Critical recurrence in the real quadratic family

TL;DR

This paper investigates the recurrence behavior of the critical orbit in the real quadratic family, establishing conditions under which the orbit frequently returns close to the critical point for almost all nonregular parameters.

Contribution

It provides a new sufficient condition on the recurrence rate that guarantees infinite returns for almost every nonregular parameter, extending previous results.

Findings

Established a recurrence condition for the critical orbit

Extended earlier results to the case where δ_n = n^{-1}

Applied to almost every nonregular parameter

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 $\vert F^n(0;a) \vert < \delta_n$ is infinite. In particular, when $\delta_n = n^{-1}$, this extends an earlier result of Avila and Moreira.