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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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