Hausdorff Dimension of the Exceptional Set in Jakobson's Theorem
by
Samuel Senti
Universite de Paris-Sud

We consider the real quadratic map P_a(x)=x²+a, where the parameter a in R. By Jakobson's theorem, there exists a measure which is invariant (by P_a) and absolutely continuous (with respect to the Lebesgue measure on the dynamical space), an a.c.i.p., for a set of parameters of positive Lebesgue measure. We compute the Hausdorff measure of the set of points which do not have nice expansion porperties: the \epsilon-exceptional set. These are the points which do not have a basis of neighborhoods that can each be sent univalently and with bounded distortion onto an interval containing an \epsilon ball centered at the critical point. We first consider \epsilon of the order of the negative fixed point \alpha. Then the exceptional set is compact and invariant by a family of maps G_i with i >= 0. An upper (resp. lower) bound c_i (resp. b_i) on the contraction ratio of G_i gives an upper (resp. lower) bound on the dimension of the compact invariant set by solving \sumc_i^d=1 (resp. \sumb_i^d'=1). Under certain conditions called strongly regular, we have good estimates on the first few c_i and b_i allowing us to compute the dimension of an approximation of the exceptional set. Additionally, these conditions imply that the dimension of the approximation and of the exceptional set are close. If we denote by M the smallest integer such that |P_a^M(0)| < |\alpha|, then the set of strongly regular parameters, f or which there exists an a.c.i.p. and for which the exceptional set's dimension is logM/M+O(loglogM/M) has positive Lebesgue measure. We then adapt the combinatorics to the case of a general \epsilon. For strongly regular parameters, there exists an a.c.i.p. and the \epsilon-exceptional set's dimension is less than log|log\epsilon|/|log\epsilon|.

Date received: March 7, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caep-15.