Exponents, attractors and Hopf decompositions for interval maps

Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following: (1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit. (2) There is a constant λT such that for Lebesgue-a.e. x. (3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy. (4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.