  We study the Lyapunov exponent for electron and phonon excitations, in pure
and random Fibonacci quasicrystal chains, using an exact real space
renormalization group method, which allows the calculation of the Lyapunov
exponent as a function of the energy. It is shown that the Lyapunov exponent on
a pure Fibonacci chain has a self-similar structure, characterized by a scaling
index that is independent of the energy for the electron excitations,
''diagonal'' or ''off-diagonal'' quasiperiodic, but is a function of the energy
for the phonon excitations. This scaling behavior implies the vanishing of the
Lyapunov exponent for the states on the spectrum, and hence the absence of
localization on the Fibonacci chain, for the various excitations considered. It
is also shown that disordered Fibonacci chains, with random tiling that
introduces phason flips at certain sites on the chain, exhibit the same
Lyapunov exponent as the pure Fibonacci chain, and hence this type of disorder
is irrelevant, either in the case of electron or phonon excitations.
