Critical recurrence in the real quadratic family

Abstract

We study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate (δn) of the critical orbit such that, for almost every nonregular parameter a, the set of n such that |F n (0; a)| < δn is infinite. In particular, when δn = n −1 , this extends an earlier result of Avila and Moreira.