## A positive Lyapunov exponent for the critical value of an $$S$$-unimodal mapping implies uniform hyperbolicity.(English)Zbl 0638.58021

A positive Lyapunov exponent for the critical value of an $$S$$-unimodal mapping implies a positive Lyapunov exponent of the backward orbit of the critical point, uniform hyperbolic structure on the set of periodic points and an exponential diminution of the length of the intervals of monotonicity. This is the proof of the Collet-Eckmann conjecture from 1981 in the general case [P. Collet and J. Eckmann, Ergodic Theory Dyn. Syst. 3, 13–46 (1983; Zbl 0532.28014)].
Reviewer: T.Nowicki

### MSC:

 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37A05 Dynamical aspects of measure-preserving transformations 28D05 Measure-preserving transformations

### Keywords:

Lyapunov exponent; hyperbolicity

Zbl 0532.28014
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### References:

 [1] Szlenk, Bol. de la Soc Mat. Mex. 24 pp 57– (1979) [2] Nowicki, Fund. Math. 126 pp 27– (1985) [3] Nowicki, Ergod. Th. & Dynam. Sys. 5 pp 611– (1985) [4] DOI: 10.2307/1971367 · Zbl 0597.58016 [5] DOI: 10.2307/1996575 · Zbl 0298.28015 [6] Collet, Iterated Maps of the Interval as Dynamical Systems (1980) · Zbl 0458.58002 [7] Collet, Ergod. Th. & Dynam. Sys. 3 pp 13– (1983) [8] Misiurewicz, Publ. Math. IHES 53 pp 17– (1981) · Zbl 0477.58020
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