×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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 Math̩matiques 53 pp 17Р(1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.