If you interpret the red-shift data to mean velocities of galaxies, then the universe actually has begun accelerating outward after a period of initial gravitational deceleration, so by this interpretation it looks like the answer is that there are no cycles.
But this is an odd way of looking at it especially since quantum mechanics has discovered that space is a sea of energy in the form of planck particle pairs. So if space itself were expanding and the zero point energy were becoming less dense, the light would actually experience a blue shift en route.
But the doppler shift interpretation of red-shift data has run into more and more problems, so other interpretations should be sought.