Dimension of Weakly Expanding Points for Quadratic Maps

— For the real quadratic map Pa(x) = x2 + a and a given > 0 a point x has good expansion properties if any interval containing x also contains a neighborhood J of x with Pn a |J univalent, with bounded distortion and B(0, ) ⊆ Pn a (J) for some n ∈ N. The -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [α,−α]. We show there is a set R of parameters with positive Lebesgue measure for which the Hausdorff dimension of the -weakly expanding set is bounded above and below by log2 M/M+O(log2 log2 M/M) for close to |α|. For arbitrary ≤ |α| the dimension is of the order of O(log2 | log2 |/| log2 |). Constants depend only on M . The Folklore Theorem then implies the existence of an absolutely continuous invariant probability measure for Pa with a ∈ R (Jakobson’s Theorem). Texte reçu le 11 février 2002, révisé le 15 juillet 2002, accepté le 16 septembre 2002 Samuel Senti, Department of Mathematics, Penn State University, University Park, PA, 16802 (USA) • E-mail : [email protected] • Url : http://www.math.psu.edu/senti 2000 Mathematics Subject Classification. — 37E05, 37D25, 37D45, 37C45.