Non-uniform hyperbolicity and universal bounds for \(S\)-unimodal maps (Q1127758)

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.