Non-uniform hyperbolicity and universal bounds for \(S\)-unimodal maps
type: Item
instance of: scholarly article
title: Non-uniform hyperbolicity and universal bounds for \(S\)-unimodal maps
zbMATH Open document ID: 0908.58016
author: Duncan Sands
author: Tomasz Nowicki
published in: Inventiones Mathematicae
publication date: 1999-03-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 1186154
zbMATH Keywords: \(S\)-unimodal map
zbMATH Keywords: Collet-Eckmann condition
zbMATH Keywords: uniform hyperbolic structure
MaRDI profile type: Publication
full work available at URL: s002220050236
OpenAlex ID: W2043895765
DOI: 10.1007/S002220050236
description: scientific article; zbMATH DE number 1186154