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Non-uniform hyperbolicity and universal bounds for S-unimodal maps
Authors:Tomasz Nowicki  Duncan Sands
Affiliation:(1) Instytut Matematyki UW, Banacha 2, Warszawa, Poland, PL;(2) Institute for Mathematical Sciences, SUNY, Stony Brook, NY 11794-3651, USA, US
Abstract: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 some renormalization of 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. Oblatum 4-VII-1996 & 4-VII-1997
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