Exponents, attractors and Hopf decompositions for interval maps

Abstract

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