  The phonon modes of the Frenkel-Kontorova model are studied both at the
pinning transition as well as in the pinned (cantorus) phase. We focus on the
minimal frequency of the phonon spectrum and the corresponding generalized
eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown
to have nontrivial scaling properties not only at the pinning transition point
but also in the cantorus regime. Therefore the phonons defy localization and
remain critical even where the associated area-preserving map has a positive
Lyapunov exponent. In this region, the critical scaling properties vary
continuously and are described by a line of renormalization limit cycles.
Interesting renormalization bifurcation diagrams are obtained by monitoring the
cycles as the parameters of the system are varied from an integrable case to
the anti-integrable limit. Both of these limits are described by a trivial
decimation fixed point. Very surprisingly we find additional special parameter
values in the cantorus regime where the renormalization limit cycle degenerates
into the above trivial fixed point. At these ``degeneracy points'' the phonon
hull is represented by an infinite series of step functions. This novel
behavior persists in the extended version of the model containing two
harmonics. Additional richnesses of this extended model are the one to two-hole
transition line, characterized by a divergence in the renormalization cycles,
nonexponentially localized phonons, and the preservation of critical behavior
all the way upto the anti-integrable limit.
