Regular or stochastic dynamics in real analytic families of unimodal maps
type: Item
instance of: scholarly article
title: Regular or stochastic dynamics in real analytic families of unimodal maps
zbMATH Open document ID: 1050.37018
author: Artur Avila
author: Mikhail Lyubich
author: Welington de Melo
published in: Inventiones Mathematicae
publication date: 2004-07-07T00:00:00Z
review text: Arguably, the most remarkable achievement in interval dynamics during the last quarter of 20th century has been the unravelling of the ``a.e. point-a.e. parameter'' dynamical structure of the quadratic family \(q_\tau(x)=\tau-1-\tau x^2\), \(\tau\in [1/2,2]\). This fruitful trend of work starts with \textit{J. Guckenheimer} [Commun. Math. Phys. 70, 133--160 (1979; Zbl 0429.58012)] and \textit{M. V. Jakobson} [Commun. Math. Phys. 81, 39--88 (1981; Zbl 0497.58017)] and culminates with \textit{M. Lyubich} [Acta Math. 178, 185--297 (1997; Zbl 0908.58053)], \textit{J. Graczyk} and \textit{G. Swiątek} [Ann. Math. (2) 146, 1--52 (1997; Zbl 0936.37015)] and \textit{M. Lyubich} [Ann. Math. (2) 156, 1--78 (2002; Zbl 1160.37356)], and has succeeded in proving that the set of parameters \(\tau\) such that \(q_\tau\) is regular (this means that all periodic orbits of the map are hyperbolic and there is a periodic orbit attracting Lebesgue almost all \(x\in [-1,1]\)) is open and dense in \([1/2,2]\), and the set of parameters \(\tau\) such that \(q_\tau\) is stochastic (which means that the map admits an invariant measure absolutely continuous with respect to the Lebesgue measure) has positive Lebesgue measure and, moreover, full measure in the complementary of regular parameters. Hence, in particular, almost every quadratic map is either regular or stochastic. In the paper under review, the latter result is extended to many analytic families of analytic unimodal maps (including all nontrivial analytic families of unimodal maps with negative Schwarzian derivative). To this aim the space of analytic maps is foliated by codimension-one analytic submanifolds (so-called ``hybrid classes''), each containing one quadratic map \(q_\tau\), which enables the authors to export the regular-stochastic dichotomy of the quadratic family to other analytic families. In order to state precisely this result, we need a number of definitions. Let \(\widetilde{\mathcal{U}}^3\) denote the family of \(…
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37C40
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37F45
zbMATH DE Number: 2078254
zbMATH Keywords: absolutely continuous invariant measure
zbMATH Keywords: attracting periodic orbit
zbMATH Keywords: hyperbolic periodic orbit
zbMATH Keywords: negative Schwarzian derivative
zbMATH Keywords: quadratic family
zbMATH Keywords: quasiquadratic map
zbMATH Keywords: regular map
zbMATH Keywords: stochastic map
zbMATH Keywords: unimodal map
reviewed by: Víctor Jiménez López
MaRDI profile type: Publication
full work available at URL: s00222-003-0307-6
OpenAlex ID: W1991942070
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Regular or stochastic dynamics in real analytic families of unimodal maps (DOI)
DOI: 10.1007/S00222-003-0307-6

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