  We report on a study of topological properties of Fibonacci quasicrystals.
Chern numbers which label the dense set of spectral gaps, are shown to be
related to the underlying palindromic symmetry. Topological and spectral
features are related to the two independent phases of the scattering matrix:
the total phase shift describing the frequency spectrum and the chiral phase
sensitive to topological features. Conveniently designed gap modes with
spectral properties directly related to the Chern numbers allow to scan these
phases. An effective topological Fabry-Perot cavity is presented.
