integrated
density-of-states to the measure of the effective Brillouin-zone areas that are
uniquely determined by the position of the Bragg peaks. Moreover we show that
the Bragg vectors can be determined by the stability analysis of the law of
recurrence used to generate the quasicrystal. Our analysis of the gap labelling
in the quasi-momentum space opens the way to an experimental proof of the gap
labelling itself within the framework of an optics experiment, polaritons, or
with ultracold atoms.
