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Exponents, attractors and Hopf decompositions for interval maps. (English) Zbl 0715.58020
Summary: Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:
(1) There is a set $$C_ T$$ such that the $$\omega$$-limit of Lebesgue-a.e. point equals $$C_ T$$. $$C_ T$$ is a finite union of closed intervals or it coincides with the closure of the critical orbit.
(2) There is a constant $$\lambda_ T$$ such that $$\lambda_ T=\limsup_{n\to \infty}(1/n)\log | (T^ n)'(x)|$$ for Lebesgue- a.e. x.
(3) $$\lambda_ T>0$$ if and only if T has an absolutely continuous invariant measure of positive entropy.
(4) $$\lambda_ T\geq \inf \{p^{-1}\log | (T^ p)'(z)|:$$ $$T^ pz=z\}$$, i.e. uniform hyperbolicity on periodic points implies $$\lambda_ T>0$$, and $$\lambda_ T<0$$ implies the existence of a stable periodic orbit.

##### MSC:
 37A99 Ergodic theory 28D20 Entropy and other invariants
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##### References:
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