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Measurable dynamics of S-unimodal maps of the interval. (English) Zbl 0790.58024
Summary: 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 respect to the Lebesgue 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 \(\mu\), then it has positive entropy: \(h_ \rho(f) > 0\). 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.
Contents: 1. Introduction. 2. Topological picture of the dynamics. 3. Distortion lemmas. 4. Expanding lemmas. 5. The measure-theoretical attractor. 6. Ergodicity. 7. Absence of strongly wandering sets. 8. Solenoidal case: pure dissipativeness. 9. Density lemmas. 10. The conservative kernel. 11. Further topological properties of Cantor attractors. 12. The finite a.c.i. measure has positive entropy. Appendix 1. Measurable endomorphisms with a quasi-invariant measure. Appendix 2. Polymodal and smooth generalizations: survey of the results.

MSC:
37A99 Ergodic theory
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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