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 nonregular parameter $a$, the set of $n$ such that $|F^n(0;a)|<{\delta }_n$ is infinite. In particular, when ${\delta }_n=n^{-1}$, this extends an earlier result of Avila and Moreira.

Revised version following the referee's report. Lemma 4.12 was added, together with minor corrections. Accepted for publication in ETDS

References in corpus (1)

• Summability implies Collet-Eckmann almost surely