an:03844066
Zbl 0532.28014
Collet, P.; Eckmann, J.-P.
Positive Liapunov exponents and absolute continuity for maps of the interval
EN
Ergodic Theory Dyn. Syst. 3, 13-46 (1983).
0143-3857 1469-4417
1983
j
37D20 37E05 28D05 37A05
positivity of the forward and backward Lyapunov exponent of the critical point; invariant measure; absolute continuity with respect to Lebesgue measure
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
P. Collet, J.-P. Eckmann