On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps
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.
scientific article; zbMATH DE number 906561