Escape components of McMullen maps

We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$ . We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 43; no. 11; pp. 3745 - 3775
Main Authors QIU, WEIYUAN, ROESCH, PASCALE, WANG, YUEYANG
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.11.2023
Subjects
Original Article
quasicircle
hyperbolic component
McMullen map
holomorphic motion
37F10
37F45
37F15
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Summary:We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$ . We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$ .
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2022.84