Stable Orbits and Bifurcation of Maps of the Interval

Differentiable maps of class at least C³ from the unit interval to itself are shown to have a finite number of stable periodic orbits, each of which attracts the iterates of some critical point, assuming the hypothesis of everywhere negative Schwarzian derivative. An example illustrates the necessity of this hypothesis; it shows that an endomorphism of the unit interval with one critical point may possess more than one stable orbit.