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


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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