  Superconductivity was recently reported in several quasicrystalline systems.
These are materials which are structurally ordered, but since they are not
translationally invariant, the usual BCS theory does not apply. At the present
time, the underlying mechanism and the properties of the superconducting phase
are insufficiently understood. To gain a better understanding of quasiperiodic
superconductors, we consider the attractive Hubbard model on the Fibonacci
chain, and examine its low-temperature superconducting phase in detail using
the Bogoliubov-de Gennes mean-field approach. We obtain superconducting
solutions as a function of the parameters controlling the physical properties
of the system: the strength of the Hubbard attraction $U$, the chemical
potential $\mu$, and the strength of the modulation of the Fibonacci
Hamiltonian, $w$. We find that there is a bulk transition at a critical
temperature that obeys a power law in $U$. The local superconducting order
parameter is self-similar both in real and perpendicular space. The local
densities of states vary from site to site, however, the width of the
superconducting gap is the same on all sites. The interplay between the Hubbard
attraction and the intrinsic gaps of the Fibonacci chain results in a complex
zero-temperature $\mu$-$U$ phase diagram with insulating domes surrounded by
superconducting regions. Finally, we show that tuning $w$ from weak to strong
quasicrystalline modulation gives rise to qualitatively different thermodynamic
behaviors as could be observed by measuring the specific heat.
