Dimension of weakly expanding points for quadratic maps
摘要
For the real quadratic map Pa(x) = x 2 + 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 P n a | J univalent, with bounded distortion and B(0, ) P n a (J) for some n N. The -weakly expanding set is the set of points which do not have good expansion properties. Let denote the negative fixed point and M the first return time of the critical orbit to [, -]. 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 log 2 M/M +O(log 2 log 2 M/M) for close to ||. For arbitrary || the dimension is of the order of O(log 2 | log 2 |/| log 2 |). 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).