Piecewise smooth interval maps with nonvanishing derivative

We consider the dynamics of piecewise smooth interval maps f with a nowhere vanishing derivative. We show that if f is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of f are hyperbolic, then f has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if f is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of f is expanding. In this case, f admits an absolutely continuous invariant probability measure.