Measurable dynamics of $S$-unimodal maps of the interval

— In this paper we sum up our results on one-dimensional measurable dynamics reducing them to the S-unimodal case (compare Appendix 2). Let / be an S-unimodal map of the interval having no limit cycles. Then / is ergodic with respect to the Lebesgue measure, and has a unique attractor A in the sense of Milnor. This attractor coincides with the conservative kernel of/. There are no strongly wandering sets of positive measure. If / has a finite a. c. i. (absolutely continuous invariant) measure a, then it has positive entropy: h^(f)>0. This result is closely related to the following: the measure of Feigenbaum-like attractors is equal to zero. Some extra topological properties of Cantor attractors are studied.