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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

##### MSC:
 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
##### Keywords:
absolutely continuous invariant measure
Full Text:
##### References:
 [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 Mathématiques 53 pp 17– (1981)
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