  We show that in any family of stunted sawtooth maps, the set of maps whose
set of periods is the set of all powers of 2 has no interior point, i.e., the
combinatorial description of the boundary of chaos coincides with the
topological description. We also show that, under mild assumptions, smooth
multimodal maps whose set of periods is the set of all powers of 2 are
infinitely renormalizable.
