Critical recurrence in the real quadratic family

Abstract We study recurrence in the real quadratic family and give a sufficient condition on rate $(\delta _n)$ of critical orbit such that, for almost every non-regular parameter , set n 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 Moreira [Statistical properties unimodal maps: family. Ann. Math. (2) 161 (2005), 831–881].