Turnaround in cyclic cosmology

It is speculated how dark energy in a brane world can help reconcile an infinitely cyclic cosmology with the second law of thermodynamics. A cyclic model is described, in which dark energy with w <-1 equation of state leads to a turnaround at a time, extremely shortly before the would-be big rip, at which both volume and entropy of our Universe decrease by a gigantic factor, while very many independent similarly small contracting universes are spawned. The entropy of our model decreases almost to zero at turnaround but increases for the remainder of the cycle by a vanishingly small amount during contraction, empty of matter, then by a large factor during inflationary expansion.