## Institute for Mathematical Sciences

## Preprint ims90-2

** A. M. Blokh and M. Yu. Lyubich**
* Measurable Dynamics of S-Unimodal Maps of the Interval*

Abstract: In this paper we sum up our results on one-dimensional measurable dynamics reducing them to the S-unimodal case (compare Appendix 2). Let f be an S-unimodal map
of the interval having no limit cycles. Then f is ergodic with repect to the Lebesque
measure, and has a unique attractor A in the sense of Milnor. This attractor coincides
with the conservative kernel of f. There are no strongly wandering sets of positive
measure. If f has a finite a.c.i. (absolutely continuous invariant) measure u, then
it has positive entropy. 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.

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