  We solve the approximate renormalisation group found by Qiu Niu and Franco
Nori(Phys. Rev. Lett. 57 2057(1986)) for a tight-binding hamiltonian on the
Fibonacci chain. This enables us to characterize analytically as completely as
possible the spectral properties of this model, such as the spectral measure,
the bandwidth distribution, the Lebesgue measure exponent, the Hausdorff
dimension, the multifractal scaling, the gaps distribution as well as the long
time return probability. Our results, qualitatively and quantitatively complete
and unify previous works on similar models.
