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Non-uniform hyperbolicity and universal bounds for $$S$$-unimodal maps. (English) Zbl 0908.58016
An $$S$$-unimodal map $$f$$ is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then $$f$$ is said to have a uniform hyperbolic structure.
We prove that an $$S$$-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that an $$S$$-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied.
The proof uses new universal bounds that hold for any $$S$$-unimodal map without periodic attractors.

##### MSC:
 37E99 Low-dimensional dynamical systems 37B99 Topological dynamics
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