  We present a detailed analysis of the nature of electronic eigenfunctions in
one-dimensional quasi-periodic chains based on a clustering idea recently
introduced by us [Sil et al., Phys. Rev. {\bf B 48}, 4192 (1993) ], within the
framework of the real-space renormalization group approach. It is shown that
even in the absence of translational invariance, extended states arise in a
class of such lattices if they possess a certain local correlation among the
constituent atoms. We have applied these ideas to the quasi-periodic
period-doubling chain, whose spectrum is found to exhibit a rich variety of
behaviour, including a cross-over from critical to an extended regime, as a
function of the hamiltonian parameters. Contrary to prevailing ideas, the
period-doubling lattice is shown to support an infinity of extended states,
even though the polynomial invariant associated with the trace map is
non-vanishing. Results are presented for different parameter regimes, yielding
both periodic as well as non-periodic eigenfunctions. We have also extended the
present theory to a multi-band model arising from a quasi-periodically arranged
array of $\delta$-function potentials on the atomic sites. Finally, we present
a multifractal analysis of these wavefunctions following the method of Godreche
and Luck [ C. Godreche and J. M. Luck, J. Phys. A :Math. Gen. {\bf 23},
