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Some remarks on recent results about \(S\)-unimodal maps. (English) Zbl 0721.58018
Let \(f: [0,1]\to [0,1]\) be a unimodal map with negative Schwarzian derivative and critical point \(c\). Assume that \(f\) has no stable periodic orbit and is also not infinitely renormalizable. It is known that for such maps there exists a compact set \(A\) such that \(\omega (x)=A\) for almost every \(x\in [0,1]\). Combining earlier results the authors arrive at the following classification for such maps:
(I) \(A=\omega (c)\) is a Cantor attractor.
(II) \(A\) is an interval attractor and \(\omega (c)\) is of Cantor-type.
(III) \(A=\omega (c)\) is an interval attractor.
The basic theorems of the paper are the following statements: Each map of type (II) has an invariant density. There are maps of type (II) with a nonintegrable invariant density. There are maps of type (III) without integrable invariant density.
Reviewer: M.Mrozek (Kraków)

MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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