Typical behavior of the trajectories of transformations of a segment

Abstract

The class of C3-smooth transformations of the segment [0, 1], for which the number of critical points is finite and the Schwarzian is negative, is considered. It is proved that for almost all x, the Ω-limit set of trajectories fnx coincides with a cycle of a periodic point or of a transitive segment or with the limit set of some recurrent critical point. From here it follows that a transformation with d critical points has at most d + 2 indecomposable attractors in the Milnor sense. An attractor is unique if the transformation is unimodal and f∶c→1→0(c is the critical point).