The boundary of chaos for piecewise smooth maps and the boundary of positive Hausdorff dimension for survivor sets of open maps

We describe the boundary of chaos separating regions of parameter space
with positive topological entropy from those with zero topological entropy for a class of piecewise smooth maps. This coincides with the boundary of positive Hausdorff dimension for the survivor sets of a class of open maps. There are precisely two codimension one types of transition across the boundaries: one of these involves heteroclinic connections, and in this case there is a finite number of periodic orbits at the transition point, and the other involves an infinite cascade of bifurcations creating infinitely many periodic orbits.