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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
28D05 Measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
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