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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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