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Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. I: The case of negative Schwarzian derivative. (English) Zbl 0665.58023
Let $${\mathcal O}$$ be the class of $$C^ 3$$-smooth one dimensional maps f: $$M\to M$$ (of the interval or the circle) with non-degenerate critical points and negative Schwarzian derivative. An interval $$J<M$$ is called wandering if $$f^ nJ\cap f^ mJ=\emptyset$$ $$(n>m\geq 0)$$ and the orbit $$\{f^ nJ\}^{\infty}_{n=0}$$ does not tend to a limit cycle. The problem of the existence of wandering intervals goes back to works of Poincaré and Denjoy. For unimodal maps $$f\in {\mathcal O}$$ (i.e. with a unique extremum) it was solved 10 years ago by J. Guckenheimer. In the paper under review the problem is solved for arbitrary polymodal $$f\in {\mathcal O}.$$
Theorem. A map $$f\in {\mathcal O}$$ has no wandering intervals. The theorem implies the full description of the behaviour of a generic orbit (i.e. outside the set of the first Baire category). More specifically, for a generic $$x\in M$$ the $$\omega$$-limit set $$\omega$$ (x) coincides with either a limit cycle, or a cycle of intervals, or a solenoidal attractor.
The proof of the theorem is based upon new technique of unimodal decompositions and estimating the distortion of $$f^ n$$ along chains of intervals. It turns out very useful in a lot of other problems concerning one dimensional dynamics (decomposition of f into ergodic components, description of measure-theoretic attractors etc.).
Reviewer: M.Yu.Lyubich

##### MSC:
 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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