On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps
type: Item
instance of: scholarly article
title: On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps
zbMATH Open document ID: 0856.58027
DOI: 10.1007/BF02102427
author: Masato Tsujii
published in: Communications in Mathematical Physics
publication date: 1997-02-24T00:00:00Z
review text: The celebrated theorem of Jakobson asserts that in the logistic family \(x\to ax(1- x)\) those parameters that correspond to transformations with absolutely continuous invariant measures (acim) fill out a set \(F\) so large that \(a= 4\) is the density point for \(E\cup (4, \infty)\). Later, it was proved that the acim condition can be strengthened to the Collet-Eckmann condition, [see \textit{P. Thieullen}, \textit{C. Tresser} and \textit{L. S. Young}, C. R. Acad. Sci., Paris, Sér. I 315, No. 1, 69-72 (1992; Zbl 0786.58026)]. The author of the present paper subsequently proved that other conditions can be added. For the logistic family, based on the recent result of Sands about the measure \(0\) for Misiurewicz parameters, these extra conditions follow from Collet-Eckmann almost everywhere. The author proceeds to use these hyperbolicity conditions to derive the continuity of Bowen-Ruelle-Sinai measures at the point \(a= 4\) on a large set of parameters, again having \(a= 4\) as the one-sided density point. The setting of the paper is actually much more general, including analogous results for circle mappings and more general families of interval maps (poly-modal). The proof of the main result is relatively simple. The paper presents an important step in the detailed understanding of the ergodic theory of unimodal mappings. Let me finally note that the statement of the ``backward Collet-Eckmann condition'', CE (ii) in Section 2 is incorrect as the derivative of the left-hand side is usually \(0\). The correct statement can be found in the earlier work by the same author, cited as [7] in his list of references.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 906561
zbMATH Keywords: Jakobson's theorem
zbMATH Keywords: Bowen-Ruelle-Sinai measures
zbMATH Keywords: circle mappings
zbMATH Keywords: interval maps
reviewed by: Grzegorz Świątek
MaRDI profile type: Publication
cites work: Measurable dynamics of $S$-unimodal maps of the interval
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cites work: Regularity and other properties of absolutely continuous invariant measures for the quadratic family
cites work: Positive Lyapunov exponents in families of one dimensional dynamical systems
cites work: Q5288794
full work available at URL: https://doi.org/10.1007/bf02102427
OpenAlex ID: W1991679144
description: scientific article; zbMATH DE number 906561