A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity

19 September 1988

journal article
research article
Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems

Vol. 8 (3) , 425-435
https://doi.org/10.1017/s0143385700004569

Abstract

A positive Liapunov exponent for the critical value of an S-unimodal mapping implies a positive Liapunov 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.

Keywords

This publication has 8 references indexed in Scilit:

Symmetric S-unimodal mappings and positive Liapunov exponents
Ergodic Theory and Dynamical Systems, 1985
On Iterations of 1 - ax 2 on (- 1,1)
Annals of Mathematics, 1985
On some dynamical properties of S-unimodal maps on an interval
Fundamenta Mathematicae, 1985
Positive Liapunov exponents and absolute continuity for maps of the interval
Ergodic Theory and Dynamical Systems, 1983
Some Dynamical Properties of Certain Differentiable Mappings of an Interval
Published by Springer Nature ,1982
Absolutely continuous measures for certain maps of an interval
Publications mathématiques de l'IHÉS, 1981
On the Existence of Invariant Measures for Piecewise Monotonic Transformations
Transactions of the American Mathematical Society, 1973
On the existence of invariant measures for piecewise monotonic transformations
Transactions of the American Mathematical Society, 1973