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 $(\delta _n)$ of the critical orbit such that, for almost every non-regular parameter a, the set of n such that $\vert F^n(0;a) \vert < \delta _n$ is infinite. In particular, when $\delta _n = n^{-1}$ , this extends an earlier result by Avila and Moreira [Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2)161(2) (2005), 831–881].