  We present an exact real-space renormalization group (RSRG) method for
evaluating the dynamic structure factor of an infinite one-dimensional
quasiperiodic period-doubling (PD) lattice. We observe that for every normal
mode frequency of the chain, the dynamic structure factor $S(q,\omega)$ always
exhibits periodicity with respect to the wave vector $q$ and the presence of
such periodicity even in absence of translational invariance in the system is
quite surprising. Our analysis shows that this periodicity in $S(q,\omega)$
actually indicates the presence of delocalized phonon modes in the PD chain.
The Brillouin Zones of the lattice are found to have a hierarchical structure
and the dispersion relation gives both the acoustic as well as optical
branches. The phonon dispersion curves have a nested structure and we have
shown that it is actually the superposition of the dispersion curves of an
infinite set of periodic lattices.
