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Another proof of Jakobson’s theorem and related results. (English) Zbl 0671.58019
Author’s abstract: The author shows that any family \(C^2\)-close to \(f_{\alpha}(x)=1-\alpha x^2\) \((2-\varepsilon \leq \alpha \leq 2)\) satisfies Jakobson’s theorem [M. V. Jakobson, Commun. Math. Phys. 81, 39–88 (1981; Zbl 0497.58017), see also M. Benedicks and L. Carleson, Ann. Math. (2) 122, No. 1, 1–25 (1985; Zbl 0597.58016)]: For a positive measure set of \(\alpha\) the transformation \(f_{\alpha}\) has an absolutely continuous invariant measure.
He also indicates some generalizations.
Reviewer: G.Warnecke

37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
28D10 One-parameter continuous families of measure-preserving transformations
Full Text: DOI
[1] Rychlik, Studia Math 76 pp 69– (1983)
[2] DOI: 10.2307/1971367 · Zbl 0597.58016
[3] DOI: 10.1007/BF01941800 · Zbl 0497.58017
[4] Misiurewicz, I.H.E.S. Publications Math̩matiques 53 pp 17Р(1981)
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