Critical recurrence in the real quadratic family

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 =...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 43; no. 10; pp. 3255 - 3287
Main Author BYLUND, MATS
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.10.2023
Subjects
critical recurrence
Matematik
Matematisk analys
Mathematical Analysis
Mathematics
Natural Sciences
Naturvetenskap
Original Article
parameter exclusion
real quadratic family
real quadratic family
critical recurrence
37B20
parameter exclusion
37E05
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Summary: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].
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2022.78