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Critical recurrence in the real quadratic family
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 10 / October 2023
- Published online by Cambridge University Press:
- 08 November 2022, pp. 3255-3287
- Print publication:
- October 2023
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- Article
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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].
