  New types of self-similar states are found in quasiperiodic systems
characterized by topological invariants-- the Chern numbers. We show that the
topology introduces a competing length in the self-similar band edge states
transforming peaks into doublets of size equal to the Chern number. This length
intertwines with the quasiperiodicity and introduces an intrinsic scale,
producing Chern-beats and nested regions where the fractal structure becomes
smooth. Cherns also influence the zero-energy mode, that for quasiperiodic
systems which exhibit exponential localization, is related to the ghost of the
Majorana; the delocalized state at the onset to topological transition. The
Chern and the Majorana, two distinct types of topological edge modes, exist in
quasiperiodic superconducting wires.
