Summability implies Collet-Eckmann almost surely
type: Item
instance of: scholarly article
zbMATH Open document ID: 1316.37023
author: Bing Gao
author: Weixiao Shen
published in: Ergodic Theory and Dynamical Systems
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.3720
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37A10
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 6359409
zbMATH Keywords: Collet-Eckmann condition
zbMATH Keywords: generic family
zbMATH Keywords: non-degeneracy condition
MaRDI profile type: Publication
OpenAlex ID: W2016672814
arXiv ID: 1111.3720
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DOI: 10.1017/ETDS.2012.173
title: Summability implies Collet-Eckmann almost surely
review text: The main result of this paper is a generalization of the famous theorem of \textit{M. Jakobson} [Commun. Math. Phys. 81, 39--88 (1981; Zbl 0497.58017)]. For a family of interval maps \(\{f_t\}\) such that \(f_0\) satisfies a summability condition, the authors prove that \(t=0\) is a Lebesgue density point of parameters corresponding to maps satisfying the Collet-Eckmann condition and the strong polynomial recurrence condition.NEWLINENEWLINEThis is done using adapted results of \textit{W. Shen} [Proc. Lond. Math. Soc. (3) 107, No. 5, 1091--1134 (2013; Zbl 1290.37022)] and \textit{M. Tsujii} [Invent. Math. 111, No. 1, 113--137 (1993; Zbl 0787.58029)]. Through a theorem of \textit{G. Levin} [``Perturbations of weakly expanding critical orbits'', in: Frontiers in complex dynamics: in celebration of John Milnor's 80th birthday. Princeton: Princeton University Press. 163--196 (2014)], the main result is extended to higher dimensions.
reviewed by: Steve Pederson
description: scientific article; zbMATH DE number 6359409