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Positive Liapunov exponents and absolute continuity for maps of the interval. (English) Zbl 0532.28014
The following theorem is proven. Let $$f$$ be a unimodal map of the interval with negative Schwarzian derivative satisfying $$xf'(x)<0$$ $$\forall x\neq 0$$ and non-degenerate critical point at 0. Assume there are constants $$C>0$$, $$\theta>0$$ so that $|(\frac{d}{dx}f^ n)(f(0))| \geq \exp(n\theta)\quad \text{and}\quad |(\frac{d}{dx}f^ m)(z)| \geq C\exp(m\theta),$ for all $$z, m$$ for which $$f^ m(z)=0$$. Then $$f$$ has an invariant measure which is absolutely continuous with respect to Lebesgue measure

##### MSC:
 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth) 28D05 Measure-preserving transformations 37A05 Dynamical aspects of measure-preserving transformations
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