A proof of Benedicks-Carleson-Jacobson theorem
The author considers the family of mappings \[ Q_ a(x):= 1- ax^ 2,\quad a\in [0,2], \] of \([-1,1]\) into itself and proves that the set \(E\subset [0,2]\) of parameters for which the Collet-Eckmann condition \[ \liminf_{n\to\infty} \textstyle{{1\over n}} \log| DQ^ n_ a(1)|> 0 \] holds, has Lebesgue measure bigger than \(\varepsilon- \varepsilon^ a\) for all \(\varepsilon>0\) sufficiently small, where \(a\) is an explicit constant \(>1\). In particular, the measure of \(E\) is positive, a result due to \textit{M. Benedicks} and \textit{L. Carleson} [Ann. Math., II. Ser. 133, No. 1, 73-169 (1991; Zbl 0724.58042)]. Since the Collet-Eckmann condition implies the existence of a finite \(Q_ a\)- invariant measure absolutely continuous to Lebesgue measure, Jakobson's theorem [\textit{M. V. Jakobson}, Commun. Math. Phys. 81, 39-88 (1981; Zbl 0497.58017)] follows.
scientific article; zbMATH DE number 558839