A classical result in solid state physics tells us that the gaps in the electronic spectrum of a one dimensional crystal are located at those values for the quasi momentum k which belong to the Bragg spectrum of the crystal. If we label a gap with the integrated density of states of energies up to that gap we obtain an order preserving map from the positive Bragg spectrum to the set of gap labels.
The purpose of the talk is to explain a similar map for aperiodic solids. The perturbation argument used by physicists to obtain the result for crystals completely brakes down in this case and a more refined approach using a perturbation expansion of the Liapounov exponent of an associated dynamical system sometimes predicts too many gaps. Our argument is based on K-theory and sometimes predicts too few gaps.