## Absolutely continuous invariant measures for $$C^ 2$$ unimodal maps satisfying the Collet-Eckmann conditions.(English)Zbl 0659.58034

Let f: [0,1]$$\to [0,1]$$ be a $$C^ 2$$-unimodal map with a non-flat critical point c (i.e. f is $$C^{\ell +1}$$ near c for some $$\ell \geq 2$$ and $$D^{\ell}f(c)=0)$$. Consider the following conditions: (*) all periodic orbits of f are hyperbolic and repelling, (CE1) there are $$K>0$$, $$\lambda >1$$ such that $$| Df^ n(f(c))| \geq K\lambda^ n$$ for all $$n>0$$, (CE2) there are $$K>0$$, $$\lambda >1$$ such that $$f^ n(z)=c\Rightarrow | Df^ n(z)| \geq K\lambda^ n$$ for all $$n>0.$$
P. Collet and J. Eckmann [Ergodic Theory Dyn. Syst. 3, 13-46 (1983; Zbl 0532.28014)] proved that if f has negative Schwarzian derivative, then (CE1) and (CE2) imply the existence of an absolutely continuous invariant measure. ((*) follows from (CE1) in this case, and T. Nowicki [ibid. 8, 425-435 (1988; Zbl 0638.58021)] showed that also (CE2) is implied by (CE1).) In the present paper the authors show that, also without the assumption on the Schwarzian derivative, (*), (CE1) and (CE2) imply the existence of a unique absolutely continuous invariant measure of positive entropy. For this result, (CE2) can also be replaced by the assumption that there is $$S<\infty$$ such that for any $$n>0$$ and any interval T for which $$f^ n|_ T$$ is monotone one has $$\sum^{n-1}_{i=0}| f^ i(T)| \leq S$$.
Reviewer: G.Keller

### MSC:

 37A99 Ergodic theory 28D05 Measure-preserving transformations 26A18 Iteration of real functions in one variable

### Citations:

Zbl 0647.58035; Zbl 0532.28014; Zbl 0638.58021
Full Text:

### References:

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