  The evolution of the spectrum of cosmological fluctuations from one cycle to
the next is studied. It is pointed out that each cycle leads to a reddening of
the spectrum. This opens up new ways to generate a scale-invariant spectrum of
curvature perturbations. The large increase in the amplitude of the
fluctuations quickly leads to a breakdown of the linear theory. More generaly,
we see that, after including linearized cosmological perturbations, a cyclic
universe cannot be truly cyclic.
