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.

5 pages, 1 figure