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

Nowicki, T.; van Strien, Sebastian

Invent. Math. 93, No. 3, 619-635 (1988).

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

Keywords:

Collet-Eckmann condition; non-flat critical point; \(C^ 2\)-unimodal map; periodic orbits; invariant measure; Schwarzian derivative

Citations:

Zbl 0647.58035; Zbl 0532.28014; Zbl 0638.58021

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