Hunting the spectra via the underlying dynamics

What can the underlying dynamics and geometry tell us about spectral properties of associated Schrödinger operators? In recent developments, the continuous deformation of the corresponding dynamical system within a suitable topology was proven to be equivalent to the continuous change of the associated spectra of such operators. If unique ergodicity is additionally assumed, measurable quantities also behave continuously such as the density of states.

The first part is devoted to present and explain these results along an example, where no background on spectral theory is required. In the second part, we focus on related questions studying the unique ergodicity of linearly repetitive Delone sets.