  Topological properties of energy spectra of general one-dimensional
quasiperiodic systems, describing also Bloch electrons in magnetic fields, are
studied for an infinity of irrational modulation frequencies corresponding to
irrational numbers of flux quanta per unit cell. These frequencies include
well-known ones considered in works on Fibonacci quasicrystals. It is shown
that the spectrum for any such frequency exhibits a self-similar hierarchy of
clusters characterized by universal (system-independent) values of Chern
topological integers which are exactly determined. The cluster hierarchy
provides a simple and systematic organization of all the spectral gaps, labeled
by universal topological numbers which are exactly determinable, thus avoiding
their numerical evaluation using rational approximants of the irrational
frequency. These numbers give both the quantum Hall conductance of the system
and the winding number of the edge-state energy traversing a gap as a Bloch
quasimomentum is varied.
