  A goal in the study of dynamics on the interval is to understand the
transition to positive topological entropy. There is a conjecture from the
1980's that the only route to positive topological entropy is through a cascade
of period doubling bifurcations. We prove this conjecture in natural families
of smooth interval maps, and use it to study the structure of the boundary of
mappings with positive entropy. In particular, we show that in families of
mappings with a fixed number of critical points the boundary is locally
connected, and for analytic mappings that it is a cellular set.
